Subgroups — C₇×#24524 #13201740

Order 1792.

884 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3094	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#449	C₂	1	1	yes	maximal, characteristic
3	C₇×#2924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2896	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2064	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2063	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1977	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1930	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1900	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1855	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1785	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
17	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
19	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₇×#121	C₂²	1	1	yes	characteristic
21	C₂×C₇×#117	C₂²	1	1	yes	characteristic
22	C₇×#482	—	2	1	no	—
23	C₇×#482	—	2	1	no	—
24	C₇×#481	—	2	1	no	—
25	C₇×#480	—	2	1	no	—
26	C₇×#449	—	2	1	no	—
27	C₇×#449	—	2	1	no	—
28	C₇×#449	—	2	1	no	—
29	C₇×#449	—	2	1	no	—
30	C₇×#433	—	2	1	no	—
31	C₇×#433	—	2	1	no	—
32	C₇×#432	—	2	1	no	—
33	C₇×#431	—	2	1	no	—
34	C₂×C₇×#108	C₂²	1	1	yes	characteristic
35	C₂×C₇×#108	C₂²	1	1	yes	characteristic
36	C₇×#428	C₂²	1	1	yes	characteristic
37	C₇×#428	C₂²	1	1	yes	characteristic
38	C₇×#424	C₂²	1	1	yes	characteristic
39	C₇×#424	C₂²	1	1	yes	characteristic
40	C₇×#419	C₂²	1	1	yes	characteristic
41	C₇×#419	C₂²	1	1	yes	characteristic
42	C₇×#418	C₂²	1	1	yes	characteristic
43	C₇×#418	C₂²	1	1	yes	characteristic
44	C₇×#418	C₂²	1	1	yes	characteristic
45	C₇×#418	C₂²	1	1	yes	characteristic
46	C₇×#417	—	2	1	no	—
47	C₇×#417	—	2	1	no	—
48	C₇×#416	—	2	1	no	—
49	C₇×#416	—	2	1	no	—
50	C₂×C₇×#103	C₂²	1	1	yes	characteristic
51	C₂×C₇×#102	C₂²	1	1	yes	characteristic
52	C₂×C₇×#101	C₂²	1	1	yes	characteristic
53	C₂×C₇×#101	C₂²	1	1	yes	characteristic
54	C₂×C₇×#100	C₂²	1	1	yes	characteristic
55	C₂×C₇×#100	C₂²	1	1	yes	characteristic
56	C₂×C₇×#100	C₂²	1	1	yes	characteristic
57	C₂×C₇×#100	C₂²	1	1	yes	characteristic
58	C₇×#407	C₂²	1	1	yes	characteristic
59	C₇×#407	C₂²	1	1	yes	characteristic
60	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
61	C₇×#405	C₂²	1	1	yes	characteristic
62	C₇×#405	C₂²	1	1	yes	characteristic
63	C₇×#343	C₂²	1	1	yes	characteristic
64	C₇×#343	C₂²	1	1	yes	characteristic
65	C₇×#341	C₂²	1	1	yes	characteristic
66	C₇×#341	C₂²	1	1	yes	characteristic
67	#24524	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	2	no	—
69	C₂×C₇×#41	—	2	2	no	—
70	C₂×C₇×#41	—	2	1	no	—
71	C₂×C₇×#41	—	2	1	no	—
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	—	2	1	no	—
74	C₂×C₇×#41	—	4	1	no	—
75	C₂×C₇×#41	—	4	1	no	—
76	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×D₄	—	2	1	no	—
78	C₂²×C₇×D₄	—	2	1	no	—
79	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
