Subgroups — C₇×#17691 #13194907

Order 1792.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3510	C₂	1	1	yes	maximal, characteristic
2	C₇×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2970	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1535	C₂	1	2	yes	maximal, complemented
6	C₇×#1532	C₂	1	2	yes	maximal, complemented
7	C₇×#1529	C₂	1	2	yes	maximal, complemented
8	C₇×#1527	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1527	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1524	C₂	1	2	yes	maximal, complemented
12	C₇×#559	—	2	1	no	—
13	C₇×#559	—	2	1	no	—
14	C₇×#556	—	2	1	no	—
15	C₇×#556	—	2	1	no	—
16	C₇×#551	—	2	2	no	—
17	C₇×#551	—	2	2	no	—
18	C₇×#542	C₄	1	2	yes	—
19	C₇×#542	C₄	1	2	yes	—
20	C₇×#536	C₄	1	1	yes	characteristic
21	C₇×#535	C₄	1	1	yes	characteristic
22	C₇×#533	C₂²	1	1	yes	characteristic
23	C₇×#529	C₂²	1	2	yes	—
24	C₇×#529	C₂²	1	2	yes	—
25	C₂×C₇×#124	C₂²	1	1	yes	characteristic
26	C₇×#519	C₄	1	1	yes	characteristic
27	C₇×#518	C₄	1	1	yes	characteristic
28	C₄×C₇×#41	C₂²	1	1	yes	characteristic
29	C₇×#446	—	2	1	no	—
30	C₇×C₈×Q₈	—	2	1	no	—
31	C₇×#436	—	2	1	no	—
32	C₇×#436	—	2	2	no	—
33	C₇×#435	—	2	2	no	—
34	C₇×C₈×D₄	—	2	1	no	—
35	C₇×#432	C₂²	1	1	yes	characteristic
36	C₇×#432	C₂²	1	1	yes	characteristic
37	C₇×#432	C₂²	1	2	yes	complemented
38	C₇×#432	C₂²	1	2	yes	complemented
39	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
41	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
42	C₂×C₇×#97	C₂²	1	2	yes	complemented
43	C₇×#330	C₂²	1	1	yes	characteristic, complemented
44	C₇×#330	C₂²	1	2	yes	complemented
45	C₇×#330	C₂²	1	1	yes	characteristic, complemented
46	C₇×#330	C₂²	1	1	yes	characteristic, complemented
47	C₇×#330	C₂²	1	1	yes	characteristic, complemented
48	C₇×#330	C₂²	1	2	yes	complemented
49	C₇×#329	C₂²	1	2	yes	—
50	C₇×#329	C₂²	1	2	yes	—
51	C₇×#329	C₂²	1	2	yes	—
52	C₇×#329	C₂²	1	2	yes	—
53	#17691	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×M₁₆	—	2	1	no	—
58	C₂×C₇×M₁₆	—	2	1	no	—
59	C₂×C₇×M₁₆	—	2	2	no	—
60	C₂²×C₇×C₈	—	2	2	no	—
61	C₇×C₄⋊Q₈	—	2	1	no	—
62	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
64	C₇×#127	—	2	1	no	—
65	C₇×#126	—	2	2	no	—
66	C₇×#126	—	2	2	no	—
67	C₇×#126	—	2	2	no	—
68	C₇×#126	—	2	2	no	—
69	C₇×#125	C₂³	1	1	yes	characteristic
70	C₇×#125	C₂³	1	1	yes	characteristic
71	C₇×#124	—	2	1	no	—
72	C₇×#124	—	2	1	no	—
73	C₇×#124	—	2	2	no	—
74	C₇×#124	—	2	2	no	—
75	C₇×#124	—	2	1	no	—
76	C₇×#124	—	2	1	no	—
77	C₇×#124	C₂×C₄	1	2	yes	—
78	C₇×#124	C₂³	1	2	yes	—
79	C₇×#123	—	2	2	no	—
80	C₇×#123	—	2	2	no	—
81	C₇×#123	—	2	2	no	—
82	C₇×#123	—	2	2	no	—
83	C₇×#123	—	2	2	no	—
84	C₇×C₂²⋊Q₈	—	2	1	no	—
85	C₇×C₂²⋊Q₈	—	2	1	no	—
86	C₇×C₂²⋊Q₈	—	2	2	no	—
87	C₇×C₂²⋊Q₈	—	2	2	no	—
88	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
89	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
90	C₇×#121	C₂×C₄	1	2	yes	—
91	C₇×#121	C₂×C₄	1	2	yes	—
92	C₇×#120	—	2	2	no	—
