Subgroups — C₇×#24926 #13202142

Order 1792.

870 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3099	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3013	C₂	1	1	yes	maximal, characteristic
3	C₇×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2898	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2039	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1930	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1929	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1882	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1876	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1859	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1815	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1793	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂×C₇×#137	C₂²	1	1	yes	characteristic
18	C₂×C₇×#135	C₂²	1	1	yes	characteristic
19	C₂×C₇×#131	C₂²	1	1	yes	characteristic
20	C₇×#537	C₂²	1	1	yes	characteristic
21	C₇×#520	C₂²	1	1	yes	characteristic
22	C₇×#520	C₂²	1	1	yes	characteristic
23	C₇×#485	—	2	1	no	—
24	C₇×#483	—	2	1	no	—
25	C₇×#482	—	2	1	no	—
26	C₇×#481	—	2	1	no	—
27	C₇×#452	—	2	1	no	—
28	C₇×#451	—	2	1	no	—
29	C₇×#449	—	2	1	no	—
30	C₇×#448	—	2	1	no	—
31	C₇×#442	—	2	1	no	—
32	C₇×#441	—	2	1	no	—
33	C₄×C₇×SD₁₆	—	2	1	no	—
34	C₄×C₇×SD₁₆	—	2	1	no	—
35	C₂×C₇×#108	C₂²	1	1	yes	characteristic
36	C₇×#428	C₂²	1	1	yes	characteristic
37	C₂×C₇×#106	C₂²	1	1	yes	characteristic
38	C₂×C₇×#106	C₂²	1	1	yes	characteristic
39	C₂×C₇×#104	C₂²	1	1	yes	characteristic
40	C₂×C₇×#104	C₂²	1	1	yes	characteristic
41	C₂×C₇×#104	C₂²	1	1	yes	characteristic
42	C₇×#419	—	2	1	no	—
43	C₇×#419	C₂²	1	1	yes	characteristic
44	C₇×#417	—	2	1	no	—
45	C₇×#417	C₂²	1	1	yes	characteristic
46	C₇×#417	C₂²	1	1	yes	characteristic
47	C₇×#417	C₂²	1	1	yes	characteristic
48	C₇×#417	C₂²	1	1	yes	characteristic
49	C₇×#417	C₂²	1	1	yes	characteristic
50	C₇×#416	—	2	1	no	—
51	C₇×#416	C₂²	1	1	yes	characteristic
52	C₂×C₇×#103	—	2	1	no	—
53	C₂×C₇×#103	C₂²	1	1	yes	characteristic
54	C₂×C₇×#103	C₂²	1	1	yes	characteristic
55	C₂×C₇×#101	C₂²	1	1	yes	characteristic
56	C₇×#408	C₂²	1	1	yes	characteristic
57	C₇×#408	C₂²	1	1	yes	characteristic
58	C₇×#405	C₂²	1	1	yes	characteristic
59	C₇×#391	C₂²	1	1	yes	characteristic
60	C₇×#384	C₂²	1	1	yes	characteristic
61	C₇×#343	C₂²	1	1	yes	characteristic
62	C₇×#343	C₂²	1	1	yes	characteristic
63	C₇×#343	C₂²	1	1	yes	characteristic
64	C₇×#341	C₂²	1	1	yes	characteristic
65	C₇×#339	C₂²	1	1	yes	characteristic
66	C₇×#336	C₂²	1	1	yes	characteristic
67	#24926	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₇×2₋¹⁺⁴	—	2	2	no	—
69	C₇×2₋¹⁺⁴	—	2	2	no	—