80	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₇	—	2	1	no	—
82	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
83	C₂×C₇×SD₁₆	—	2	2	no	—
84	C₂×C₇×SD₁₆	—	2	1	no	—
85	C₂×C₇×SD₁₆	—	2	1	no	—
86	C₂×C₇×SD₁₆	—	4	1	no	—
87	C₂×C₇×D₈	—	2	2	no	—
88	C₂×C₇×D₈	—	2	1	no	—
89	C₂×C₇×D₈	—	2	1	no	—
90	C₂×C₇×D₈	—	4	1	no	—
91	C₂×C₇×M₁₆	—	2	1	no	—
92	C₂×C₇×M₁₆	—	4	1	no	—
93	C₂×C₇×M₁₆	—	4	1	no	—
94	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
95	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
96	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
97	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
98	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
99	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
100	C₂²×C₇×C₈	—	2	1	no	—
101	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
102	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
103	C₇×#121	—	2	1	no	—
104	C₇×#121	—	2	1	no	—
105	C₇×#121	—	2	1	no	—
106	C₇×#121	—	2	1	no	—
107	C₇×#117	—	2	1	no	—
108	C₇×#117	—	2	1	no	—
109	C₇×#117	D₄	1	1	yes	characteristic
110	C₇×#117	D₄	1	1	yes	characteristic
111	C₇×#117	C₂³	1	1	yes	characteristic
112	C₇×#117	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
115	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
116	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
118	C₂×C₄²×C₇	—	2	1	no	—
119	C₇×#108	—	4	1	no	—
120	C₇×#108	—	4	1	no	—
121	C₇×#107	—	2	1	no	—
122	C₇×#107	—	2	1	no	—
123	C₇×#106	—	2	1	no	—
124	C₇×#106	—	2	1	no	—
125	C₇×C₄⋊C₈	—	2	1	no	—
126	C₇×C₄⋊C₈	—	2	1	no	—
127	C₇×C₄⋊C₈	—	2	1	no	—
128	C₇×C₄⋊C₈	—	2	1	no	—
129	C₇×#103	—	2	1	no	—
130	C₇×#103	—	2	1	no	—
131	C₇×#103	—	2	1	no	—
132	C₇×#103	—	2	1	no	—
133	C₇×#103	—	2	1	no	—
134	C₇×#103	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	2	1	no	—
137	C₇×#102	—	2	1	no	—
138	C₇×#102	—	2	1	no	—
139	C₇×#102	—	2	1	no	—
140	C₇×#102	—	2	1	no	—
141	C₇×#102	—	2	1	no	—
142	C₇×#102	—	2	1	no	—
143	C₇×#102	—	2	1	no	—
144	C₇×#102	—	2	1	no	—
145	C₇×#101	—	2	2	no	—
146	C₇×#101	—	4	1	no	—
147	C₇×#100	—	2	2	no	—
148	C₇×#100	—	4	1	no	—
149	C₇×#100	—	4	1	no	—
150	C₇×#100	—	4	1	no	—
151	C₇×C₂²⋊C₈	—	2	1	no	—
152	C₇×C₂²⋊C₈	—	2	1	no	—
153	C₇×C₂²⋊C₈	—	2	1	no	—
154	C₇×C₂²⋊C₈	—	2	1	no	—
155	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
156	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
157	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
158	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
159	C₇×#97	—	2	1	no	—
160	C₇×#97	—	2	1	no	—
161	C₄×C₇×C₈	—	2	1	no	—
162	C₄×C₇×C₈	—	2	1	no	—
163	#3094	C₂×C₇	1	1	yes	characteristic, complemented
164	C₂×#449	C₂×C₇	1	1	yes	characteristic
165	#2924	C₂×C₇	1	1	yes	characteristic, complemented
166	#2896	C₂×C₇	1	1	yes	characteristic, complemented
167	#2064	C₂×C₇	1	1	yes	characteristic, complemented