93	C₄×C₇×Q₈	—	2	1	no	—
94	C₄×C₇×Q₈	—	2	1	no	—
95	C₄×C₇×Q₈	—	2	1	no	—
96	C₄×C₇×Q₈	—	2	1	no	—
97	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
98	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
99	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
100	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
101	C₄×C₇×D₄	—	2	2	no	—
102	C₄×C₇×D₄	—	2	2	no	—
103	C₄×C₇×D₄	—	2	1	no	—
104	C₄×C₇×D₄	—	2	1	no	—
105	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
106	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
107	C₄×C₇×D₄	D₄	1	2	yes	—
108	C₄×C₇×D₄	D₄	1	2	yes	—
109	C₄×C₇×D₄	D₄	1	1	yes	characteristic
110	C₄×C₇×D₄	D₄	1	1	yes	characteristic
111	C₇×#117	—	2	2	no	—
112	C₇×#117	—	2	2	no	—
113	C₇×#117	—	2	2	no	—
114	C₇×#117	C₂×C₄	1	1	yes	characteristic
115	C₇×#117	C₂×C₄	1	1	yes	characteristic
116	C₇×#117	C₂×C₄	1	1	yes	characteristic
117	C₇×#117	C₂×C₄	1	2	yes	—
118	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₇	C₂³	1	2	yes	—
123	C₇×C₄⋊C₈	—	2	1	no	—
124	C₇×C₄⋊C₈	—	2	1	no	—
125	C₇×C₄⋊C₈	—	2	2	no	—
126	C₇×C₄⋊C₈	—	2	2	no	—
127	C₇×C₂²⋊C₈	—	2	1	no	—
128	C₇×C₂²⋊C₈	—	2	1	no	—
129	C₇×C₂²⋊C₈	—	2	2	no	—
130	C₇×C₂²⋊C₈	—	2	2	no	—
131	C₇×#97	—	2	2	no	—
132	C₇×#97	C₂³	1	1	yes	characteristic
133	C₇×#97	C₂³	1	2	yes	—
134	C₇×#97	C₂³	1	1	yes	characteristic
135	C₇×#97	C₂³	1	1	yes	characteristic
136	C₇×#97	C₂³	1	1	yes	characteristic
137	C₇×#97	C₂³	1	2	yes	—
138	C₄×C₇×C₈	—	2	1	no	—
139	C₄×C₇×C₈	—	2	1	no	—
140	C₄×C₇×C₈	—	2	2	no	—
141	#3510	C₂×C₇	1	1	yes	characteristic
142	#2971	C₂×C₇	1	1	yes	characteristic, complemented
143	#2970	C₂×C₇	1	1	yes	characteristic, complemented
144	#1537	C₂×C₇	1	1	yes	characteristic, complemented
145	#1535	C₂×C₇	1	2	yes	complemented
146	#1532	C₂×C₇	1	2	yes	complemented
147	#1529	C₂×C₇	1	2	yes	complemented
148	#1527	C₂×C₇	1	1	yes	characteristic, complemented
149	#1527	C₂×C₇	1	1	yes	characteristic, complemented
150	#1526	C₂×C₇	1	1	yes	characteristic, complemented
151	#1524	C₂×C₇	1	2	yes	complemented
152	C₂⁴×C₇	—	2	1	no	—
153	C₇×#41	—	4	2	no	—
154	C₂×C₇×Q₈	—	2	1	no	—
155	C₂×C₇×Q₈	—	2	1	no	—
156	C₂×C₇×Q₈	—	2	1	no	—
157	C₂×C₇×Q₈	—	4	2	no	—
158	C₂×C₇×Q₈	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	2	no	—
162	C₂×C₇×D₄	—	2	1	no	—
163	C₂×C₇×D₄	—	4	2	no	—
164	C₂×C₇×D₄	C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₇×D₄	C₂²⋊C₄	1	2	yes	—
166	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₇	—	2	1	no	—
168	C₂²×C₄×C₇	—	2	2	no	—
169	C₂²×C₄×C₇	—	2	2	no	—
170	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
172	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
176	C₇×M₁₆	—	4	2	no	—
177	C₇×M₁₆	—	4	2	no	—
178	C₇×M₁₆	—	4	4	no	—
179	C₂×C₇×C₈	—	2	1	no	—
180	C₂×C₇×C₈	—	2	1	no	—
181	C₂×C₇×C₈	—	2	1	no	—
182	C₂×C₇×C₈	—	2	1	no	—
183	C₂×C₇×C₈	—	2	2	no	—
184	C₂×C₇×C₈	—	2	2	no	—
185	C₂×C₇×C₈	—	4	4	no	—
186	C₇×C₄⋊C₄	—	2	2	no	—
187	C₇×C₄⋊C₄	—	2	2	no	—
