Subgroups — C₇×#25186 #13202402

Order 1792.

974 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3044	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#456	C₂	1	1	yes	maximal, characteristic
4	C₇×#3011	C₂	1	1	yes	maximal, characteristic
5	C₇×#2947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2898	C₂	1	1	yes	maximal, characteristic
7	C₇×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2052	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2038	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#2016	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1929	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1886	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1882	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1866	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×#137	C₂²	1	1	yes	characteristic
19	C₂×C₇×#137	C₂²	1	1	yes	characteristic
20	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₇×#131	C₂²	1	1	yes	characteristic
23	C₇×#536	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₇×#520	C₂²	1	1	yes	characteristic
26	C₇×#519	C₂²	1	1	yes	characteristic
27	C₇×#473	—	2	4	no	—
28	C₇×#464	—	2	1	no	—
29	C₇×#464	—	2	1	no	—
30	C₇×#462	—	2	1	no	—
31	C₇×#462	—	2	1	no	—
32	C₇×#456	—	2	4	no	—
33	C₇×#452	—	2	1	no	—
34	C₇×#452	—	2	1	no	—
35	C₇×#451	—	2	1	no	—
36	C₇×#451	—	2	1	no	—
37	C₇×#441	—	2	1	no	—
38	C₇×#440	—	2	1	no	—
39	C₄×C₇×Q₁₆	—	2	1	no	—
40	C₄×C₇×SD₁₆	—	2	1	no	—
41	C₂×C₇×#108	C₂²	1	1	yes	characteristic
42	C₂×C₇×#108	C₂²	1	1	yes	characteristic
43	C₇×#428	C₂²	1	1	yes	characteristic
44	C₇×#424	C₂²	1	1	yes	characteristic
45	C₂×C₇×#104	C₂²	1	1	yes	characteristic
46	C₂×C₇×#104	C₂²	1	1	yes	characteristic
47	C₇×#419	—	2	1	no	—
48	C₇×#419	C₂²	1	1	yes	characteristic
49	C₇×#419	C₂²	1	1	yes	characteristic
50	C₇×#419	C₂²	1	1	yes	characteristic
51	C₇×#417	—	2	1	no	—
52	C₇×#417	C₂²	1	1	yes	characteristic
53	C₇×#416	—	2	1	no	—
54	C₇×#416	C₂²	1	1	yes	characteristic
55	C₇×#416	C₂²	1	1	yes	characteristic
56	C₂×C₇×#103	—	2	1	no	—
57	C₂×C₇×#103	C₂²	1	1	yes	characteristic
58	C₂×C₇×#103	C₂²	1	1	yes	characteristic
59	C₂×C₇×#101	C₂²	1	1	yes	characteristic
60	C₂×C₇×#101	C₂²	1	1	yes	characteristic
61	C₂×C₇×#101	C₂²	1	1	yes	characteristic
62	C₇×#408	C₂²	1	1	yes	characteristic
63	C₇×#405	C₂²	1	1	yes	characteristic
64	C₇×#391	C₂²	1	1	yes	characteristic
65	C₇×#343	C₂²	1	1	yes	characteristic
66	C₇×#339	C₂²	1	1	yes	characteristic
67	C₇×#339	C₂²	1	1	yes	characteristic
68	C₇×#336	C₂²	1	1	yes	characteristic
69	#25186	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₇×2₋¹⁺⁴	—	2	2	no	—
71	C₇×2₋¹⁺⁴	—	2	2	no	—
72	C₇×2₋¹⁺⁴	—	2	2	no	—
73	C₇×2₋¹⁺⁴	—	4	1	no	—
74	C₂×C₇×#41	—	2	1	no	—
75	C₂×C₇×#41	—	2	1	no	—
76	C₂×C₇×#41	—	4	1	no	—
77	C₂×C₇×#41	C₂³	1	1	yes	characteristic
78	C₂×C₇×#41	C₂³	1	1	yes	characteristic
79	C₂²×C₇×Q₈	—	2	1	no	—
80	C₂²×C₇×Q₈	—	2	1	no	—
81	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