168	#2063	C₂×C₇	1	1	yes	characteristic, complemented
169	#2046	C₂×C₇	1	1	yes	characteristic, complemented
170	#2045	C₂×C₇	1	1	yes	characteristic, complemented
171	#1978	C₂×C₇	1	1	yes	characteristic, complemented
172	#1977	C₂×C₇	1	1	yes	characteristic, complemented
173	#1931	C₂×C₇	1	1	yes	characteristic, complemented
174	#1930	C₂×C₇	1	1	yes	characteristic, complemented
175	#1900	C₂×C₇	1	1	yes	characteristic, complemented
176	#1855	C₂×C₇	1	1	yes	characteristic, complemented
177	#1785	C₂×C₇	1	1	yes	characteristic, complemented
178	C₂⁴×C₇	—	2	1	no	—
179	C₂⁴×C₇	—	2	1	no	—
180	C₂⁴×C₇	#41	1	1	yes	characteristic
181	C₇×#41	—	2	4	no	—
182	C₇×#41	—	2	4	no	—
183	C₇×#41	—	4	2	no	—
184	C₇×#41	—	4	2	no	—
185	C₇×#41	—	4	2	no	—
186	C₇×#41	—	4	2	no	—
187	C₇×#41	—	4	2	no	—
188	C₇×#41	—	4	2	no	—
189	C₂×C₇×Q₈	—	2	1	no	—
190	C₂×C₇×Q₈	—	2	1	no	—
191	C₂×C₇×Q₈	—	2	1	no	—
192	C₂×C₇×Q₈	—	2	1	no	—
193	C₂×C₇×Q₈	—	2	1	no	—
194	C₂×C₇×Q₈	#41	1	2	yes	—
195	C₂×C₇×D₄	—	2	1	no	—
196	C₂×C₇×D₄	—	2	1	no	—
197	C₂×C₇×D₄	—	2	1	no	—
198	C₂×C₇×D₄	—	2	1	no	—
199	C₂×C₇×D₄	—	2	1	no	—
200	C₂×C₇×D₄	—	2	1	no	—
201	C₂×C₇×D₄	—	2	1	no	—
202	C₂×C₇×D₄	—	2	1	no	—
203	C₂×C₇×D₄	—	2	1	no	—
204	C₂×C₇×D₄	—	2	1	no	—
205	C₂×C₇×D₄	—	2	1	no	—
206	C₂×C₇×D₄	—	4	2	no	—
207	C₂×C₇×D₄	—	4	2	no	—
208	C₂×C₇×D₄	—	4	1	no	—
209	C₂×C₇×D₄	—	4	1	no	—
210	C₂×C₇×D₄	—	4	1	no	—
211	C₂×C₇×D₄	—	4	1	no	—
212	C₂×C₇×D₄	—	8	2	no	—
213	C₂×C₇×D₄	#41	1	2	yes	—
214	C₂²×C₄×C₇	—	2	1	no	—
215	C₂²×C₄×C₇	—	2	1	no	—
216	C₂²×C₄×C₇	—	2	1	no	—
217	C₂²×C₄×C₇	—	2	1	no	—
218	C₂²×C₄×C₇	—	2	1	no	—
219	C₂²×C₄×C₇	—	2	1	no	—
220	C₂²×C₄×C₇	—	2	1	no	—
221	C₂²×C₄×C₇	—	2	1	no	—
222	C₂²×C₄×C₇	—	4	2	no	—
223	C₂²×C₄×C₇	—	4	2	no	—
224	C₂²×C₄×C₇	—	4	1	no	—
225	C₂²×C₄×C₇	—	4	1	no	—
226	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
229	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
230	C₂²×C₄×C₇	#41	1	1	yes	characteristic
231	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
232	C₇×SD₁₆	—	4	2	no	—
233	C₇×SD₁₆	—	4	2	no	—
234	C₇×D₈	—	4	2	no	—
235	C₇×D₈	—	4	2	no	—
236	C₇×M₁₆	—	2	1	no	—
237	C₇×M₁₆	—	2	1	no	—
238	C₇×M₁₆	—	2	1	no	—
239	C₇×M₁₆	—	2	1	no	—
240	C₇×M₁₆	—	2	1	no	—
241	C₇×M₁₆	—	2	1	no	—
242	C₇×M₁₆	—	2	1	no	—
243	C₇×M₁₆	—	2	1	no	—
244	C₇×M₁₆	—	2	1	no	—
245	C₇×M₁₆	—	2	1	no	—
246	C₇×M₁₆	—	2	1	no	—
247	C₇×M₁₆	—	2	1	no	—
248	C₇×M₁₆	—	4	2	no	—
249	C₂×C₇×C₈	—	2	1	no	—
250	C₂×C₇×C₈	—	2	1	no	—
251	C₂×C₇×C₈	—	2	1	no	—
252	C₂×C₇×C₈	—	2	1	no	—
253	C₂×C₇×C₈	—	2	1	no	—
254	C₂×C₇×C₈	—	2	1	no	—
255	C₂×C₇×C₈	—	2	1	no	—
256	C₂×C₇×C₈	—	2	1	no	—
257	C₂×C₇×C₈	—	2	1	no	—
258	C₂×C₇×C₈	—	2	1	no	—
259	C₂×C₇×C₈	—	4	2	no	—
260	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
261	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
262	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
263	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
264	C₇×C₄⋊C₄	—	2	1	no	—
265	C₇×C₄⋊C₄	—	2	1	no	—
