Subgroups — C₇×#22165 #13199381

Order 1792.

764 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3035	C₂	1	1	yes	maximal, characteristic
2	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1980	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1974	C₂	1	2	yes	maximal, complemented
6	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1937	C₂	1	2	yes	maximal, complemented
8	C₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₇×#1866	C₂	1	2	yes	maximal, complemented
10	C₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	2	yes	—
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	1	yes	characteristic
15	C₄×C₇×#41	C₂²	1	1	yes	characteristic
16	C₇×#464	C₄	1	2	yes	complemented
17	C₇×#463	C₄	1	2	yes	complemented
18	C₇×#462	C₄	1	1	yes	characteristic, complemented
19	C₇×#461	C₄	1	1	yes	characteristic, complemented
20	C₇×#460	C₄	1	1	yes	characteristic, complemented
21	C₇×#459	C₄	1	1	yes	characteristic, complemented
22	C₇×#436	—	2	1	no	—
23	C₇×#436	—	2	1	no	—
24	C₇×C₈×D₄	—	2	1	no	—
25	C₇×C₈×D₄	—	2	1	no	—
26	C₇×#433	—	2	1	no	—
27	C₇×#433	—	2	1	no	—
28	C₇×#431	—	2	1	no	—
29	C₇×#431	—	2	1	no	—
30	C₂×C₇×#108	C₂²	1	2	yes	—
31	C₇×#424	C₂²	1	1	yes	characteristic
32	C₇×#424	C₂²	1	1	yes	characteristic
33	C₇×#423	C₂²	1	1	yes	characteristic
34	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₇×#104	C₂²	1	2	yes	—
36	C₂×C₇×#104	C₂²	1	2	yes	—
37	C₂×C₇×#104	C₂²	1	2	yes	—
38	C₇×#416	C₂²	1	2	yes	—
39	C₇×#408	C₂²	1	1	yes	characteristic
40	C₇×#408	C₂²	1	2	yes	—
41	C₇×#408	C₂²	1	1	yes	characteristic
42	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
44	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
45	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
46	C₇×#341	C₂²	1	2	yes	—
47	C₇×#339	C₂²	1	2	yes	—
48	C₇×#336	C₂²	1	2	yes	—
49	#22165	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₇×#41	C₂³	1	1	yes	characteristic
51	C₂×C₇×#41	C₂³	1	2	yes	—
52	C₇×#135	—	4	2	no	—
53	C₇×#135	—	4	2	no	—
54	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
55	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
56	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
57	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
58	C₇×#131	—	4	2	no	—
59	C₇×#131	—	4	2	no	—
60	C₂×C₇×M₁₆	—	2	2	no	—
61	C₂×C₇×M₁₆	C₂³	1	2	yes	—
62	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
64	C₂²×C₇×C₈	—	2	2	no	—
65	C₂²×C₇×C₈	C₂³	1	2	yes	—
66	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
69	C₇×#127	C₂×C₄	1	1	yes	characteristic
70	C₇×#124	C₂×C₄	1	2	yes	—
71	C₇×#121	—	2	1	no	—
72	C₇×#121	—	2	1	no	—
73	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₇×D₄	—	2	1	no	—
76	C₄×C₇×D₄	—	2	1	no	—
77	C₄×C₇×D₄	—	2	1	no	—
78	C₄×C₇×D₄	—	2	1	no	—
79	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
80	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
81	C₇×#117	—	2	1	no	—
82	C₇×#117	C₂³	1	1	yes	characteristic
83	C₂×C₇×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄²×C₇	—	2	1	no	—
85	C₂×C₄²×C₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₇×#108	—	4	2	no	—
88	C₇×C₄⋊C₈	—	2	1	no	—
89	C₇×C₄⋊C₈	—	2	1	no	—
90	C₇×C₄⋊C₈	—	2	1	no	—
91	C₇×C₄⋊C₈	—	2	1	no	—
92	C₇×C₄⋊C₈	—	2	1	no	—
93	C₇×C₄⋊C₈	—	2	1	no	—
94	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
