Subgroups — C₇×#22831 #13200047

Order 1792.

616 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₇×#1985	C₂	1	2	yes	maximal
3	C₇×#1917	C₂	1	2	yes	maximal
4	C₇×#1916	C₂	1	2	yes	maximal
5	C₇×#1840	C₂	1	2	yes	maximal
6	C₇×#1831	C₂	1	1	yes	maximal, characteristic
7	C₇×#1771	C₂	1	2	yes	maximal
8	C₇×#1767	C₂	1	1	yes	maximal, characteristic
9	C₇×#1746	C₂	1	2	yes	maximal
10	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×#97	C₂²	1	2	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#391	C₄	1	1	yes	characteristic
17	C₇×#391	C₄	1	1	yes	characteristic
18	C₇×#389	C₄	1	1	yes	characteristic
19	C₇×#389	C₄	1	1	yes	characteristic
20	C₇×#387	C₄	1	2	yes	—
21	C₇×#387	C₄	1	2	yes	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₇×#376	C₂²	1	2	yes	—
25	C₄³×C₇	C₂²	1	1	yes	characteristic
26	C₇×#336	C₂²	1	2	yes	—
27	C₇×#336	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	1	yes	characteristic
31	C₇×#336	C₂²	1	2	yes	—
32	C₇×#336	C₂²	1	1	yes	characteristic
33	C₇×#336	C₂²	1	2	yes	—
34	C₇×#336	C₂²	1	2	yes	—
35	C₇×#336	C₂²	1	1	yes	characteristic
36	C₇×#336	C₂²	1	1	yes	characteristic
37	#22831	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₂²×C₇×C₈	C₂³	1	2	yes	—
42	C₂²×C₇×C₈	C₂³	1	2	yes	—
43	C₄×C₇×Q₈	—	2	2	no	—
44	C₄×C₇×Q₈	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
60	C₇×C₄⋊C₈	—	2	4	no	—
61	C₇×C₄⋊C₈	—	2	4	no	—
62	C₇×#97	—	2	4	no	—
63	C₇×#97	—	2	4	no	—
64	C₄×C₇×C₈	—	2	4	no	—
65	C₇×#95	C₂×C₄	1	2	yes	—
66	C₇×#95	C₂×C₄	1	2	yes	—
67	C₇×#95	C₂×C₄	1	2	yes	—
68	C₇×#95	C₂×C₄	1	2	yes	—
69	C₇×#95	C₂×C₄	1	2	yes	—
70	C₇×#95	C₂×C₄	1	2	yes	—
71	C₇×#95	C₂×C₄	1	2	yes	—
72	#2295	C₂×C₇	1	1	yes	characteristic
73	#1985	C₂×C₇	1	2	yes	—
74	#1917	C₂×C₇	1	2	yes	—
75	#1916	C₂×C₇	1	2	yes	—
76	#1840	C₂×C₇	1	2	yes	—
77	#1831	C₂×C₇	1	1	yes	characteristic
78	#1771	C₂×C₇	1	2	yes	—
79	#1767	C₂×C₇	1	1	yes	characteristic
80	#1746	C₂×C₇	1	2	yes	—
81	C₂×C₇×Q₈	—	2	2	no	—
82	C₂×C₇×Q₈	—	4	1	no	—
83	C₂×C₇×Q₈	—	4	1	no	—
84	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	#41	1	4	yes	—
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	2	4	no	—
109	C₇×C₄⋊C₄	—	2	4	no	—
110	C₇×C₄⋊C₄	—	4	1	no	—
111	C₇×C₄⋊C₄	—	4	1	no	—
112	C₇×C₄⋊C₄	—	4	2	no	—
113	C₄²×C₇	—	2	2	no	—
114	C₄²×C₇	—	2	2	no	—
115	C₄²×C₇	—	2	2	no	—
116	C₄²×C₇	—	2	2	no	—
117	C₄²×C₇	—	2	2	no	—
118	C₄²×C₇	—	2	2	no	—
119	C₄²×C₇	—	2	2	no	—
120	C₄²×C₇	—	4	1	no	—
121	C₄²×C₇	—	4	1	no	—
122	C₄²×C₇	—	4	1	no	—
123	C₄²×C₇	#41	1	2	yes	—
124	C₄²×C₇	#41	1	2	yes	—
125	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
128	C₂×#97	C₂²×C₇	1	2	yes	—
129	C₂×#97	C₂²×C₇	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
131	#391	C₄×C₇	1	1	yes	characteristic
132	#391	C₄×C₇	1	1	yes	characteristic
133	#389	C₄×C₇	1	1	yes	characteristic
134	#389	C₄×C₇	1	1	yes	characteristic
135	#387	C₄×C₇	1	2	yes	—
136	#387	C₄×C₇	1	2	yes	—
137	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
138	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
139	#376	C₂²×C₇	1	2	yes	—
140	C₄³	C₂²×C₇	1	1	yes	characteristic
141	#336	C₂²×C₇	1	2	yes	—
142	#336	C₂²×C₇	1	1	yes	characteristic
143	#336	C₂²×C₇	1	1	yes	characteristic
144	#336	C₂²×C₇	1	1	yes	characteristic
145	#336	C₂²×C₇	1	2	yes	—
146	#336	C₂²×C₇	1	2	yes	—
147	#336	C₂²×C₇	1	2	yes	—
148	#336	C₂²×C₇	1	2	yes	—
149	#336	C₂²×C₇	1	2	yes	—
150	#336	C₂²×C₇	1	2	yes	—
151	#336	C₂²×C₇	1	1	yes	characteristic
152	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
153	C₇×Q₈	—	4	4	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	1	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	2	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	2	1	no	—