266	C₇×C₄⋊C₄	—	2	1	no	—
267	C₇×C₄⋊C₄	—	2	1	no	—
268	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
269	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
270	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
271	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
272	C₇×C₂²⋊C₄	—	2	1	no	—
273	C₇×C₂²⋊C₄	—	2	1	no	—
274	C₇×C₂²⋊C₄	—	2	1	no	—
275	C₇×C₂²⋊C₄	—	2	1	no	—
276	C₇×C₂²⋊C₄	—	4	1	no	—
277	C₇×C₂²⋊C₄	—	4	1	no	—
278	C₄²×C₇	—	2	1	no	—
279	C₄²×C₇	—	2	1	no	—
280	C₄²×C₇	—	2	1	no	—
281	C₄²×C₇	—	2	1	no	—
282	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
283	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
284	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
285	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
286	C₂×#121	C₂²×C₇	1	1	yes	characteristic
287	C₂×#117	C₂²×C₇	1	1	yes	characteristic
288	#482	—	2	1	no	—
289	#482	—	2	1	no	—
290	#481	—	2	1	no	—
291	#480	—	2	1	no	—
292	#449	—	2	1	no	—
293	#449	—	2	1	no	—
294	#449	—	2	1	no	—
295	#449	—	2	1	no	—
296	#433	—	2	1	no	—
297	#433	—	2	1	no	—
298	#432	—	2	1	no	—
299	#431	—	2	1	no	—
300	C₂×#108	C₂²×C₇	1	1	yes	characteristic
301	C₂×#108	C₂²×C₇	1	1	yes	characteristic
302	#428	C₂²×C₇	1	1	yes	characteristic
303	#428	C₂²×C₇	1	1	yes	characteristic
304	#424	C₂²×C₇	1	1	yes	characteristic
305	#424	C₂²×C₇	1	1	yes	characteristic
306	#419	C₂²×C₇	1	1	yes	characteristic
307	#419	C₂²×C₇	1	1	yes	characteristic
308	#418	C₂²×C₇	1	1	yes	characteristic
309	#418	C₂²×C₇	1	1	yes	characteristic
310	#418	C₂²×C₇	1	1	yes	characteristic
311	#418	C₂²×C₇	1	1	yes	characteristic
312	#417	—	2	1	no	—
313	#417	—	2	1	no	—
314	#416	—	2	1	no	—
315	#416	—	2	1	no	—
316	C₂×#103	C₂²×C₇	1	1	yes	characteristic
317	C₂×#102	C₂²×C₇	1	1	yes	characteristic
318	C₂×#101	C₂²×C₇	1	1	yes	characteristic
319	C₂×#101	C₂²×C₇	1	1	yes	characteristic
320	C₂×#100	C₂²×C₇	1	1	yes	characteristic
321	C₂×#100	C₂²×C₇	1	1	yes	characteristic
322	C₂×#100	C₂²×C₇	1	1	yes	characteristic
323	C₂×#100	C₂²×C₇	1	1	yes	characteristic
324	#407	C₂²×C₇	1	1	yes	characteristic
325	#407	C₂²×C₇	1	1	yes	characteristic
326	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
327	#405	C₂²×C₇	1	1	yes	characteristic
328	#405	C₂²×C₇	1	1	yes	characteristic
329	#343	C₂²×C₇	1	1	yes	characteristic
330	#343	C₂²×C₇	1	1	yes	characteristic
331	#341	C₂²×C₇	1	1	yes	characteristic
332	#341	C₂²×C₇	1	1	yes	characteristic
333	C₂³×C₇	—	2	1	no	—
334	C₂³×C₇	—	2	1	no	—
335	C₂³×C₇	—	2	2	no	—
336	C₂³×C₇	—	4	1	no	—
337	C₂³×C₇	—	4	1	no	—
338	C₂³×C₇	—	4	1	no	—
339	C₂³×C₇	—	4	1	no	—
340	C₂³×C₇	—	4	1	no	—
341	C₂³×C₇	—	4	1	no	—
342	C₂³×C₇	—	4	1	no	—
343	C₂³×C₇	—	8	1	no	—
344	C₂³×C₇	—	8	1	no	—
345	C₂³×C₇	—	8	1	no	—
346	C₂³×C₇	—	8	1	no	—
347	C₂³×C₇	#121	1	1	yes	characteristic
348	C₂³×C₇	#121	1	1	yes	characteristic
349	C₂³×C₇	C₂×#41	1	1	yes	characteristic
350	C₇×Q₈	—	2	2	no	—
351	C₇×Q₈	—	2	2	no	—