97	C₇×#104	—	4	2	no	—
98	C₇×#104	—	4	2	no	—
99	C₇×#104	—	4	2	no	—
100	C₇×#103	C₂×C₄	1	2	yes	—
101	C₇×#103	C₂×C₄	1	2	yes	—
102	C₇×#102	C₂×C₄	1	2	yes	—
103	C₇×#102	C₂×C₄	1	2	yes	—
104	C₇×C₂²⋊C₈	—	2	1	no	—
105	C₇×C₂²⋊C₈	—	2	1	no	—
106	C₇×C₂²⋊C₈	—	2	1	no	—
107	C₇×C₂²⋊C₈	—	2	1	no	—
108	C₇×C₂²⋊C₈	—	2	2	no	—
109	C₇×C₂²⋊C₈	—	2	2	no	—
110	C₇×#97	—	2	1	no	—
111	C₄×C₇×C₈	—	2	1	no	—
112	C₄×C₇×C₈	—	4	2	no	—
113	C₄×C₇×C₈	D₄	1	1	yes	characteristic
114	C₄×C₇×C₈	D₄	1	1	yes	characteristic
115	C₄×C₇×C₈	D₄	1	1	yes	characteristic
116	C₄×C₇×C₈	D₄	1	1	yes	characteristic
117	C₇×#95	—	2	1	no	—
118	#3035	C₂×C₇	1	1	yes	characteristic
119	#2931	C₂×C₇	1	1	yes	characteristic, complemented
120	#2922	C₂×C₇	1	1	yes	characteristic, complemented
121	#1980	C₂×C₇	1	1	yes	characteristic, complemented
122	#1974	C₂×C₇	1	2	yes	complemented
123	#1950	C₂×C₇	1	1	yes	characteristic, complemented
124	#1937	C₂×C₇	1	2	yes	complemented
125	#1876	C₂×C₇	1	2	yes	complemented
126	#1866	C₂×C₇	1	2	yes	complemented
127	#1806	C₂×C₇	1	1	yes	characteristic, complemented
128	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
129	C₇×#41	—	2	1	no	—
130	C₇×#41	—	2	2	no	—
131	C₇×#41	—	2	2	no	—
132	C₇×#41	—	2	2	no	—
133	C₇×#41	—	2	1	no	—
134	C₇×#41	—	2	1	no	—
135	C₇×#41	—	2	1	no	—
136	C₇×#41	—	2	2	no	—
137	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
138	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₇×D₄	—	2	1	no	—
140	C₂×C₇×D₄	—	2	2	no	—
141	C₂×C₇×D₄	—	2	1	no	—
142	C₂×C₇×D₄	—	2	1	no	—
143	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
144	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	—	2	2	no	—
146	C₂²×C₄×C₇	—	2	1	no	—
147	C₂²×C₄×C₇	—	2	1	no	—
148	C₂²×C₄×C₇	—	2	1	no	—
149	C₂²×C₄×C₇	—	2	2	no	—
150	C₂²×C₄×C₇	#41	1	1	yes	characteristic
151	C₂²×C₄×C₇	#41	1	1	yes	characteristic
152	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
153	C₇×Q₁₆	—	4	2	no	—
154	C₇×SD₁₆	—	4	2	no	—
155	C₇×SD₁₆	—	4	2	no	—
156	C₇×D₈	—	4	2	no	—
157	C₇×M₁₆	—	2	1	no	—
158	C₇×M₁₆	—	2	1	no	—
159	C₇×M₁₆	—	2	2	no	—
160	C₇×M₁₆	—	2	1	no	—
161	C₇×M₁₆	—	2	2	no	—
162	C₇×M₁₆	—	2	1	no	—
163	C₇×M₁₆	—	4	4	no	—
164	C₂×C₇×C₈	—	2	1	no	—
165	C₂×C₇×C₈	—	2	2	no	—
166	C₂×C₇×C₈	—	2	2	no	—
167	C₂×C₇×C₈	—	2	1	no	—
168	C₂×C₇×C₈	—	2	1	no	—
169	C₂×C₇×C₈	—	2	1	no	—
170	C₂×C₇×C₈	—	2	1	no	—
171	C₂×C₇×C₈	—	2	1	no	—
172	C₂×C₇×C₈	—	2	2	no	—
173	C₂×C₇×C₈	—	4	4	no	—
174	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
175	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
176	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
177	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
178	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
180	C₇×C₄⋊C₄	—	2	4	no	—
181	C₇×C₄⋊C₄	—	2	1	no	—
182	C₇×C₄⋊C₄	—	2	1	no	—
183	C₇×C₄⋊C₄	—	2	1	no	—
184	C₇×C₄⋊C₄	—	2	1	no	—
185	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
186	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
187	C₇×C₂²⋊C₄	—	2	1	no	—
188	C₇×C₂²⋊C₄	—	2	1	no	—
189	C₇×C₂²⋊C₄	—	2	1	no	—
190	C₇×C₂²⋊C₄	—	2	2	no	—
191	C₇×C₂²⋊C₄	—	2	2	no	—
192	C₄²×C₇	—	2	1	no	—
193	C₄²×C₇	—	2	1	no	—
194	C₄²×C₇	—	2	1	no	—
195	C₄²×C₇	—	2	2	no	—
196	C₄²×C₇	—	2	2	no	—
197	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
198	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