188	C₇×C₄⋊C₄	—	2	2	no	—
189	C₇×C₄⋊C₄	—	2	2	no	—
190	C₇×C₄⋊C₄	—	2	1	no	—
191	C₇×C₄⋊C₄	—	2	1	no	—
192	C₇×C₄⋊C₄	—	2	1	no	—
193	C₇×C₄⋊C₄	—	2	1	no	—
194	C₇×C₄⋊C₄	—	2	1	no	—
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
197	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
198	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
199	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
200	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
201	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
202	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
203	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
204	C₇×C₂²⋊C₄	—	2	1	no	—
205	C₇×C₂²⋊C₄	—	2	1	no	—
206	C₇×C₂²⋊C₄	—	2	2	no	—
207	C₇×C₂²⋊C₄	—	2	2	no	—
208	C₇×C₂²⋊C₄	—	2	2	no	—
209	C₇×C₂²⋊C₄	—	2	2	no	—
210	C₇×C₂²⋊C₄	—	2	2	no	—
211	C₇×C₂²⋊C₄	—	2	2	no	—
212	C₇×C₂²⋊C₄	—	2	2	no	—
213	C₇×C₂²⋊C₄	—	2	2	no	—
214	C₇×C₂²⋊C₄	—	2	2	no	—
215	C₇×C₂²⋊C₄	—	2	2	no	—
216	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
217	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
218	C₇×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
219	C₄²×C₇	—	2	2	no	—
220	C₄²×C₇	—	2	2	no	—
221	C₄²×C₇	—	2	1	no	—
222	C₄²×C₇	—	2	1	no	—
223	C₄²×C₇	C₂²×C₄	1	2	yes	—
224	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₇	C₂×D₄	1	2	yes	—
226	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₇	C₂×D₄	1	2	yes	—
229	C₄²×C₇	C₂⁴	1	1	yes	characteristic
230	#559	—	2	1	no	—
231	#559	—	2	1	no	—
232	#556	—	2	1	no	—
233	#556	—	2	1	no	—
234	#551	—	2	2	no	—
235	#551	—	2	2	no	—
236	#542	C₄×C₇	1	2	yes	—
237	#542	C₄×C₇	1	2	yes	—
238	#536	C₄×C₇	1	1	yes	characteristic
239	#535	C₄×C₇	1	1	yes	characteristic
240	#533	C₂²×C₇	1	1	yes	characteristic
241	#529	C₂²×C₇	1	2	yes	—
242	#529	C₂²×C₇	1	2	yes	—
243	C₂×#124	C₂²×C₇	1	1	yes	characteristic
244	#519	C₄×C₇	1	1	yes	characteristic
245	#518	C₄×C₇	1	1	yes	characteristic
246	C₄×#41	C₂²×C₇	1	1	yes	characteristic
247	#446	—	2	1	no	—
248	C₈×Q₈	—	2	1	no	—
249	#436	—	2	1	no	—
250	#436	—	2	2	no	—
251	#435	—	2	2	no	—
252	C₈×D₄	—	2	1	no	—
253	#432	C₂²×C₇	1	1	yes	characteristic
254	#432	C₂²×C₇	1	1	yes	characteristic
255	#432	C₂²×C₇	1	2	yes	complemented
256	#432	C₂²×C₇	1	2	yes	complemented
257	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
258	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
259	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
260	C₂×#97	C₂²×C₇	1	2	yes	complemented
261	#330	C₂²×C₇	1	2	yes	complemented
262	#330	C₂²×C₇	1	2	yes	complemented
263	#330	C₂²×C₇	1	1	yes	characteristic, complemented
264	#330	C₂²×C₇	1	1	yes	characteristic, complemented
265	#330	C₂²×C₇	1	1	yes	characteristic, complemented
266	#330	C₂²×C₇	1	1	yes	characteristic, complemented
267	#329	C₂²×C₇	1	2	yes	—
268	#329	C₂²×C₇	1	2	yes	—
269	#329	C₂²×C₇	1	2	yes	—
270	#329	C₂²×C₇	1	2	yes	—
271	C₂³×C₇	—	2	2	no	—
272	C₂³×C₇	—	4	2	no	—
273	C₂³×C₇	—	8	1	no	—