82	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
83	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
84	C₇×#137	—	2	2	no	—
85	C₇×#137	—	2	2	no	—
86	C₇×#137	—	2	2	no	—
87	C₇×#137	—	2	2	no	—
88	C₇×#137	—	4	1	no	—
89	C₇×#137	D₄	1	4	yes	—
90	C₂×C₇×Q₁₆	—	2	1	no	—
91	C₂×C₇×Q₁₆	—	2	2	no	—
92	C₂×C₇×Q₁₆	—	2	1	no	—
93	C₂×C₇×Q₁₆	—	2	1	no	—
94	C₂×C₇×Q₁₆	—	2	2	no	—
95	C₂×C₇×Q₁₆	—	2	1	no	—
96	C₂×C₇×Q₁₆	—	2	1	no	—
97	C₂×C₇×Q₁₆	—	2	1	no	—
98	C₂×C₇×Q₁₆	—	2	1	no	—
99	C₂×C₇×Q₁₆	—	4	1	no	—
100	C₂×C₇×Q₁₆	—	4	1	no	—
101	C₂×C₇×SD₁₆	—	2	1	no	—
102	C₂×C₇×SD₁₆	—	2	1	no	—
103	C₂×C₇×SD₁₆	—	2	1	no	—
104	C₇×#131	—	2	2	no	—
105	C₇×#131	—	4	1	no	—
106	C₂×C₇×M₁₆	—	2	1	no	—
107	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
108	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
109	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
111	C₂²×C₇×C₈	—	2	1	no	—
112	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
113	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
114	C₇×C₄⋊Q₈	—	2	2	no	—
115	C₇×C₄⋊Q₈	—	2	1	no	—
116	C₇×C₄⋊Q₈	—	2	2	no	—
117	C₇×#124	—	2	1	no	—
118	C₇×#123	—	4	1	no	—
119	C₇×C₂²⋊Q₈	—	2	1	no	—
120	C₇×C₂²⋊Q₈	—	2	1	no	—
121	C₄×C₇×Q₈	—	2	1	no	—
122	C₄×C₇×Q₈	—	2	1	no	—
123	C₄×C₇×Q₈	—	2	1	no	—
124	C₄×C₇×Q₈	—	2	1	no	—
125	C₄×C₇×Q₈	—	2	1	no	—
126	C₄×C₇×D₄	—	2	1	no	—
127	C₄×C₇×D₄	—	4	1	no	—
128	C₇×#117	—	2	1	no	—
129	C₇×#117	—	4	1	no	—
130	C₇×#117	C₂³	1	1	yes	characteristic
131	C₇×#117	C₂³	1	1	yes	characteristic
132	C₂×C₇×C₄⋊C₄	—	2	1	no	—
133	C₂×C₇×C₄⋊C₄	—	4	1	no	—
134	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
136	C₇×#108	—	2	2	no	—
137	C₇×#108	—	4	1	no	—
138	C₇×#107	—	2	1	no	—
139	C₇×#106	—	2	1	no	—
140	C₇×C₄⋊C₈	—	2	1	no	—
141	C₇×C₄⋊C₈	—	2	1	no	—
142	C₇×#104	—	2	2	no	—
143	C₇×#104	—	2	2	no	—
144	C₇×#103	—	2	1	no	—
145	C₇×#103	—	2	1	no	—
146	C₇×#103	—	2	1	no	—
147	C₇×#103	—	2	1	no	—
148	C₇×#103	—	2	1	no	—
149	C₇×#103	—	2	1	no	—
150	C₇×#103	—	2	1	no	—
151	C₇×#103	—	2	1	no	—
152	C₇×#103	—	2	1	no	—
153	C₇×#103	—	2	1	no	—
154	C₇×#103	—	2	1	no	—
155	C₇×#103	D₄	1	1	yes	characteristic
156	C₇×#103	D₄	1	1	yes	characteristic
157	C₇×#103	D₄	1	1	yes	characteristic
158	C₇×#103	D₄	1	1	yes	characteristic
159	C₇×#102	—	2	1	no	—
160	C₇×#102	D₄	1	1	yes	characteristic
161	C₇×#102	D₄	1	1	yes	characteristic
162	C₇×#102	D₄	1	1	yes	characteristic
163	C₇×#102	D₄	1	1	yes	characteristic
164	C₇×#101	—	2	2	no	—
165	C₇×#101	—	2	2	no	—
166	C₇×#101	—	2	2	no	—
167	C₇×C₂²⋊C₈	—	2	1	no	—
168	C₇×C₂²⋊C₈	—	2	1	no	—
169	C₇×#97	—	2	1	no	—
170	C₄×C₇×C₈	—	2	1	no	—
171	C₇×#95	—	4	1	no	—
172	C₂×#473	C₂×C₇	1	1	yes	characteristic, complemented
173	#3044	C₂×C₇	1	1	yes	characteristic, complemented
174	C₂×#456	C₂×C₇	1	1	yes	characteristic
175	#3011	C₂×C₇	1	1	yes	characteristic
176	#2947	C₂×C₇	1	1	yes	characteristic, complemented
177	#2898	C₂×C₇	1	1	yes	characteristic
178	#2064	C₂×C₇	1	1	yes	characteristic, complemented