199	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
201	C₂×#135	C₂²×C₇	1	2	yes	—
202	C₂×#131	C₂²×C₇	1	2	yes	—
203	#532	C₂²×C₇	1	1	yes	characteristic
204	C₄×#41	C₂²×C₇	1	1	yes	characteristic
205	#464	C₄×C₇	1	2	yes	complemented
206	#463	C₄×C₇	1	2	yes	complemented
207	#462	C₄×C₇	1	1	yes	characteristic, complemented
208	#461	C₄×C₇	1	1	yes	characteristic, complemented
209	#460	C₄×C₇	1	1	yes	characteristic, complemented
210	#459	C₄×C₇	1	1	yes	characteristic, complemented
211	#436	—	2	1	no	—
212	#436	—	2	1	no	—
213	C₈×D₄	—	2	1	no	—
214	C₈×D₄	—	2	1	no	—
215	#433	—	2	1	no	—
216	#433	—	2	1	no	—
217	#431	—	2	1	no	—
218	#431	—	2	1	no	—
219	C₂×#108	C₂²×C₇	1	2	yes	—
220	#424	C₂²×C₇	1	1	yes	characteristic
221	#424	C₂²×C₇	1	1	yes	characteristic
222	#423	C₂²×C₇	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
224	C₂×#104	C₂²×C₇	1	2	yes	—
225	C₂×#104	C₂²×C₇	1	2	yes	—
226	C₂×#104	C₂²×C₇	1	2	yes	—
227	#416	C₂²×C₇	1	2	yes	—
228	#408	C₂²×C₇	1	1	yes	characteristic
229	#408	C₂²×C₇	1	1	yes	characteristic
230	#408	C₂²×C₇	1	2	yes	—
231	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
232	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
234	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
235	#341	C₂²×C₇	1	2	yes	—
236	#339	C₂²×C₇	1	2	yes	—
237	#336	C₂²×C₇	1	2	yes	—
238	C₂³×C₇	—	2	1	no	—
239	C₂³×C₇	—	2	2	no	—
240	C₂³×C₇	C₂×#41	1	1	yes	characteristic
241	C₇×Q₈	—	2	1	no	—
242	C₇×Q₈	—	2	2	no	—
243	C₇×Q₈	—	2	2	no	—
244	C₇×Q₈	—	2	1	no	—
245	C₇×D₄	—	2	1	no	—
246	C₇×D₄	—	2	2	no	—
247	C₇×D₄	—	2	2	no	—
248	C₇×D₄	—	2	1	no	—
249	C₇×D₄	—	4	2	no	—
250	C₇×D₄	—	4	1	no	—
251	C₇×D₄	—	4	2	no	—
252	C₇×D₄	—	4	1	no	—
253	C₇×D₄	—	4	2	no	—
254	C₇×D₄	—	4	2	no	—
255	C₂×C₄×C₇	—	2	2	no	—
256	C₂×C₄×C₇	—	2	4	no	—
257	C₂×C₄×C₇	—	2	1	no	—
258	C₂×C₄×C₇	—	2	1	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	2	2	no	—
261	C₂×C₄×C₇	—	4	1	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	4	2	no	—
264	C₂×C₄×C₇	—	4	2	no	—
265	C₂×C₄×C₇	—	4	2	no	—
266	C₂×C₄×C₇	—	4	1	no	—
267	C₂×C₄×C₇	—	4	1	no	—
268	C₂×C₄×C₇	—	4	2	no	—
269	C₂×C₄×C₇	—	4	2	no	—
270	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
271	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
274	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
275	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
277	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
279	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
280	C₇×C₈	—	2	2	no	—
281	C₇×C₈	—	2	2	no	—
282	C₇×C₈	—	2	1	no	—
283	C₇×C₈	—	2	1	no	—
284	C₇×C₈	—	2	1	no	—
285	C₇×C₈	—	2	1	no	—
286	C₇×C₈	—	4	2	no	—
287	C₇×C₈	—	4	1	no	—
288	C₇×C₈	—	4	1	no	—
289	C₂×#41	C₂³×C₇	1	1	yes	characteristic
290	C₂×#41	C₂³×C₇	1	2	yes	—
291	#135	—	4	2	no	—
292	#135	—	4	2	no	—
293	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
294	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
295	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
296	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
297	#131	—	4	2	no	—
298	#131	—	4	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	C₂³×C₇	1	2	yes	—
301	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
302	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	C₂³×C₇	1	2	yes	—
305	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