352	C₇×Q₈	—	4	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×D₄	—	2	2	no	—
355	C₇×D₄	—	2	2	no	—
356	C₇×D₄	—	4	4	no	—
357	C₇×D₄	—	4	4	no	—
358	C₇×D₄	—	4	1	no	—
359	C₇×D₄	—	4	1	no	—
360	C₇×D₄	—	4	1	no	—
361	C₇×D₄	—	4	1	no	—
362	C₇×D₄	—	4	1	no	—
363	C₇×D₄	—	4	1	no	—
364	C₇×D₄	—	8	2	no	—
365	C₇×D₄	—	8	2	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	2	1	no	—
373	C₂×C₄×C₇	—	2	1	no	—
374	C₂×C₄×C₇	—	2	1	no	—
375	C₂×C₄×C₇	—	2	1	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	2	1	no	—
378	C₂×C₄×C₇	—	2	1	no	—
379	C₂×C₄×C₇	—	2	1	no	—
380	C₂×C₄×C₇	—	2	2	no	—
381	C₂×C₄×C₇	—	4	4	no	—
382	C₂×C₄×C₇	—	4	4	no	—
383	C₂×C₄×C₇	—	4	2	no	—
384	C₂×C₄×C₇	—	4	1	no	—
385	C₂×C₄×C₇	—	4	1	no	—
386	C₂×C₄×C₇	—	4	1	no	—
387	C₂×C₄×C₇	—	8	1	no	—
388	C₂×C₄×C₇	—	8	1	no	—
389	C₂×C₄×C₇	#121	1	1	yes	characteristic
390	C₂×C₄×C₇	#121	1	1	yes	characteristic
391	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
393	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
394	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
395	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
396	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
397	C₇×C₈	—	2	1	no	—
398	C₇×C₈	—	2	1	no	—
399	C₇×C₈	—	2	1	no	—
400	C₇×C₈	—	2	1	no	—
401	C₇×C₈	—	4	1	no	—
402	C₇×C₈	—	4	1	no	—
403	C₇×C₈	—	4	1	no	—
404	C₇×C₈	—	4	1	no	—
405	C₇×C₈	—	4	1	no	—
406	C₇×C₈	—	4	1	no	—
407	C₂×#41	—	2	2	no	—
408	C₂×#41	—	2	2	no	—
409	C₂×#41	—	2	1	no	—
410	C₂×#41	—	2	1	no	—
411	C₂×#41	—	2	1	no	—
412	C₂×#41	—	2	1	no	—
413	C₂×#41	—	4	1	no	—
414	C₂×#41	—	4	1	no	—
415	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
416	C₂²×D₄	—	2	1	no	—
417	C₂²×D₄	—	2	1	no	—
418	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
419	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
422	C₂×SD₁₆	—	2	2	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×SD₁₆	—	4	1	no	—
426	C₂×D₈	—	2	2	no	—
427	C₂×D₈	—	2	1	no	—
428	C₂×D₈	—	2	1	no	—
429	C₂×D₈	—	4	1	no	—
430	C₂×M₁₆	—	2	1	no	—
431	C₂×M₁₆	—	4	1	no	—
432	C₂×M₁₆	—	4	1	no	—
433	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
434	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
435	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
436	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
437	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
438	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
439	C₂²×C₈	—	2	1	no	—
440	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
441	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
442	#121	—	2	1	no	—
443	#121	—	2	1	no	—
444	#121	—	2	1	no	—
445	#121	—	2	1	no	—
446	#117	—	2	1	no	—
447	#117	—	2	1	no	—
448	#117	C₇×D₄	1	1	yes	characteristic
449	#117	C₇×D₄	1	1	yes	characteristic
450	#117	C₂³×C₇	1	1	yes	characteristic
451	#117	C₂³×C₇	1	1	yes	characteristic
452	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