179	#2052	C₂×C₇	1	1	yes	characteristic, complemented
180	#2038	C₂×C₇	1	1	yes	characteristic, complemented
181	#2016	C₂×C₇	1	1	yes	characteristic, complemented
182	#1979	C₂×C₇	1	1	yes	characteristic, complemented
183	#1929	C₂×C₇	1	1	yes	characteristic, complemented
184	#1886	C₂×C₇	1	1	yes	characteristic, complemented
185	#1882	C₂×C₇	1	1	yes	characteristic, complemented
186	#1866	C₂×C₇	1	1	yes	characteristic, complemented
187	C₇×#41	—	2	1	no	—
188	C₇×#41	—	2	1	no	—
189	C₇×#41	—	2	4	no	—
190	C₇×#41	—	2	1	no	—
191	C₇×#41	—	2	1	no	—
192	C₇×#41	—	2	1	no	—
193	C₇×#41	—	2	1	no	—
194	C₇×#41	—	4	2	no	—
195	C₇×#41	—	4	4	no	—
196	C₇×#41	—	4	2	no	—
197	C₇×#41	—	4	4	no	—
198	C₇×#41	—	4	2	no	—
199	C₇×#41	C₂×D₄	1	2	yes	—
200	C₇×#41	#41	1	2	yes	—
201	C₂×C₇×Q₈	—	2	1	no	—
202	C₂×C₇×Q₈	—	2	1	no	—
203	C₂×C₇×Q₈	—	2	4	no	—
204	C₂×C₇×Q₈	—	2	1	no	—
205	C₂×C₇×Q₈	—	2	1	no	—
206	C₂×C₇×Q₈	—	2	1	no	—
207	C₂×C₇×Q₈	—	2	1	no	—
208	C₂×C₇×Q₈	—	2	1	no	—
209	C₂×C₇×Q₈	—	2	1	no	—
210	C₂×C₇×Q₈	—	2	1	no	—
211	C₂×C₇×Q₈	—	2	1	no	—
212	C₂×C₇×Q₈	—	2	1	no	—
213	C₂×C₇×Q₈	—	2	1	no	—
214	C₂×C₇×Q₈	—	2	1	no	—
215	C₂×C₇×Q₈	—	2	1	no	—
216	C₂×C₇×Q₈	—	2	1	no	—
217	C₂×C₇×Q₈	—	2	1	no	—
218	C₂×C₇×Q₈	—	4	1	no	—
219	C₂×C₇×Q₈	—	4	2	no	—
220	C₂×C₇×Q₈	—	4	2	no	—
221	C₂×C₇×Q₈	—	4	2	no	—
222	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
223	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
224	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
225	C₂×C₇×Q₈	#41	1	2	yes	—
226	C₂×C₇×D₄	—	2	1	no	—
227	C₂×C₇×D₄	—	2	1	no	—
228	C₂×C₇×D₄	—	4	1	no	—
229	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
230	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
231	C₂²×C₄×C₇	—	2	1	no	—
232	C₂²×C₄×C₇	—	2	1	no	—
233	C₂²×C₄×C₇	—	2	1	no	—
234	C₂²×C₄×C₇	—	2	1	no	—
235	C₂²×C₄×C₇	—	2	1	no	—
236	C₂²×C₄×C₇	—	2	1	no	—
237	C₂²×C₄×C₇	—	2	1	no	—
238	C₂²×C₄×C₇	—	4	1	no	—
239	C₂²×C₄×C₇	—	4	1	no	—
240	C₂²×C₄×C₇	—	4	1	no	—
241	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
242	C₇×Q₁₆	—	2	4	no	—
243	C₇×Q₁₆	—	4	2	no	—
244	C₇×Q₁₆	—	4	2	no	—
245	C₇×Q₁₆	—	4	2	no	—
246	C₇×Q₁₆	—	4	2	no	—
247	C₇×Q₁₆	—	4	2	no	—
248	C₇×Q₁₆	—	4	2	no	—
249	C₇×SD₁₆	—	2	4	no	—
250	C₇×SD₁₆	—	4	2	no	—
251	C₇×SD₁₆	—	4	2	no	—
252	C₇×M₁₆	—	2	1	no	—
253	C₇×M₁₆	—	2	1	no	—
254	C₇×M₁₆	—	2	1	no	—
255	C₇×M₁₆	—	2	1	no	—
256	C₇×M₁₆	—	2	1	no	—
257	C₇×M₁₆	—	2	1	no	—
258	C₇×M₁₆	—	2	1	no	—
259	C₇×M₁₆	—	4	2	no	—
260	C₇×M₁₆	C₂×D₄	1	2	yes	—
261	C₂×C₇×C₈	—	2	1	no	—
262	C₂×C₇×C₈	—	2	1	no	—
263	C₂×C₇×C₈	—	2	1	no	—
264	C₂×C₇×C₈	—	2	1	no	—
265	C₂×C₇×C₈	—	2	1	no	—
266	C₂×C₇×C₈	—	2	1	no	—
267	C₂×C₇×C₈	—	2	1	no	—
268	C₂×C₇×C₈	—	2	1	no	—
269	C₂×C₇×C₈	—	4	2	no	—
270	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
271	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
272	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
273	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