70	C₇×2₋¹⁺⁴	—	2	2	no	—
71	C₇×2₋¹⁺⁴	—	4	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	C₂³	1	1	yes	characteristic
76	C₂×C₇×#41	C₂³	1	1	yes	characteristic
77	C₂×C₇×#41	C₂³	1	1	yes	characteristic
78	C₂²×C₇×Q₈	—	2	1	no	—
79	C₂²×C₇×Q₈	—	2	1	no	—
80	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
81	C₇×#137	—	2	2	no	—
82	C₇×#137	—	4	1	no	—
83	C₇×#135	—	2	2	no	—
84	C₇×#135	—	2	2	no	—
85	C₂×C₇×Q₁₆	—	2	1	no	—
86	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
87	C₂×C₇×SD₁₆	—	2	1	no	—
88	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
89	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
90	C₂×C₇×D₈	D₄	1	1	yes	characteristic
91	C₇×#131	—	2	2	no	—
92	C₇×#131	—	4	1	no	—
93	C₂×C₇×M₁₆	—	2	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	C₂²×C₇×C₈	—	2	1	no	—
99	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
100	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
101	C₇×C₂²⋊Q₈	—	2	1	no	—
102	C₇×C₂²⋊Q₈	—	2	1	no	—
103	C₇×#121	—	2	1	no	—
104	C₇×#121	—	2	1	no	—
105	C₇×#121	—	4	1	no	—
106	C₄×C₇×Q₈	—	2	1	no	—
107	C₄×C₇×Q₈	—	2	1	no	—
108	C₄×C₇×D₄	—	2	1	no	—
109	C₄×C₇×D₄	—	2	1	no	—
110	C₄×C₇×D₄	—	4	1	no	—
111	C₄×C₇×D₄	—	4	1	no	—
112	C₇×#117	—	2	1	no	—
113	C₇×#117	—	2	1	no	—
114	C₇×#117	C₂³	1	1	yes	characteristic
115	C₂×C₇×C₄⋊C₄	—	2	1	no	—
116	C₂×C₇×C₄⋊C₄	—	2	1	no	—
117	C₂×C₇×C₄⋊C₄	—	4	1	no	—
118	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
120	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₇×#108	—	4	1	no	—
123	C₇×#107	—	2	1	no	—
124	C₇×#106	—	2	1	no	—
125	C₇×#106	—	2	1	no	—
126	C₇×#106	—	2	1	no	—
127	C₇×#106	—	2	1	no	—
128	C₇×#106	—	2	1	no	—
129	C₇×#104	—	2	2	no	—
130	C₇×#104	—	4	1	no	—
131	C₇×#104	—	4	1	no	—
132	C₇×#103	—	2	1	no	—
133	C₇×#103	—	2	1	no	—
134	C₇×#103	—	2	1	no	—
135	C₇×#103	—	2	1	no	—
136	C₇×#103	—	2	1	no	—
137	C₇×#103	—	2	1	no	—
138	C₇×#103	—	2	1	no	—
139	C₇×#103	—	2	1	no	—
140	C₇×#103	—	2	1	no	—
141	C₇×#103	—	2	1	no	—
142	C₇×#103	—	2	1	no	—
143	C₇×#103	D₄	1	1	yes	characteristic
144	C₇×#103	D₄	1	1	yes	characteristic
145	C₇×#102	—	2	1	no	—
146	C₇×#102	—	2	1	no	—
147	C₇×#102	—	2	1	no	—
148	C₇×#102	—	2	1	no	—
149	C₇×#102	—	2	1	no	—
150	C₇×#102	D₄	1	1	yes	characteristic
151	C₇×#102	D₄	1	1	yes	characteristic
152	C₇×#101	—	2	2	no	—
153	C₇×C₂²⋊C₈	—	2	1	no	—
154	C₇×C₂²⋊C₈	—	2	1	no	—
155	C₇×C₂²⋊C₈	—	2	1	no	—
156	C₇×C₂²⋊C₈	—	2	1	no	—
157	C₇×#97	—	2	1	no	—
158	C₄×C₇×C₈	—	2	1	no	—
159	C₇×#95	—	4	1	no	—
160	#3099	C₂×C₇	1	1	yes	characteristic, complemented
161	#3013	C₂×C₇	1	1	yes	characteristic
162	#2952	C₂×C₇	1	1	yes	characteristic, complemented
163	#2898	C₂×C₇	1	1	yes	characteristic, complemented