274	C₂³×C₇	—	8	1	no	—
275	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
277	C₇×Q₈	—	2	2	no	—
278	C₇×Q₈	—	4	2	no	—
279	C₇×Q₈	—	4	2	no	—
280	C₇×Q₈	—	8	1	no	—
281	C₇×Q₈	—	8	1	no	—
282	C₇×D₄	—	2	2	no	—
283	C₇×D₄	—	4	2	no	—
284	C₇×D₄	—	4	2	no	—
285	C₇×D₄	—	4	2	no	—
286	C₇×D₄	—	8	2	no	—
287	C₇×D₄	—	8	2	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	1	no	—
290	C₂×C₄×C₇	—	2	1	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	1	no	—
293	C₂×C₄×C₇	—	2	1	no	—
294	C₂×C₄×C₇	—	4	2	no	—
295	C₂×C₄×C₇	—	4	2	no	—
296	C₂×C₄×C₇	—	4	1	no	—
297	C₂×C₄×C₇	—	4	1	no	—
298	C₂×C₄×C₇	—	4	2	no	—
299	C₂×C₄×C₇	—	4	2	no	—
300	C₂×C₄×C₇	—	8	2	no	—
301	C₂×C₄×C₇	—	8	2	no	—
302	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
303	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
304	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
307	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
309	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
311	C₇×C₈	—	2	2	no	—
312	C₇×C₈	—	2	2	no	—
313	C₇×C₈	—	4	2	no	—
314	C₇×C₈	—	4	1	no	—
315	C₇×C₈	—	4	1	no	—
316	C₇×C₈	—	4	2	no	—
317	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
318	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
319	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
320	C₂×M₁₆	—	2	1	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂²×C₈	—	2	2	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
326	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
327	#127	—	2	1	no	—
328	#126	—	2	2	no	—
329	#126	—	2	2	no	—
330	#126	—	2	2	no	—
331	#126	—	2	2	no	—
332	#125	C₂³×C₇	1	1	yes	characteristic
333	#125	C₂³×C₇	1	1	yes	characteristic
334	#124	—	2	1	no	—
335	#124	—	2	1	no	—
336	#124	—	2	2	no	—
337	#124	—	2	2	no	—
338	#124	—	2	1	no	—
339	#124	—	2	1	no	—
340	#124	C₂×C₄×C₇	1	2	yes	—
341	#124	C₂³×C₇	1	2	yes	—
342	#123	—	2	2	no	—
343	#123	—	2	2	no	—
344	#123	—	2	2	no	—
345	#123	—	2	2	no	—
346	#123	—	2	2	no	—
347	C₂²⋊Q₈	—	2	1	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	2	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
352	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
353	#121	C₂×C₄×C₇	1	2	yes	—
354	#121	C₂×C₄×C₇	1	2	yes	—
355	#120	—	2	2	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
361	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
362	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
363	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
369	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
370	C₄×D₄	C₇×D₄	1	2	yes	—
371	C₄×D₄	C₇×D₄	1	2	yes	—
372	C₄×D₄	C₇×D₄	1	1	yes	characteristic
373	C₄×D₄	C₇×D₄	1	1	yes	characteristic
374	#117	—	2	2	no	—
375	#117	—	2	2	no	—
376	#117	—	2	2	no	—
377	#117	C₂×C₄×C₇	1	1	yes	characteristic
378	#117	C₂×C₄×C₇	1	1	yes	characteristic
379	#117	C₂×C₄×C₇	1	1	yes	characteristic
380	#117	C₂×C₄×C₇	1	2	yes	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₇	1	2	yes	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	2	no	—