274	C₇×C₄⋊C₄	—	2	1	no	—
275	C₇×C₄⋊C₄	—	2	1	no	—
276	C₇×C₄⋊C₄	—	4	1	no	—
277	C₇×C₄⋊C₄	—	4	4	no	—
278	C₇×C₄⋊C₄	—	4	1	no	—
279	C₇×C₄⋊C₄	—	4	1	no	—
280	C₇×C₄⋊C₄	—	4	1	no	—
281	C₇×C₄⋊C₄	—	4	1	no	—
282	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
283	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
284	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
285	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
286	C₇×C₂²⋊C₄	—	2	1	no	—
287	C₇×C₂²⋊C₄	—	2	1	no	—
288	C₇×C₂²⋊C₄	—	4	1	no	—
289	C₇×C₂²⋊C₄	—	4	1	no	—
290	C₇×C₂²⋊C₄	—	4	1	no	—
291	C₄²×C₇	—	2	1	no	—
292	C₄²×C₇	—	2	1	no	—
293	C₄²×C₇	—	2	1	no	—
294	C₄²×C₇	—	2	1	no	—
295	C₄²×C₇	—	4	1	no	—
296	C₄²×C₇	—	4	1	no	—
297	C₄²×C₇	—	4	1	no	—
298	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
299	C₂×#137	C₂²×C₇	1	1	yes	characteristic
300	C₂×#137	C₂²×C₇	1	1	yes	characteristic
301	C₂×#137	C₂²×C₇	1	1	yes	characteristic
302	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
303	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
304	C₂×#131	C₂²×C₇	1	1	yes	characteristic
305	#536	C₂²×C₇	1	1	yes	characteristic
306	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
307	#520	C₂²×C₇	1	1	yes	characteristic
308	#519	C₂²×C₇	1	1	yes	characteristic
309	#473	—	2	4	no	—
310	#464	—	2	1	no	—
311	#464	—	2	1	no	—
312	#462	—	2	1	no	—
313	#462	—	2	1	no	—
314	#456	—	2	4	no	—
315	#452	—	2	1	no	—
316	#452	—	2	1	no	—
317	#451	—	2	1	no	—
318	#451	—	2	1	no	—
319	#441	—	2	1	no	—
320	#440	—	2	1	no	—
321	C₄×Q₁₆	—	2	1	no	—
322	C₄×SD₁₆	—	2	1	no	—
323	C₂×#108	C₂²×C₇	1	1	yes	characteristic
324	C₂×#108	C₂²×C₇	1	1	yes	characteristic
325	#428	C₂²×C₇	1	1	yes	characteristic
326	#424	C₂²×C₇	1	1	yes	characteristic
327	C₂×#104	C₂²×C₇	1	1	yes	characteristic
328	C₂×#104	C₂²×C₇	1	1	yes	characteristic
329	#419	—	2	1	no	—
330	#419	C₂²×C₇	1	1	yes	characteristic
331	#419	C₂²×C₇	1	1	yes	characteristic
332	#419	C₂²×C₇	1	1	yes	characteristic
333	#417	—	2	1	no	—
334	#417	C₂²×C₇	1	1	yes	characteristic
335	#416	—	2	1	no	—
336	#416	C₂²×C₇	1	1	yes	characteristic
337	#416	C₂²×C₇	1	1	yes	characteristic
338	C₂×#103	—	2	1	no	—
339	C₂×#103	C₂²×C₇	1	1	yes	characteristic
340	C₂×#103	C₂²×C₇	1	1	yes	characteristic
341	C₂×#101	C₂²×C₇	1	1	yes	characteristic
342	C₂×#101	C₂²×C₇	1	1	yes	characteristic
343	C₂×#101	C₂²×C₇	1	1	yes	characteristic
344	#408	C₂²×C₇	1	1	yes	characteristic
345	#405	C₂²×C₇	1	1	yes	characteristic
346	#391	C₂²×C₇	1	1	yes	characteristic
347	#343	C₂²×C₇	1	1	yes	characteristic
348	#339	C₂²×C₇	1	1	yes	characteristic
349	#339	C₂²×C₇	1	1	yes	characteristic
350	#336	C₂²×C₇	1	1	yes	characteristic
351	C₂³×C₇	—	2	1	no	—
352	C₂³×C₇	—	2	1	no	—
353	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
354	C₇×Q₈	—	2	1	no	—
355	C₇×Q₈	—	2	2	no	—
356	C₇×Q₈	—	2	1	no	—
357	C₇×Q₈	—	2	2	no	—
358	C₇×Q₈	—	2	2	no	—
359	C₇×Q₈	—	2	1	no	—
360	C₇×Q₈	—	2	2	no	—
361	C₇×Q₈	—	4	4	no	—
362	C₇×Q₈	—	4	1	no	—
363	C₇×Q₈	—	4	1	no	—