164	#2096	C₂×C₇	1	1	yes	characteristic, complemented
165	#2079	C₂×C₇	1	1	yes	characteristic, complemented
166	#2052	C₂×C₇	1	1	yes	characteristic, complemented
167	#2039	C₂×C₇	1	1	yes	characteristic, complemented
168	#1930	C₂×C₇	1	1	yes	characteristic, complemented
169	#1929	C₂×C₇	1	1	yes	characteristic, complemented
170	#1882	C₂×C₇	1	1	yes	characteristic, complemented
171	#1876	C₂×C₇	1	1	yes	characteristic, complemented
172	#1859	C₂×C₇	1	1	yes	characteristic, complemented
173	#1815	C₂×C₇	1	1	yes	characteristic, complemented
174	#1793	C₂×C₇	1	1	yes	characteristic, complemented
175	C₇×#41	—	2	1	no	—
176	C₇×#41	—	2	1	no	—
177	C₇×#41	—	2	4	no	—
178	C₇×#41	—	2	1	no	—
179	C₇×#41	—	2	1	no	—
180	C₇×#41	—	2	1	no	—
181	C₇×#41	—	2	1	no	—
182	C₇×#41	—	2	1	no	—
183	C₇×#41	—	2	1	no	—
184	C₇×#41	—	2	1	no	—
185	C₇×#41	—	2	1	no	—
186	C₇×#41	—	4	2	no	—
187	C₇×#41	—	4	4	no	—
188	C₇×#41	—	4	2	no	—
189	C₇×#41	—	4	4	no	—
190	C₇×#41	—	4	2	no	—
191	C₇×#41	#41	1	2	yes	—
192	C₇×#41	#41	1	2	yes	—
193	C₂×C₇×Q₈	—	2	1	no	—
194	C₂×C₇×Q₈	—	2	4	no	—
195	C₂×C₇×Q₈	—	2	1	no	—
196	C₂×C₇×Q₈	—	2	1	no	—
197	C₂×C₇×Q₈	—	2	1	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	4	1	no	—
200	C₂×C₇×Q₈	—	4	2	no	—
201	C₂×C₇×Q₈	—	4	2	no	—
202	C₂×C₇×Q₈	—	4	2	no	—
203	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
204	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×Q₈	#41	1	2	yes	—
207	C₂×C₇×D₄	—	2	1	no	—
208	C₂×C₇×D₄	—	2	1	no	—
209	C₂×C₇×D₄	—	2	1	no	—
210	C₂×C₇×D₄	—	4	1	no	—
211	C₂×C₇×D₄	—	4	1	no	—
212	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
223	C₂²×C₄×C₇	—	4	1	no	—
224	C₂²×C₄×C₇	—	4	1	no	—
225	C₂²×C₄×C₇	—	4	1	no	—
226	C₂²×C₄×C₇	—	4	1	no	—
227	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
228	C₇×Q₁₆	—	2	2	no	—
229	C₇×Q₁₆	—	4	2	no	—
230	C₇×SD₁₆	—	2	2	no	—
231	C₇×SD₁₆	—	2	2	no	—
232	C₇×SD₁₆	—	4	2	no	—
233	C₇×D₈	—	2	2	no	—
234	C₇×M₁₆	—	2	1	no	—
235	C₇×M₁₆	—	2	1	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	—	4	2	no	—
243	C₂×C₇×C₈	—	2	1	no	—
244	C₂×C₇×C₈	—	2	1	no	—
245	C₂×C₇×C₈	—	2	1	no	—
246	C₂×C₇×C₈	—	2	1	no	—
247	C₂×C₇×C₈	—	2	1	no	—
248	C₂×C₇×C₈	—	2	1	no	—
249	C₂×C₇×C₈	—	2	1	no	—
250	C₂×C₇×C₈	—	2	1	no	—
251	C₂×C₇×C₈	—	4	2	no	—
252	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
253	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
254	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
255	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	—	2	1	no	—
257	C₇×C₄⋊C₄	—	2	1	no	—
258	C₇×C₄⋊C₄	—	2	1	no	—
259	C₇×C₄⋊C₄	—	2	1	no	—
260	C₇×C₄⋊C₄	—	2	1	no	—
261	C₇×C₄⋊C₄	—	2	1	no	—
262	C₇×C₄⋊C₄	—	4	1	no	—