453	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
455	C₂×C₂²⋊C₄	—	2	1	no	—
456	C₂×C₂²⋊C₄	—	2	1	no	—
457	C₂×C₄²	—	2	1	no	—
458	#108	—	4	1	no	—
459	#108	—	4	1	no	—
460	#107	—	2	1	no	—
461	#107	—	2	1	no	—
462	#106	—	2	1	no	—
463	#106	—	2	1	no	—
464	C₄⋊C₈	—	2	1	no	—
465	C₄⋊C₈	—	2	1	no	—
466	C₄⋊C₈	—	2	1	no	—
467	C₄⋊C₈	—	2	1	no	—
468	#103	—	2	1	no	—
469	#103	—	2	1	no	—
470	#103	—	2	1	no	—
471	#103	—	2	1	no	—
472	#103	—	2	1	no	—
473	#103	—	2	1	no	—
474	#102	—	2	1	no	—
475	#102	—	2	1	no	—
476	#102	—	2	1	no	—
477	#102	—	2	1	no	—
478	#102	—	2	1	no	—
479	#102	—	2	1	no	—
480	#102	—	2	1	no	—
481	#102	—	2	1	no	—
482	#102	—	2	1	no	—
483	#102	—	2	1	no	—
484	#101	—	2	2	no	—
485	#101	—	4	1	no	—
486	#100	—	2	2	no	—
487	#100	—	4	1	no	—
488	#100	—	4	1	no	—
489	#100	—	4	1	no	—
490	C₂²⋊C₈	—	2	1	no	—
491	C₂²⋊C₈	—	2	1	no	—
492	C₂²⋊C₈	—	2	1	no	—
493	C₂²⋊C₈	—	2	1	no	—
494	C₂²⋊C₈	C₇×D₄	1	1	yes	characteristic
495	C₂²⋊C₈	C₇×D₄	1	1	yes	characteristic
496	C₂²⋊C₈	C₇×D₄	1	1	yes	characteristic
497	C₂²⋊C₈	C₇×D₄	1	1	yes	characteristic
498	#97	—	2	1	no	—
499	#97	—	2	1	no	—
500	C₄×C₈	—	2	1	no	—
501	C₄×C₈	—	2	1	no	—
502	C₂²×C₇	—	2	1	no	—
503	C₂²×C₇	—	2	1	no	—
504	C₂²×C₇	—	2	1	no	—
505	C₂²×C₇	—	2	1	no	—
506	C₂²×C₇	—	2	1	no	—
507	C₂²×C₇	—	2	1	no	—
508	C₂²×C₇	—	2	1	no	—
509	C₂²×C₇	—	2	1	no	—
510	C₂²×C₇	—	4	2	no	—
511	C₂²×C₇	—	8	1	no	—
512	C₂²×C₇	—	8	1	no	—
513	C₂²×C₇	—	8	2	no	—
514	C₂²×C₇	—	8	2	no	—
515	C₂²×C₇	—	8	1	no	—
516	C₂²×C₇	—	8	1	no	—
517	C₂²×C₇	—	8	1	no	—
518	C₂²×C₇	—	8	1	no	—
519	C₂²×C₇	—	8	1	no	—
520	C₂²×C₇	—	16	1	no	—
521	C₂²×C₇	—	16	1	no	—
522	C₂²×C₇	C₂×#121	1	1	yes	characteristic
523	C₂²×C₇	#551	1	1	yes	characteristic
524	C₂²×C₇	#551	1	1	yes	characteristic
525	C₄×C₇	—	2	1	no	—
526	C₄×C₇	—	2	1	no	—
527	C₄×C₇	—	4	2	no	—
528	C₄×C₇	—	4	1	no	—
529	C₄×C₇	—	4	1	no	—
530	C₄×C₇	—	4	1	no	—
531	C₄×C₇	—	4	1	no	—
532	C₄×C₇	—	8	1	no	—
533	C₄×C₇	#525	1	1	yes	characteristic
534	C₄×C₇	#535	1	1	yes	characteristic
535	C₄×C₇	#546	1	1	yes	characteristic
536	C₄×C₇	#546	1	1	yes	characteristic
537	C₂⁴	—	2	1	no	—
538	C₂⁴	—	2	1	no	—
539	C₂⁴	C₇×#41	1	1	yes	characteristic
540	#41	—	2	4	no	—
541	#41	—	2	4	no	—
542	#41	—	4	2	no	—
543	#41	—	4	2	no	—
544	#41	—	4	2	no	—
545	#41	—	4	2	no	—
546	#41	—	4	2	no	—
547	#41	—	4	2	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	2	1	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	C₇×#41	1	2	yes	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	—	2	1	no	—
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	—	2	1	no	—
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	4	2	no	—
566	C₂×D₄	—	4	2	no	—
567	C₂×D₄	—	4	1	no	—
568	C₂×D₄	—	4	1	no	—
569	C₂×D₄	—	4	1	no	—
570	C₂×D₄	—	4	1	no	—
571	C₂×D₄	—	8	2	no	—
572	C₂×D₄	C₇×#41	1	2	yes	—
573	C₂²×C₄	—	2	1	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	2	1	no	—
579	C₂²×C₄	—	2	1	no	—
580	C₂²×C₄	—	2	1	no	—
581	C₂²×C₄	—	4	2	no	—