307	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
308	#127	C₂×C₄×C₇	1	1	yes	characteristic
309	#124	C₂×C₄×C₇	1	2	yes	—
310	#121	—	2	1	no	—
311	#121	—	2	1	no	—
312	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
313	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
319	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
320	#117	—	2	1	no	—
321	#117	C₂³×C₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	C₂³×C₇	1	2	yes	—
325	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
326	#108	—	4	2	no	—
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
334	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
335	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
336	#104	—	4	2	no	—
337	#104	—	4	2	no	—
338	#104	—	4	2	no	—
339	#103	C₂×C₄×C₇	1	2	yes	—
340	#103	C₂×C₄×C₇	1	2	yes	—
341	#102	C₂×C₄×C₇	1	2	yes	—
342	#102	C₂×C₄×C₇	1	2	yes	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂²⋊C₈	—	2	2	no	—
349	#97	—	2	1	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	—	4	2	no	—
352	C₄×C₈	C₇×D₄	1	1	yes	characteristic
353	C₄×C₈	C₇×D₄	1	1	yes	characteristic
354	C₄×C₈	C₇×D₄	1	1	yes	characteristic
355	C₄×C₈	C₇×D₄	1	1	yes	characteristic
356	#95	—	2	1	no	—
357	C₂²×C₇	—	2	1	no	—
358	C₂²×C₇	—	2	1	no	—
359	C₂²×C₇	—	4	2	no	—
360	C₂²×C₇	—	4	1	no	—
361	C₂²×C₇	—	4	1	no	—
362	C₂²×C₇	—	4	2	no	—
363	C₂²×C₇	—	4	1	no	—
364	C₂²×C₇	—	4	2	no	—
365	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
366	C₂²×C₇	#540	1	1	yes	characteristic
367	C₂²×C₇	#542	1	1	yes	characteristic
368	C₄×C₇	—	2	1	no	—
369	C₄×C₇	—	2	1	no	—
370	C₄×C₇	—	2	1	no	—
371	C₄×C₇	—	4	4	no	—
372	C₄×C₇	—	4	1	no	—
373	C₄×C₇	—	4	2	no	—
374	C₄×C₇	—	4	1	no	—
375	C₄×C₇	—	4	1	no	—
376	C₄×C₇	#443	1	2	yes	—
377	C₄×C₇	#518	1	1	yes	characteristic
378	C₄×C₇	#520	1	1	yes	characteristic
379	C₄×C₇	#535	1	1	yes	characteristic
380	C₄×C₇	#537	1	1	yes	characteristic
381	#41	—	2	1	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	—	2	1	no	—
386	#41	—	2	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	2	no	—
389	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
390	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
396	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	C₇×#41	1	1	yes	characteristic
403	C₂²×C₄	C₇×#41	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
405	Q₁₆	—	4	2	no	—
406	SD₁₆	—	4	2	no	—
407	SD₁₆	—	4	2	no	—
408	D₈	—	4	2	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	1	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	4	4	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
427	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
428	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
429	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
430	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
431	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
438	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
439	C₂²⋊C₄	—	2	1	no	—
440	C₂²⋊C₄	—	2	1	no	—
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
450	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
451	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
452	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂×C₇	—	2	1	no	—
454	C₂×C₇	—	2	1	no	—