263	C₇×C₄⋊C₄	—	4	1	no	—
264	C₇×C₄⋊C₄	—	4	1	no	—
265	C₇×C₄⋊C₄	—	8	2	no	—
266	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
267	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
268	C₇×C₂²⋊C₄	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	4	1	no	—
270	C₇×C₂²⋊C₄	—	4	1	no	—
271	C₇×C₂²⋊C₄	—	4	1	no	—
272	C₇×C₂²⋊C₄	—	4	1	no	—
273	C₄²×C₇	—	2	1	no	—
274	C₄²×C₇	—	2	1	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	4	1	no	—
277	C₄²×C₇	—	4	1	no	—
278	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
279	C₂×#137	C₂²×C₇	1	1	yes	characteristic
280	C₂×#135	C₂²×C₇	1	1	yes	characteristic
281	C₂×#131	C₂²×C₇	1	1	yes	characteristic
282	#537	C₂²×C₇	1	1	yes	characteristic
283	#520	C₂²×C₇	1	1	yes	characteristic
284	#520	C₂²×C₇	1	1	yes	characteristic
285	#485	—	2	1	no	—
286	#483	—	2	1	no	—
287	#482	—	2	1	no	—
288	#481	—	2	1	no	—
289	#452	—	2	1	no	—
290	#451	—	2	1	no	—
291	#449	—	2	1	no	—
292	#448	—	2	1	no	—
293	#442	—	2	1	no	—
294	#441	—	2	1	no	—
295	C₄×SD₁₆	—	2	1	no	—
296	C₄×SD₁₆	—	2	1	no	—
297	C₂×#108	C₂²×C₇	1	1	yes	characteristic
298	#428	C₂²×C₇	1	1	yes	characteristic
299	C₂×#106	C₂²×C₇	1	1	yes	characteristic
300	C₂×#106	C₂²×C₇	1	1	yes	characteristic
301	C₂×#104	C₂²×C₇	1	1	yes	characteristic
302	C₂×#104	C₂²×C₇	1	1	yes	characteristic
303	C₂×#104	C₂²×C₇	1	1	yes	characteristic
304	#419	—	2	1	no	—
305	#419	C₂²×C₇	1	1	yes	characteristic
306	#417	—	2	1	no	—
307	#417	C₂²×C₇	1	1	yes	characteristic
308	#417	C₂²×C₇	1	1	yes	characteristic
309	#417	C₂²×C₇	1	1	yes	characteristic
310	#417	C₂²×C₇	1	1	yes	characteristic
311	#417	C₂²×C₇	1	1	yes	characteristic
312	#416	—	2	1	no	—
313	#416	C₂²×C₇	1	1	yes	characteristic
314	C₂×#103	—	2	1	no	—
315	C₂×#103	C₂²×C₇	1	1	yes	characteristic
316	C₂×#103	C₂²×C₇	1	1	yes	characteristic
317	C₂×#101	C₂²×C₇	1	1	yes	characteristic
318	#408	C₂²×C₇	1	1	yes	characteristic
319	#408	C₂²×C₇	1	1	yes	characteristic
320	#405	C₂²×C₇	1	1	yes	characteristic
321	#391	C₂²×C₇	1	1	yes	characteristic
322	#384	C₂²×C₇	1	1	yes	characteristic
323	#343	C₂²×C₇	1	1	yes	characteristic
324	#343	C₂²×C₇	1	1	yes	characteristic
325	#343	C₂²×C₇	1	1	yes	characteristic
326	#341	C₂²×C₇	1	1	yes	characteristic
327	#339	C₂²×C₇	1	1	yes	characteristic
328	#336	C₂²×C₇	1	1	yes	characteristic
329	C₂³×C₇	—	2	1	no	—
330	C₂³×C₇	—	2	1	no	—
331	C₂³×C₇	—	2	1	no	—
332	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
333	C₇×Q₈	—	2	1	no	—
334	C₇×Q₈	—	2	1	no	—
335	C₇×Q₈	—	2	1	no	—
336	C₇×Q₈	—	2	2	no	—
337	C₇×Q₈	—	2	2	no	—
338	C₇×Q₈	—	2	1	no	—
339	C₇×Q₈	—	2	1	no	—
340	C₇×Q₈	—	4	4	no	—
341	C₇×Q₈	—	4	1	no	—
342	C₇×Q₈	—	4	1	no	—
343	C₇×Q₈	#123	1	2	yes	—