364	C₇×Q₈	—	4	1	no	—
365	C₇×Q₈	—	4	1	no	—
366	C₇×Q₈	—	4	1	no	—
367	C₇×Q₈	—	4	1	no	—
368	C₇×Q₈	—	4	1	no	—
369	C₇×Q₈	—	4	1	no	—
370	C₇×Q₈	#121	1	2	yes	—
371	C₇×D₄	—	2	2	no	—
372	C₇×D₄	—	2	2	no	—
373	C₇×D₄	—	2	1	no	—
374	C₇×D₄	—	2	1	no	—
375	C₇×D₄	—	2	1	no	—
376	C₇×D₄	—	4	4	no	—
377	C₇×D₄	—	4	1	no	—
378	C₇×D₄	—	4	1	no	—
379	C₇×D₄	#121	1	2	yes	—
380	C₂×C₄×C₇	—	2	1	no	—
381	C₂×C₄×C₇	—	2	2	no	—
382	C₂×C₄×C₇	—	2	2	no	—
383	C₂×C₄×C₇	—	2	2	no	—
384	C₂×C₄×C₇	—	2	1	no	—
385	C₂×C₄×C₇	—	2	1	no	—
386	C₂×C₄×C₇	—	2	1	no	—
387	C₂×C₄×C₇	—	2	1	no	—
388	C₂×C₄×C₇	—	2	1	no	—
389	C₂×C₄×C₇	—	2	2	no	—
390	C₂×C₄×C₇	—	2	2	no	—
391	C₂×C₄×C₇	—	4	4	no	—
392	C₂×C₄×C₇	—	4	1	no	—
393	C₂×C₄×C₇	—	4	4	no	—
394	C₂×C₄×C₇	—	4	1	no	—
395	C₂×C₄×C₇	—	4	1	no	—
396	C₂×C₄×C₇	—	4	1	no	—
397	C₂×C₄×C₇	—	4	1	no	—
398	C₂×C₄×C₇	—	4	1	no	—
399	C₂×C₄×C₇	—	4	1	no	—
400	C₂×C₄×C₇	—	4	1	no	—
401	C₂×C₄×C₇	—	4	1	no	—
402	C₂×C₄×C₇	—	4	1	no	—
403	C₂×C₄×C₇	—	4	1	no	—
404	C₂×C₄×C₇	—	4	1	no	—
405	C₂×C₄×C₇	—	8	1	no	—
406	C₂×C₄×C₇	—	16	1	no	—
407	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
411	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
412	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
413	C₇×C₈	—	2	2	no	—
414	C₇×C₈	—	2	1	no	—
415	C₇×C₈	—	2	1	no	—
416	C₇×C₈	—	2	1	no	—
417	C₇×C₈	—	2	1	no	—
418	C₇×C₈	—	4	1	no	—
419	C₇×C₈	—	4	1	no	—
420	C₇×C₈	—	4	1	no	—
421	2₋¹⁺⁴	—	2	2	no	—
422	2₋¹⁺⁴	—	2	2	no	—
423	2₋¹⁺⁴	—	2	2	no	—
424	2₋¹⁺⁴	—	4	1	no	—
425	C₂×#41	—	2	1	no	—
426	C₂×#41	—	2	1	no	—
427	C₂×#41	—	4	1	no	—
428	C₂×#41	C₂³×C₇	1	1	yes	characteristic
429	C₂×#41	C₂³×C₇	1	1	yes	characteristic
430	C₂²×Q₈	—	2	1	no	—
431	C₂²×Q₈	—	2	1	no	—
432	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
433	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
434	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
435	#137	—	2	2	no	—
436	#137	—	2	2	no	—
437	#137	—	2	2	no	—
438	#137	—	2	2	no	—
439	#137	—	4	1	no	—
440	#137	C₇×D₄	1	4	yes	—
441	C₂×Q₁₆	—	2	1	no	—
442	C₂×Q₁₆	—	2	2	no	—
443	C₂×Q₁₆	—	2	1	no	—
444	C₂×Q₁₆	—	2	1	no	—
445	C₂×Q₁₆	—	2	2	no	—
446	C₂×Q₁₆	—	2	1	no	—
447	C₂×Q₁₆	—	2	1	no	—
448	C₂×Q₁₆	—	2	1	no	—
449	C₂×Q₁₆	—	2	1	no	—
450	C₂×Q₁₆	—	4	1	no	—
451	C₂×Q₁₆	—	4	1	no	—
452	C₂×SD₁₆	—	2	1	no	—
453	C₂×SD₁₆	—	2	1	no	—
454	C₂×SD₁₆	—	2	1	no	—
455	#131	—	2	2	no	—
456	#131	—	4	1	no	—
457	C₂×M₁₆	—	2	1	no	—
458	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
459	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
460	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
461	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
462	C₂²×C₈	—	2	1	no	—
463	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
464	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
465	C₄⋊Q₈	—	2	2	no	—
466	C₄⋊Q₈	—	2	1	no	—
467	C₄⋊Q₈	—	2	2	no	—
468	#124	—	2	1	no	—
469	#123	—	4	1	no	—
470	C₂²⋊Q₈	—	2	1	no	—
471	C₂²⋊Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	—	4	1	no	—