389	C₄⋊C₈	—	2	2	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	C₂²⋊C₈	—	2	1	no	—
392	C₂²⋊C₈	—	2	2	no	—
393	C₂²⋊C₈	—	2	2	no	—
394	#97	—	2	2	no	—
395	#97	C₂³×C₇	1	1	yes	characteristic
396	#97	C₂³×C₇	1	2	yes	—
397	#97	C₂³×C₇	1	1	yes	characteristic
398	#97	C₂³×C₇	1	1	yes	characteristic
399	#97	C₂³×C₇	1	1	yes	characteristic
400	#97	C₂³×C₇	1	2	yes	—
401	C₄×C₈	—	2	1	no	—
402	C₄×C₈	—	2	1	no	—
403	C₄×C₈	—	2	2	no	—
404	C₂²×C₇	—	2	1	no	—
405	C₂²×C₇	—	2	2	no	—
406	C₂²×C₇	—	2	2	no	—
407	C₂²×C₇	—	2	2	no	—
408	C₂²×C₇	—	2	1	no	—
409	C₂²×C₇	—	2	1	no	—
410	C₂²×C₇	—	4	2	no	—
411	C₂²×C₇	—	4	2	no	—
412	C₂²×C₇	—	4	2	no	—
413	C₂²×C₇	—	16	2	no	—
414	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
415	C₄×C₇	—	2	1	no	—
416	C₄×C₇	—	2	2	no	—
417	C₄×C₇	—	2	1	no	—
418	C₄×C₇	—	2	1	no	—
419	C₄×C₇	—	2	2	no	—
420	C₄×C₇	—	2	2	no	—
421	C₄×C₇	—	2	1	no	—
422	C₄×C₇	—	2	1	no	—
423	C₄×C₇	—	4	2	no	—
424	C₄×C₇	—	4	1	no	—
425	C₄×C₇	—	4	1	no	—
426	C₄×C₇	—	4	1	no	—
427	C₄×C₇	—	4	1	no	—
428	C₄×C₇	#413	1	2	yes	—
429	C₂⁴	—	2	1	no	—
430	#41	—	4	2	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	4	2	no	—
435	C₂×Q₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
436	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
442	C₂×D₄	C₇×C₂²⋊C₄	1	2	yes	—
443	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
449	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
450	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
452	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
453	M₁₆	—	4	2	no	—
454	M₁₆	—	4	2	no	—
455	M₁₆	—	4	4	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	4	4	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
474	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
475	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
476	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
477	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
478	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
479	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
480	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	1	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
494	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
495	C₂²⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	C₂²×C₄×C₇	1	2	yes	—
501	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
502	C₄²	C₂×C₇×D₄	1	2	yes	—
503	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
504	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
505	C₄²	C₂×C₇×D₄	1	2	yes	—
506	C₄²	C₂⁴×C₇	1	1	yes	characteristic
507	C₂×C₇	—	4	2	no	—
508	C₂×C₇	—	4	1	no	—
509	C₂×C₇	—	8	2	no	—
510	C₂×C₇	C₂×#413	1	1	yes	characteristic
511	C₂×C₇	#2901	1	1	yes	characteristic
512	C₂×C₇	#2901	1	1	yes	characteristic
513	C₂³	—	2	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	8	1	no	—
516	C₂³	—	8	1	no	—