344	C₇×D₄	—	2	2	no	—
345	C₇×D₄	—	2	2	no	—
346	C₇×D₄	—	2	1	no	—
347	C₇×D₄	—	2	1	no	—
348	C₇×D₄	—	2	1	no	—
349	C₇×D₄	—	2	1	no	—
350	C₇×D₄	—	2	1	no	—
351	C₇×D₄	—	4	4	no	—
352	C₇×D₄	—	4	1	no	—
353	C₇×D₄	—	4	1	no	—
354	C₇×D₄	—	4	1	no	—
355	C₇×D₄	—	4	1	no	—
356	C₇×D₄	—	16	1	no	—
357	C₇×D₄	#123	1	2	yes	—
358	C₂×C₄×C₇	—	2	1	no	—
359	C₂×C₄×C₇	—	2	2	no	—
360	C₂×C₄×C₇	—	2	2	no	—
361	C₂×C₄×C₇	—	2	2	no	—
362	C₂×C₄×C₇	—	2	1	no	—
363	C₂×C₄×C₇	—	2	1	no	—
364	C₂×C₄×C₇	—	2	1	no	—
365	C₂×C₄×C₇	—	2	1	no	—
366	C₂×C₄×C₇	—	2	2	no	—
367	C₂×C₄×C₇	—	4	4	no	—
368	C₂×C₄×C₇	—	4	1	no	—
369	C₂×C₄×C₇	—	4	4	no	—
370	C₂×C₄×C₇	—	4	1	no	—
371	C₂×C₄×C₇	—	4	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	1	no	—
374	C₂×C₄×C₇	—	4	1	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	1	no	—
379	C₂×C₄×C₇	—	4	1	no	—
380	C₂×C₄×C₇	—	8	1	no	—
381	C₂×C₄×C₇	—	8	1	no	—
382	C₂×C₄×C₇	—	8	1	no	—
383	C₂×C₄×C₇	—	16	1	no	—
384	C₂×C₄×C₇	—	16	1	no	—
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
387	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
388	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
389	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
390	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
391	C₇×C₈	—	2	1	no	—
392	C₇×C₈	—	2	1	no	—
393	C₇×C₈	—	2	1	no	—
394	C₇×C₈	—	2	1	no	—
395	C₇×C₈	—	4	1	no	—
396	C₇×C₈	—	4	1	no	—
397	C₇×C₈	—	4	1	no	—
398	C₇×C₈	—	4	1	no	—
399	2₋¹⁺⁴	—	2	2	no	—
400	2₋¹⁺⁴	—	2	2	no	—
401	2₋¹⁺⁴	—	2	2	no	—
402	2₋¹⁺⁴	—	4	1	no	—
403	C₂×#41	—	2	1	no	—
404	C₂×#41	—	2	1	no	—
405	C₂×#41	—	4	1	no	—
406	C₂×#41	C₂³×C₇	1	1	yes	characteristic
407	C₂×#41	C₂³×C₇	1	1	yes	characteristic
408	C₂×#41	C₂³×C₇	1	1	yes	characteristic
409	C₂²×Q₈	—	2	1	no	—
410	C₂²×Q₈	—	2	1	no	—
411	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
412	#137	—	2	2	no	—
413	#137	—	4	1	no	—
414	#135	—	2	2	no	—
415	#135	—	2	2	no	—
416	C₂×Q₁₆	—	2	1	no	—
417	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
420	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
421	C₂×D₈	C₇×D₄	1	1	yes	characteristic
422	#131	—	2	2	no	—
423	#131	—	4	1	no	—
424	C₂×M₁₆	—	2	1	no	—
425	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
426	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
427	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
428	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
429	C₂²×C₈	—	2	1	no	—
430	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	—	2	1	no	—
434	#121	—	2	1	no	—
435	#121	—	2	1	no	—
436	#121	—	4	1	no	—
437	C₄×Q₈	—	2	1	no	—
438	C₄×Q₈	—	2	1	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	—	2	1	no	—
441	C₄×D₄	—	4	1	no	—