167	C₂×C₄×C₇	—	2	1	no	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	1	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	1	no	—
174	C₂×C₄×C₇	—	2	1	no	—
175	C₂×C₄×C₇	—	2	2	no	—
176	C₂×C₄×C₇	—	2	2	no	—
177	C₂×C₄×C₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	#117	1	2	yes	—
182	C₂×C₄×C₇	#117	1	2	yes	—
183	C₂×C₄×C₇	#131	1	2	yes	—
184	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₇	C₂×#41	1	2	yes	—
186	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
188	C₇×C₈	—	2	4	no	—
189	C₇×C₈	—	4	2	no	—
190	C₇×C₈	—	4	2	no	—
191	C₇×C₈	—	4	2	no	—
192	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₇	1	2	yes	—
196	C₂²×C₈	C₂³×C₇	1	2	yes	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₇	1	2	yes	—
210	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	#97	—	2	4	no	—
217	#97	—	2	4	no	—
218	C₄×C₈	—	2	4	no	—
219	#95	C₂×C₄×C₇	1	2	yes	—
220	#95	C₂×C₄×C₇	1	2	yes	—
221	#95	C₂×C₄×C₇	1	2	yes	—
222	#95	C₂×C₄×C₇	1	2	yes	—
223	#95	C₂×C₄×C₇	1	2	yes	—
224	#95	C₂×C₄×C₇	1	2	yes	—
225	#95	C₂×C₄×C₇	1	2	yes	—
226	C₂²×C₇	C₂×#117	1	1	yes	characteristic
227	C₂²×C₇	C₄×#41	1	1	yes	characteristic
228	C₂²×C₇	#519	1	1	yes	characteristic
229	C₂²×C₇	C₂×#131	1	1	yes	characteristic
230	C₂²×C₇	#568	1	1	yes	characteristic
231	C₂²×C₇	#568	1	1	yes	characteristic
232	C₂²×C₇	#568	1	1	yes	characteristic
233	C₄×C₇	—	2	2	no	—
234	C₄×C₇	—	2	2	no	—
235	C₄×C₇	—	2	2	no	—
236	C₄×C₇	—	4	2	no	—
237	C₄×C₇	—	4	2	no	—
238	C₄×C₇	—	4	1	no	—
239	C₄×C₇	—	4	2	no	—
240	C₄×C₇	—	4	2	no	—
241	C₄×C₇	—	4	1	no	—
242	C₄×C₇	—	4	1	no	—
243	C₄×C₇	#552	1	2	yes	—
244	C₄×C₇	#556	1	2	yes	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₇×#41	1	4	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	C₇×#41	1	2	yes	—
288	C₄²	C₇×#41	1	2	yes	—
289	C₂×C₇	#2356	1	1	yes	characteristic
290	C₂×C₇	#2921	1	1	yes	characteristic
291	C₂×C₇	#2925	1	1	yes	characteristic
292	C₂×C₇	#2967	1	1	yes	characteristic
293	C₂×C₇	#2968	1	1	yes	characteristic
294	C₂×C₇	#2985	1	1	yes	characteristic
295	C₂×C₇	#2987	1	1	yes	characteristic
296	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
297	Q₈	—	4	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	C₇×#117	1	2	yes	—
326	C₂×C₄	C₇×#117	1	2	yes	—
327	C₂×C₄	C₇×#131	1	2	yes	—
328	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
329	C₂×C₄	C₂×C₇×#41	1	2	yes	—
330	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
331	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
332	C₈	—	2	4	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₇	#22831	1	1	yes	characteristic, direct product factor, Sylow
337	C₂²	C₂×C₇×#117	1	1	yes	characteristic
338	C₂²	C₄×C₇×#41	1	1	yes	characteristic
339	C₂²	C₇×#519	1	1	yes	characteristic
340	C₂²	C₂×C₇×#131	1	1	yes	characteristic
341	C₂²	C₇×#568	1	1	yes	characteristic
342	C₂²	C₇×#568	1	1	yes	characteristic
343	C₂²	C₇×#568	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	C₇×#552	1	2	yes	—
355	C₄	C₇×#556	1	2	yes	—
356	C₂	C₇×#2356	1	1	yes	characteristic
357	C₂	C₇×#2921	1	1	yes	characteristic
358	C₂	C₇×#2925	1	1	yes	characteristic
359	C₂	C₇×#2967	1	1	yes	characteristic
360	C₂	C₇×#2968	1	1	yes	characteristic
361	C₂	C₇×#2985	1	1	yes	characteristic
362	C₂	C₇×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
37. #22831	C₇	direct (N × H)
336. C₇	#22831	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×#2295	2
C₇×#1985	2
C₇×#1917	2
C₇×#1916	2
C₇×#1840	2
C₇×#1831	2
C₇×#1771	2
C₇×#1767	2
C₇×#1746	2
#22831	7

Subgroups — C7×#22831 #13200047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22831 #13200047