479	#117	—	2	1	no	—
480	#117	—	4	1	no	—
481	#117	C₂³×C₇	1	1	yes	characteristic
482	#117	C₂³×C₇	1	1	yes	characteristic
483	C₂×C₄⋊C₄	—	2	1	no	—
484	C₂×C₄⋊C₄	—	4	1	no	—
485	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
486	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
487	#108	—	2	2	no	—
488	#108	—	4	1	no	—
489	#107	—	2	1	no	—
490	#106	—	2	1	no	—
491	C₄⋊C₈	—	2	1	no	—
492	C₄⋊C₈	—	2	1	no	—
493	#104	—	2	2	no	—
494	#104	—	2	2	no	—
495	#103	—	2	1	no	—
496	#103	—	2	1	no	—
497	#103	—	2	1	no	—
498	#103	—	2	1	no	—
499	#103	—	2	1	no	—
500	#103	—	2	1	no	—
501	#103	—	2	1	no	—
502	#103	—	2	1	no	—
503	#103	—	2	1	no	—
504	#103	—	2	1	no	—
505	#103	—	2	1	no	—
506	#103	C₇×D₄	1	1	yes	characteristic
507	#103	C₇×D₄	1	1	yes	characteristic
508	#103	C₇×D₄	1	1	yes	characteristic
509	#103	C₇×D₄	1	1	yes	characteristic
510	#102	—	2	1	no	—
511	#102	C₇×D₄	1	1	yes	characteristic
512	#102	C₇×D₄	1	1	yes	characteristic
513	#102	C₇×D₄	1	1	yes	characteristic
514	#102	C₇×D₄	1	1	yes	characteristic
515	#101	—	2	2	no	—
516	#101	—	2	2	no	—
517	#101	—	2	2	no	—
518	C₂²⋊C₈	—	2	1	no	—
519	C₂²⋊C₈	—	2	1	no	—
520	#97	—	2	1	no	—
521	C₄×C₈	—	2	1	no	—
522	#95	—	4	1	no	—
523	C₂²×C₇	—	2	1	no	—
524	C₂²×C₇	—	2	1	no	—
525	C₂²×C₇	—	2	2	no	—
526	C₂²×C₇	—	4	1	no	—
527	C₂²×C₇	—	4	1	no	—
528	C₂²×C₇	—	4	1	no	—
529	C₂²×C₇	—	4	1	no	—
530	C₂²×C₇	—	4	1	no	—
531	C₂²×C₇	#538	1	1	yes	characteristic
532	C₂²×C₇	D₄²	1	1	yes	characteristic
533	C₂²×C₇	#546	1	1	yes	characteristic
534	C₄×C₇	—	2	2	no	—
535	C₄×C₇	—	4	2	no	—
536	C₄×C₇	—	4	1	no	—
537	C₄×C₇	—	4	2	no	—
538	C₄×C₇	—	4	2	no	—
539	C₄×C₇	—	4	1	no	—
540	C₄×C₇	—	4	1	no	—
541	C₄×C₇	—	4	1	no	—
542	C₄×C₇	—	4	1	no	—
543	C₄×C₇	—	8	1	no	—
544	C₄×C₇	—	8	1	no	—
545	C₄×C₇	C₂×#121	1	1	yes	characteristic
546	C₄×C₇	#525	1	1	yes	characteristic
547	C₄×C₇	D₄²	1	1	yes	characteristic
548	C₄×C₇	#546	1	1	yes	characteristic
549	#41	—	2	1	no	—
550	#41	—	2	1	no	—
551	#41	—	2	4	no	—
552	#41	—	2	1	no	—
553	#41	—	2	1	no	—
554	#41	—	2	1	no	—
555	#41	—	2	1	no	—
556	#41	—	4	2	no	—
557	#41	—	4	4	no	—
558	#41	—	4	2	no	—
559	#41	—	4	4	no	—
560	#41	—	4	2	no	—
561	#41	C₂×C₇×D₄	1	2	yes	—
562	#41	C₇×#41	1	2	yes	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	4	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	—	2	1	no	—
568	C₂×Q₈	—	2	1	no	—
569	C₂×Q₈	—	2	1	no	—
570	C₂×Q₈	—	2	1	no	—
571	C₂×Q₈	—	2	1	no	—
572	C₂×Q₈	—	2	1	no	—
573	C₂×Q₈	—	2	1	no	—
574	C₂×Q₈	—	2	1	no	—
575	C₂×Q₈	—	2	1	no	—
576	C₂×Q₈	—	2	1	no	—
577	C₂×Q₈	—	2	1	no	—
578	C₂×Q₈	—	2	1	no	—
579	C₂×Q₈	—	2	1	no	—
580	C₂×Q₈	—	4	1	no	—
581	C₂×Q₈	—	4	2	no	—
582	C₂×Q₈	—	4	2	no	—
583	C₂×Q₈	—	4	2	no	—
584	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
585	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
586	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
587	C₂×Q₈	C₇×#41	1	2	yes	—
588	C₂×D₄	—	2	1	no	—
589	C₂×D₄	—	2	1	no	—
590	C₂×D₄	—	4	1	no	—