442	C₄×D₄	—	4	1	no	—
443	#117	—	2	1	no	—
444	#117	—	2	1	no	—
445	#117	C₂³×C₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	4	1	no	—
449	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
453	#108	—	4	1	no	—
454	#107	—	2	1	no	—
455	#106	—	2	1	no	—
456	#106	—	2	1	no	—
457	#106	—	2	1	no	—
458	#106	—	2	1	no	—
459	#106	—	2	1	no	—
460	#104	—	2	2	no	—
461	#104	—	4	1	no	—
462	#104	—	4	1	no	—
463	#103	—	2	1	no	—
464	#103	—	2	1	no	—
465	#103	—	2	1	no	—
466	#103	—	2	1	no	—
467	#103	—	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	#103	C₇×D₄	1	1	yes	characteristic
475	#103	C₇×D₄	1	1	yes	characteristic
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	C₇×D₄	1	1	yes	characteristic
482	#102	C₇×D₄	1	1	yes	characteristic
483	#101	—	2	2	no	—
484	C₂²⋊C₈	—	2	1	no	—
485	C₂²⋊C₈	—	2	1	no	—
486	C₂²⋊C₈	—	2	1	no	—
487	C₂²⋊C₈	—	2	1	no	—
488	#97	—	2	1	no	—
489	C₄×C₈	—	2	1	no	—
490	#95	—	4	1	no	—
491	C₂²×C₇	—	2	1	no	—
492	C₂²×C₇	—	2	1	no	—
493	C₂²×C₇	—	2	2	no	—
494	C₂²×C₇	—	4	1	no	—
495	C₂²×C₇	—	4	1	no	—
496	C₂²×C₇	—	4	1	no	—
497	C₂²×C₇	—	4	1	no	—
498	C₂²×C₇	—	4	1	no	—
499	C₂²×C₇	—	4	1	no	—
500	C₂²×C₇	—	4	1	no	—
501	C₂²×C₇	—	4	1	no	—
502	C₂²×C₇	#538	1	1	yes	characteristic
503	C₂²×C₇	D₄²	1	1	yes	characteristic
504	C₂²×C₇	#551	1	1	yes	characteristic
505	C₄×C₇	—	2	2	no	—
506	C₄×C₇	—	4	2	no	—
507	C₄×C₇	—	4	1	no	—
508	C₄×C₇	—	4	2	no	—
509	C₄×C₇	—	4	1	no	—
510	C₄×C₇	—	4	1	no	—
511	C₄×C₇	—	4	1	no	—
512	C₄×C₇	—	8	1	no	—
513	C₄×C₇	—	8	1	no	—
514	C₄×C₇	—	8	1	no	—
515	C₄×C₇	C₂×#123	1	1	yes	characteristic
516	C₄×C₇	#525	1	1	yes	characteristic
517	C₄×C₇	#546	1	1	yes	characteristic
518	C₄×C₇	#546	1	1	yes	characteristic
519	#41	—	2	1	no	—
520	#41	—	2	1	no	—
521	#41	—	2	4	no	—
522	#41	—	2	1	no	—
523	#41	—	2	1	no	—
524	#41	—	2	1	no	—
525	#41	—	2	1	no	—
526	#41	—	2	1	no	—
527	#41	—	2	1	no	—
528	#41	—	2	1	no	—
529	#41	—	2	1	no	—
530	#41	—	4	2	no	—
531	#41	—	4	4	no	—
532	#41	—	4	2	no	—
533	#41	—	4	4	no	—
534	#41	—	4	2	no	—
535	#41	C₇×#41	1	2	yes	—
536	#41	C₇×#41	1	2	yes	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	4	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	2	1	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	4	1	no	—
544	C₂×Q₈	—	4	2	no	—
545	C₂×Q₈	—	4	2	no	—
546	C₂×Q₈	—	4	2	no	—
547	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
548	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
549	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
550	C₂×Q₈	C₇×#41	1	2	yes	—
551	C₂×D₄	—	2	1	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	4	1	no	—
555	C₂×D₄	—	4	1	no	—
556	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