517	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
518	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
519	Q₈	—	2	2	no	—
520	Q₈	—	4	2	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	8	1	no	—
523	Q₈	—	8	1	no	—
524	D₄	—	2	2	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	8	2	no	—
529	D₄	—	8	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	—	8	2	no	—
544	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
545	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
546	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
547	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
548	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
549	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
550	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
552	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
553	C₈	—	2	2	no	—
554	C₈	—	2	2	no	—
555	C₈	—	4	2	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	2	no	—
559	C₇	#17691	1	1	yes	characteristic, direct product factor, Sylow
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	16	2	no	—
570	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	2	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	2	no	—
584	C₄	C₇×#413	1	2	yes	—
585	C₂	—	4	1	no	—
586	C₂	—	4	2	no	—
587	C₂	—	8	2	no	—
588	C₂	C₂×C₇×#413	1	1	yes	characteristic
589	C₂	C₇×#2901	1	1	yes	characteristic
590	C₂	C₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2971	C₂	semidirect
3. C₇×#2970	C₂	semidirect
4. C₇×#1537	C₂	semidirect
5. C₇×#1535	C₂	semidirect
6. C₇×#1532	C₂	semidirect
7. C₇×#1529	C₂	semidirect
8. C₇×#1527	C₂	semidirect
9. C₇×#1527	C₂	semidirect
10. C₇×#1526	C₂	semidirect
11. C₇×#1524	C₂	semidirect
37. C₇×#432	C₂²	semidirect
38. C₇×#432	C₂²	semidirect
41. C₄×C₇×M₁₆	C₂²	semidirect
42. C₂×C₇×#97	C₂²	semidirect
43. C₇×#330	C₂²	semidirect
44. C₇×#330	C₂²	semidirect
45. C₇×#330	C₂²	semidirect
46. C₇×#330	C₂²	semidirect
47. C₇×#330	C₂²	semidirect
48. C₇×#330	C₂²	semidirect
53. #17691	C₇	direct (N × H)
142. #2971	C₂×C₇	semidirect
143. #2970	C₂×C₇	semidirect
144. #1537	C₂×C₇	semidirect
145. #1535	C₂×C₇	semidirect
146. #1532	C₂×C₇	semidirect
147. #1529	C₂×C₇	semidirect
148. #1527	C₂×C₇	semidirect
149. #1527	C₂×C₇	semidirect
150. #1526	C₂×C₇	semidirect
151. #1524	C₂×C₇	semidirect
255. #432	C₂²×C₇	semidirect
256. #432	C₂²×C₇	semidirect
259. C₄×M₁₆	C₂²×C₇	semidirect
260. C₂×#97	C₂²×C₇	semidirect
261. #330	C₂²×C₇	semidirect
262. #330	C₂²×C₇	semidirect
263. #330	C₂²×C₇	semidirect
264. #330	C₂²×C₇	semidirect
265. #330	C₂²×C₇	semidirect
266. #330	C₂²×C₇	semidirect
559. C₇	#17691	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×#3510	2
C₇×#2971	2
C₇×#2970	2
C₇×#1537	2
C₇×#1535	2
C₇×#1532	2
C₇×#1529	2
C₇×#1527	2
C₇×#1526	2
C₇×#1524	2
#17691	7

Subgroups — C7×#17691 #13194907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17691 #13194907