455	C₂×C₇	—	8	1	no	—
456	C₂×C₇	—	8	2	no	—
457	C₂×C₇	#2324	1	1	yes	characteristic
458	C₂×C₇	C₂×#443	1	1	yes	characteristic
459	C₂×C₇	#2955	1	1	yes	characteristic
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	C₂×C₇×#41	1	1	yes	characteristic
463	Q₈	—	2	1	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	1	no	—
467	D₄	—	2	1	no	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	1	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	1	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	1	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
493	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
494	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
495	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
496	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
497	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
498	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
500	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
501	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
502	C₈	—	2	2	no	—
503	C₈	—	2	2	no	—
504	C₈	—	2	1	no	—
505	C₈	—	2	1	no	—
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	4	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	1	no	—
511	C₇	#22165	1	1	yes	characteristic, direct product factor, Sylow
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	1	no	—
520	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
521	C₂²	C₇×#540	1	1	yes	characteristic
522	C₂²	C₇×#542	1	1	yes	characteristic
523	C₄	—	2	1	no	—
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	4	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	2	no	—
531	C₄	C₇×#443	1	2	yes	—
532	C₄	C₇×#518	1	1	yes	characteristic
533	C₄	C₇×#520	1	1	yes	characteristic
534	C₄	C₇×#535	1	1	yes	characteristic
535	C₄	C₇×#537	1	1	yes	characteristic
536	C₂	—	2	1	no	—
537	C₂	—	2	1	no	—
538	C₂	—	8	1	no	—
539	C₂	—	8	2	no	—
540	C₂	C₇×#2324	1	1	yes	characteristic
541	C₂	C₂×C₇×#443	1	1	yes	characteristic
542	C₂	C₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2931	C₂	semidirect
3. C₇×#2922	C₂	semidirect
4. C₇×#1980	C₂	semidirect
5. C₇×#1974	C₂	semidirect
6. C₇×#1950	C₂	semidirect
7. C₇×#1937	C₂	semidirect
8. C₇×#1876	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1806	C₂	semidirect
11. C₇×C₈×C₄⋊C₄	C₂	semidirect
16. C₇×#464	C₄	semidirect
17. C₇×#463	C₄	semidirect
18. C₇×#462	C₄	semidirect
19. C₇×#461	C₄	semidirect
20. C₇×#460	C₄	semidirect
21. C₇×#459	C₄	semidirect
49. #22165	C₇	direct (N × H)
119. #2931	C₂×C₇	semidirect
120. #2922	C₂×C₇	semidirect
121. #1980	C₂×C₇	semidirect
122. #1974	C₂×C₇	semidirect
123. #1950	C₂×C₇	semidirect
124. #1937	C₂×C₇	semidirect
125. #1876	C₂×C₇	semidirect
126. #1866	C₂×C₇	semidirect
127. #1806	C₂×C₇	semidirect
128. C₈×C₄⋊C₄	C₂×C₇	semidirect
205. #464	C₄×C₇	semidirect
206. #463	C₄×C₇	semidirect
207. #462	C₄×C₇	semidirect
208. #461	C₄×C₇	semidirect
209. #460	C₄×C₇	semidirect
210. #459	C₄×C₇	semidirect
511. C₇	#22165	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×#3035	2
C₇×#2931	2
C₇×#2922	2
C₇×#1980	2
C₇×#1974	2
C₇×#1950	2
C₇×#1937	2
C₇×#1876	2
C₇×#1866	2
C₇×#1806	2
C₇×C₈×C₄⋊C₄	2
#22165	7

Subgroups — C7×#22165 #13199381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22165 #13199381