582	C₂²×C₄	—	4	2	no	—
583	C₂²×C₄	—	4	1	no	—
584	C₂²×C₄	—	4	1	no	—
585	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
586	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
587	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
588	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
589	C₂²×C₄	C₇×#41	1	1	yes	characteristic
590	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
591	SD₁₆	—	4	2	no	—
592	SD₁₆	—	4	2	no	—
593	D₈	—	4	2	no	—
594	D₈	—	4	2	no	—
595	M₁₆	—	2	1	no	—
596	M₁₆	—	2	1	no	—
597	M₁₆	—	2	1	no	—
598	M₁₆	—	2	1	no	—
599	M₁₆	—	2	1	no	—
600	M₁₆	—	2	1	no	—
601	M₁₆	—	2	1	no	—
602	M₁₆	—	2	1	no	—
603	M₁₆	—	2	1	no	—
604	M₁₆	—	2	1	no	—
605	M₁₆	—	2	1	no	—
606	M₁₆	—	2	1	no	—
607	M₁₆	—	4	2	no	—
608	C₂×C₈	—	2	1	no	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₂×C₈	—	2	1	no	—
613	C₂×C₈	—	2	1	no	—
614	C₂×C₈	—	2	1	no	—
615	C₂×C₈	—	2	1	no	—
616	C₂×C₈	—	2	1	no	—
617	C₂×C₈	—	2	1	no	—
618	C₂×C₈	—	4	2	no	—
619	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
620	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
621	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
622	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	—	2	1	no	—
627	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
628	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
629	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
630	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
631	C₂²⋊C₄	—	2	1	no	—
632	C₂²⋊C₄	—	2	1	no	—
633	C₂²⋊C₄	—	2	1	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	4	1	no	—
636	C₂²⋊C₄	—	4	1	no	—
637	C₄²	—	2	1	no	—
638	C₄²	—	2	1	no	—
639	C₄²	—	2	1	no	—
640	C₄²	—	2	1	no	—
641	C₂×C₇	—	2	1	no	—
642	C₂×C₇	—	2	1	no	—
643	C₂×C₇	—	4	1	no	—
644	C₂×C₇	—	4	1	no	—
645	C₂×C₇	—	8	2	no	—
646	C₂×C₇	—	16	1	no	—
647	C₂×C₇	#2411	1	1	yes	characteristic
648	C₂×C₇	#3067	1	1	yes	characteristic
649	C₂×C₇	#3068	1	1	yes	characteristic
650	C₂³	—	2	1	no	—
651	C₂³	—	2	1	no	—
652	C₂³	—	2	2	no	—
653	C₂³	—	4	1	no	—
654	C₂³	—	4	1	no	—
655	C₂³	—	4	1	no	—
656	C₂³	—	4	1	no	—
657	C₂³	—	4	1	no	—
658	C₂³	—	4	1	no	—
659	C₂³	—	4	1	no	—
660	C₂³	—	8	1	no	—
661	C₂³	—	8	1	no	—
662	C₂³	—	8	1	no	—
663	C₂³	—	8	1	no	—
664	C₂³	C₇×#121	1	1	yes	characteristic
665	C₂³	C₇×#121	1	1	yes	characteristic
666	C₂³	C₂×C₇×#41	1	1	yes	characteristic
667	Q₈	—	2	2	no	—
668	Q₈	—	2	2	no	—
669	Q₈	—	4	1	no	—
670	Q₈	—	4	1	no	—
671	D₄	—	2	2	no	—
672	D₄	—	2	2	no	—
673	D₄	—	4	4	no	—
674	D₄	—	4	4	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	4	1	no	—
678	D₄	—	4	1	no	—
679	D₄	—	4	1	no	—
680	D₄	—	4	1	no	—
681	D₄	—	8	2	no	—
682	D₄	—	8	2	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	2	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	2	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	8	1	no	—
705	C₂×C₄	—	8	1	no	—
706	C₂×C₄	C₇×#121	1	1	yes	characteristic