557	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
558	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
559	C₂²×C₄	—	2	1	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	4	1	no	—
568	C₂²×C₄	—	4	1	no	—
569	C₂²×C₄	—	4	1	no	—
570	C₂²×C₄	—	4	1	no	—
571	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
572	Q₁₆	—	2	2	no	—
573	Q₁₆	—	4	2	no	—
574	SD₁₆	—	2	2	no	—
575	SD₁₆	—	2	2	no	—
576	SD₁₆	—	4	2	no	—
577	D₈	—	2	2	no	—
578	M₁₆	—	2	1	no	—
579	M₁₆	—	2	1	no	—
580	M₁₆	—	2	1	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	4	2	no	—
587	C₂×C₈	—	2	1	no	—
588	C₂×C₈	—	2	1	no	—
589	C₂×C₈	—	2	1	no	—
590	C₂×C₈	—	2	1	no	—
591	C₂×C₈	—	2	1	no	—
592	C₂×C₈	—	2	1	no	—
593	C₂×C₈	—	2	1	no	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	4	2	no	—
596	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
597	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
598	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
599	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	4	1	no	—
607	C₄⋊C₄	—	4	1	no	—
608	C₄⋊C₄	—	4	1	no	—
609	C₄⋊C₄	—	8	2	no	—
610	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
611	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	4	1	no	—
614	C₂²⋊C₄	—	4	1	no	—
615	C₂²⋊C₄	—	4	1	no	—
616	C₂²⋊C₄	—	4	1	no	—
617	C₄²	—	2	1	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	4	1	no	—
621	C₄²	—	4	1	no	—
622	C₂×C₇	—	2	1	no	—
623	C₂×C₇	—	2	1	no	—
624	C₂×C₇	—	4	2	no	—
625	C₂×C₇	—	8	1	no	—
626	C₂×C₇	—	8	1	no	—
627	C₂×C₇	#2418	1	1	yes	characteristic
628	C₂×C₇	#3289	1	1	yes	characteristic
629	C₂×C₇	#3291	1	1	yes	characteristic
630	C₂³	—	2	1	no	—
631	C₂³	—	2	1	no	—
632	C₂³	—	2	1	no	—
633	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
634	Q₈	—	2	1	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	2	1	no	—
637	Q₈	—	2	2	no	—
638	Q₈	—	2	2	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	4	4	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	C₇×#123	1	2	yes	—
645	D₄	—	2	2	no	—
646	D₄	—	2	2	no	—
647	D₄	—	2	1	no	—
648	D₄	—	2	1	no	—
649	D₄	—	2	1	no	—
650	D₄	—	2	1	no	—
651	D₄	—	2	1	no	—
652	D₄	—	4	4	no	—
653	D₄	—	4	1	no	—
654	D₄	—	4	1	no	—
655	D₄	—	4	1	no	—
656	D₄	—	4	1	no	—
657	D₄	—	16	1	no	—
658	D₄	C₇×#123	1	2	yes	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	4	4	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	4	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	8	1	no	—
682	C₂×C₄	—	8	1	no	—
683	C₂×C₄	—	8	1	no	—
684	C₂×C₄	—	16	1	no	—
685	C₂×C₄	—	16	1	no	—
686	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
687	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