591	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
592	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
593	C₂²×C₄	—	2	1	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	1	no	—
596	C₂²×C₄	—	2	1	no	—
597	C₂²×C₄	—	2	1	no	—
598	C₂²×C₄	—	2	1	no	—
599	C₂²×C₄	—	2	1	no	—
600	C₂²×C₄	—	4	1	no	—
601	C₂²×C₄	—	4	1	no	—
602	C₂²×C₄	—	4	1	no	—
603	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
604	Q₁₆	—	2	4	no	—
605	Q₁₆	—	4	2	no	—
606	Q₁₆	—	4	2	no	—
607	Q₁₆	—	4	2	no	—
608	Q₁₆	—	4	2	no	—
609	Q₁₆	—	4	2	no	—
610	Q₁₆	—	4	2	no	—
611	SD₁₆	—	2	4	no	—
612	SD₁₆	—	4	2	no	—
613	SD₁₆	—	4	2	no	—
614	M₁₆	—	2	1	no	—
615	M₁₆	—	2	1	no	—
616	M₁₆	—	2	1	no	—
617	M₁₆	—	2	1	no	—
618	M₁₆	—	2	1	no	—
619	M₁₆	—	2	1	no	—
620	M₁₆	—	2	1	no	—
621	M₁₆	—	4	2	no	—
622	M₁₆	C₂×C₇×D₄	1	2	yes	—
623	C₂×C₈	—	2	1	no	—
624	C₂×C₈	—	2	1	no	—
625	C₂×C₈	—	2	1	no	—
626	C₂×C₈	—	2	1	no	—
627	C₂×C₈	—	2	1	no	—
628	C₂×C₈	—	2	1	no	—
629	C₂×C₈	—	2	1	no	—
630	C₂×C₈	—	2	1	no	—
631	C₂×C₈	—	4	2	no	—
632	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
633	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
634	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
635	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
636	C₄⋊C₄	—	2	1	no	—
637	C₄⋊C₄	—	2	1	no	—
638	C₄⋊C₄	—	4	1	no	—
639	C₄⋊C₄	—	4	4	no	—
640	C₄⋊C₄	—	4	1	no	—
641	C₄⋊C₄	—	4	1	no	—
642	C₄⋊C₄	—	4	1	no	—
643	C₄⋊C₄	—	4	1	no	—
644	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
645	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
646	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
647	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
648	C₂²⋊C₄	—	2	1	no	—
649	C₂²⋊C₄	—	2	1	no	—
650	C₂²⋊C₄	—	4	1	no	—
651	C₂²⋊C₄	—	4	1	no	—
652	C₂²⋊C₄	—	4	1	no	—
653	C₄²	—	2	1	no	—
654	C₄²	—	2	1	no	—
655	C₄²	—	2	1	no	—
656	C₄²	—	2	1	no	—
657	C₄²	—	4	1	no	—
658	C₄²	—	4	1	no	—
659	C₄²	—	4	1	no	—
660	C₂×C₇	—	2	1	no	—
661	C₂×C₇	—	2	1	no	—
662	C₂×C₇	—	4	2	no	—
663	C₂×C₇	—	8	1	no	—
664	C₂×C₇	#2424	1	1	yes	characteristic
665	C₂×C₇	#3289	1	1	yes	characteristic
666	C₂×C₇	#3293	1	1	yes	characteristic
667	C₂³	—	2	1	no	—
668	C₂³	—	2	1	no	—
669	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
670	Q₈	—	2	1	no	—
671	Q₈	—	2	2	no	—
672	Q₈	—	2	1	no	—
673	Q₈	—	2	2	no	—
674	Q₈	—	2	2	no	—
675	Q₈	—	2	1	no	—
676	Q₈	—	2	2	no	—
677	Q₈	—	4	4	no	—
678	Q₈	—	4	1	no	—
679	Q₈	—	4	1	no	—
680	Q₈	—	4	1	no	—
681	Q₈	—	4	1	no	—
682	Q₈	—	4	1	no	—
683	Q₈	—	4	1	no	—
684	Q₈	—	4	1	no	—
685	Q₈	—	4	1	no	—
686	Q₈	C₇×#121	1	2	yes	—
687	D₄	—	2	2	no	—
688	D₄	—	2	2	no	—
689	D₄	—	2	1	no	—
690	D₄	—	2	1	no	—
691	D₄	—	2	1	no	—
692	D₄	—	4	4	no	—
693	D₄	—	4	1	no	—
694	D₄	—	4	1	no	—
695	D₄	C₇×#121	1	2	yes	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	2	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	1	no	—
701	C₂×C₄	—	2	1	no	—
702	C₂×C₄	—	2	1	no	—
703	C₂×C₄	—	2	1	no	—
704	C₂×C₄	—	2	1	no	—
705	C₂×C₄	—	2	2	no	—