707	C₂×C₄	C₇×#121	1	1	yes	characteristic
708	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
709	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
710	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
711	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
712	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
713	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
714	C₈	—	2	1	no	—
715	C₈	—	2	1	no	—
716	C₈	—	2	1	no	—
717	C₈	—	2	1	no	—
718	C₈	—	4	1	no	—
719	C₈	—	4	1	no	—
720	C₈	—	4	1	no	—
721	C₈	—	4	1	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₇	#24524	1	1	yes	characteristic, direct product factor, Sylow
725	C₂²	—	2	1	no	—
726	C₂²	—	2	1	no	—
727	C₂²	—	2	1	no	—
728	C₂²	—	2	1	no	—
729	C₂²	—	2	1	no	—
730	C₂²	—	2	1	no	—
731	C₂²	—	2	1	no	—
732	C₂²	—	2	1	no	—
733	C₂²	—	4	2	no	—
734	C₂²	—	8	1	no	—
735	C₂²	—	8	1	no	—
736	C₂²	—	8	2	no	—
737	C₂²	—	8	2	no	—
738	C₂²	—	8	1	no	—
739	C₂²	—	8	1	no	—
740	C₂²	—	8	1	no	—
741	C₂²	—	8	1	no	—
742	C₂²	—	8	1	no	—
743	C₂²	—	16	1	no	—
744	C₂²	—	16	1	no	—
745	C₂²	C₂×C₇×#121	1	1	yes	characteristic
746	C₂²	C₇×#551	1	1	yes	characteristic
747	C₂²	C₇×#551	1	1	yes	characteristic
748	C₄	—	2	1	no	—
749	C₄	—	2	1	no	—
750	C₄	—	4	2	no	—
751	C₄	—	4	1	no	—
752	C₄	—	4	1	no	—
753	C₄	—	4	1	no	—
754	C₄	—	4	1	no	—
755	C₄	—	8	1	no	—
756	C₄	C₇×#525	1	1	yes	characteristic
757	C₄	C₇×#535	1	1	yes	characteristic
758	C₄	C₇×#546	1	1	yes	characteristic
759	C₄	C₇×#546	1	1	yes	characteristic
760	C₂	—	2	1	no	—
761	C₂	—	2	1	no	—
762	C₂	—	4	1	no	—
763	C₂	—	4	1	no	—
764	C₂	—	8	2	no	—
765	C₂	—	16	1	no	—
766	C₂	C₇×#2411	1	1	yes	characteristic
767	C₂	C₇×#3067	1	1	yes	characteristic
768	C₂	C₇×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3094	C₂	semidirect
3. C₇×#2924	C₂	semidirect
4. C₇×#2896	C₂	semidirect
5. C₇×#2064	C₂	semidirect
6. C₇×#2063	C₂	semidirect
7. C₇×#2046	C₂	semidirect
8. C₇×#2045	C₂	semidirect
9. C₇×#1978	C₂	semidirect
10. C₇×#1977	C₂	semidirect
11. C₇×#1931	C₂	semidirect
12. C₇×#1930	C₂	semidirect
13. C₇×#1900	C₂	semidirect
14. C₇×#1855	C₂	semidirect
15. C₇×#1785	C₂	semidirect
67. #24524	C₇	direct (N × H)
163. #3094	C₂×C₇	semidirect
165. #2924	C₂×C₇	semidirect
166. #2896	C₂×C₇	semidirect
167. #2064	C₂×C₇	semidirect
168. #2063	C₂×C₇	semidirect
169. #2046	C₂×C₇	semidirect
170. #2045	C₂×C₇	semidirect
171. #1978	C₂×C₇	semidirect
172. #1977	C₂×C₇	semidirect
173. #1931	C₂×C₇	semidirect
174. #1930	C₂×C₇	semidirect
175. #1900	C₂×C₇	semidirect
176. #1855	C₂×C₇	semidirect
177. #1785	C₂×C₇	semidirect
724. C₇	#24524	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3094	2
C₂×C₇×#449	2
C₇×#2924	2
C₇×#2896	2
C₇×#2064	2
C₇×#2063	2
C₇×#2046	2
C₇×#2045	2
C₇×#1978	2
C₇×#1977	2
C₇×#1931	2
C₇×#1930	2
C₇×#1900	2
C₇×#1855	2
C₇×#1785	2
#24524	7

Subgroups — C7×#24524 #13201740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24524 #13201740