688	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
689	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
690	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
691	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
692	C₈	—	2	1	no	—
693	C₈	—	2	1	no	—
694	C₈	—	2	1	no	—
695	C₈	—	2	1	no	—
696	C₈	—	4	1	no	—
697	C₈	—	4	1	no	—
698	C₈	—	4	1	no	—
699	C₈	—	4	1	no	—
700	C₇	#24926	1	1	yes	characteristic, direct product factor, Sylow
701	C₂²	—	2	1	no	—
702	C₂²	—	2	1	no	—
703	C₂²	—	2	2	no	—
704	C₂²	—	4	1	no	—
705	C₂²	—	4	1	no	—
706	C₂²	—	4	1	no	—
707	C₂²	—	4	1	no	—
708	C₂²	—	4	1	no	—
709	C₂²	—	4	1	no	—
710	C₂²	—	4	1	no	—
711	C₂²	—	4	1	no	—
712	C₂²	C₇×#538	1	1	yes	characteristic
713	C₂²	C₇×D₄²	1	1	yes	characteristic
714	C₂²	C₇×#551	1	1	yes	characteristic
715	C₄	—	2	2	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	1	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	1	no	—
720	C₄	—	4	1	no	—
721	C₄	—	4	1	no	—
722	C₄	—	8	1	no	—
723	C₄	—	8	1	no	—
724	C₄	—	8	1	no	—
725	C₄	C₂×C₇×#123	1	1	yes	characteristic
726	C₄	C₇×#525	1	1	yes	characteristic
727	C₄	C₇×#546	1	1	yes	characteristic
728	C₄	C₇×#546	1	1	yes	characteristic
729	C₂	—	2	1	no	—
730	C₂	—	2	1	no	—
731	C₂	—	4	2	no	—
732	C₂	—	8	1	no	—
733	C₂	—	8	1	no	—
734	C₂	C₇×#2418	1	1	yes	characteristic
735	C₂	C₇×#3289	1	1	yes	characteristic
736	C₂	C₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₇×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3099	C₂	semidirect
3. C₇×#2952	C₂	semidirect
4. C₇×#2898	C₂	semidirect
5. C₇×#2096	C₂	semidirect
6. C₇×#2079	C₂	semidirect
7. C₇×#2052	C₂	semidirect
8. C₇×#2039	C₂	semidirect
9. C₇×#1930	C₂	semidirect
10. C₇×#1929	C₂	semidirect
11. C₇×#1882	C₂	semidirect
12. C₇×#1876	C₂	semidirect
13. C₇×#1859	C₂	semidirect
14. C₇×#1815	C₂	semidirect
15. C₇×#1793	C₂	semidirect
67. #24926	C₇	direct (N × H)
160. #3099	C₂×C₇	semidirect
162. #2952	C₂×C₇	semidirect
163. #2898	C₂×C₇	semidirect
164. #2096	C₂×C₇	semidirect
165. #2079	C₂×C₇	semidirect
166. #2052	C₂×C₇	semidirect
167. #2039	C₂×C₇	semidirect
168. #1930	C₂×C₇	semidirect
169. #1929	C₂×C₇	semidirect
170. #1882	C₂×C₇	semidirect
171. #1876	C₂×C₇	semidirect
172. #1859	C₂×C₇	semidirect
173. #1815	C₂×C₇	semidirect
174. #1793	C₂×C₇	semidirect
700. C₇	#24926	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×#3099	2
C₇×#3013	2
C₇×#2952	2
C₇×#2898	2
C₇×#2096	2
C₇×#2079	2
C₇×#2052	2
C₇×#2039	2
C₇×#1930	2
C₇×#1929	2
C₇×#1882	2
C₇×#1876	2
C₇×#1859	2
C₇×#1815	2
C₇×#1793	2
#24926	7

Subgroups — C7×#24926 #13202142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24926 #13202142