706	C₂×C₄	—	2	2	no	—
707	C₂×C₄	—	4	4	no	—
708	C₂×C₄	—	4	1	no	—
709	C₂×C₄	—	4	4	no	—
710	C₂×C₄	—	4	1	no	—
711	C₂×C₄	—	4	1	no	—
712	C₂×C₄	—	4	1	no	—
713	C₂×C₄	—	4	1	no	—
714	C₂×C₄	—	4	1	no	—
715	C₂×C₄	—	4	1	no	—
716	C₂×C₄	—	4	1	no	—
717	C₂×C₄	—	4	1	no	—
718	C₂×C₄	—	4	1	no	—
719	C₂×C₄	—	4	1	no	—
720	C₂×C₄	—	4	1	no	—
721	C₂×C₄	—	8	1	no	—
722	C₂×C₄	—	16	1	no	—
723	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
724	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
725	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
726	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
727	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
728	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
729	C₈	—	2	2	no	—
730	C₈	—	2	1	no	—
731	C₈	—	2	1	no	—
732	C₈	—	2	1	no	—
733	C₈	—	2	1	no	—
734	C₈	—	4	1	no	—
735	C₈	—	4	1	no	—
736	C₈	—	4	1	no	—
737	C₇	#25186	1	1	yes	characteristic, direct product factor, Sylow
738	C₂²	—	2	1	no	—
739	C₂²	—	2	1	no	—
740	C₂²	—	2	2	no	—
741	C₂²	—	4	1	no	—
742	C₂²	—	4	1	no	—
743	C₂²	—	4	1	no	—
744	C₂²	—	4	1	no	—
745	C₂²	—	4	1	no	—
746	C₂²	C₇×#538	1	1	yes	characteristic
747	C₂²	C₇×D₄²	1	1	yes	characteristic
748	C₂²	C₇×#546	1	1	yes	characteristic
749	C₄	—	2	2	no	—
750	C₄	—	4	2	no	—
751	C₄	—	4	1	no	—
752	C₄	—	4	2	no	—
753	C₄	—	4	2	no	—
754	C₄	—	4	1	no	—
755	C₄	—	4	1	no	—
756	C₄	—	4	1	no	—
757	C₄	—	4	1	no	—
758	C₄	—	8	1	no	—
759	C₄	—	8	1	no	—
760	C₄	C₂×C₇×#121	1	1	yes	characteristic
761	C₄	C₇×#525	1	1	yes	characteristic
762	C₄	C₇×D₄²	1	1	yes	characteristic
763	C₄	C₇×#546	1	1	yes	characteristic
764	C₂	—	2	1	no	—
765	C₂	—	2	1	no	—
766	C₂	—	4	2	no	—
767	C₂	—	8	1	no	—
768	C₂	C₇×#2424	1	1	yes	characteristic
769	C₂	C₇×#3289	1	1	yes	characteristic
770	C₂	C₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×2₋¹⁺⁴	C₂²
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. C₂×C₇×#473	C₂	semidirect
2. C₇×#3044	C₂	semidirect
5. C₇×#2947	C₂	semidirect
7. C₇×#2064	C₂	semidirect
8. C₇×#2052	C₂	semidirect
9. C₇×#2038	C₂	semidirect
10. C₇×#2016	C₂	semidirect
11. C₇×#1979	C₂	semidirect
12. C₇×#1929	C₂	semidirect
13. C₇×#1886	C₂	semidirect
14. C₇×#1882	C₂	semidirect
15. C₇×#1866	C₂	semidirect
69. #25186	C₇	direct (N × H)
172. C₂×#473	C₂×C₇	semidirect
173. #3044	C₂×C₇	semidirect
176. #2947	C₂×C₇	semidirect
178. #2064	C₂×C₇	semidirect
179. #2052	C₂×C₇	semidirect
180. #2038	C₂×C₇	semidirect
181. #2016	C₂×C₇	semidirect
182. #1979	C₂×C₇	semidirect
183. #1929	C₂×C₇	semidirect
184. #1886	C₂×C₇	semidirect
185. #1882	C₂×C₇	semidirect
186. #1866	C₂×C₇	semidirect
737. C₇	#25186	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#473	2
C₇×#3044	2
C₂×C₇×#456	2
C₇×#3011	2
C₇×#2947	2
C₇×#2898	2
C₇×#2064	2
C₇×#2052	2
C₇×#2038	2
C₇×#2016	2
C₇×#1979	2
C₇×#1929	2
C₇×#1886	2
C₇×#1882	2
C₇×#1866	2
#25186	7

Subgroups — C7×#25186 #13202402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#25186 #13202402