Subgroups — C₇×#23843 #13201059

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2330	C₂	1	2	yes	maximal
2	C₇×#2084	C₂	1	2	yes	maximal
3	C₇×#2083	C₂	1	2	yes	maximal
4	C₇×#2059	C₂	1	2	yes	maximal
5	C₇×#2059	C₂	1	2	yes	maximal
6	C₇×#1776	C₂	1	2	yes	maximal
7	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic
8	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
9	C₇×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₂²	1	2	yes	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#106	C₂²	1	2	yes	—
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#389	—	2	2	no	—
18	C₇×#389	—	2	2	no	—
19	C₇×#389	C₂²	1	2	yes	—
20	C₇×#389	C₂²	1	2	yes	—
21	C₇×#387	—	2	2	no	—
22	C₇×#387	—	2	2	no	—
23	C₇×#384	C₂²	1	2	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#376	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	1	yes	characteristic
29	C₇×#340	C₂²	1	1	yes	characteristic
30	C₇×#340	C₂²	1	1	yes	characteristic
31	C₇×#340	C₂²	1	2	yes	—
32	C₇×#340	C₂²	1	2	yes	—
33	C₇×#336	C₂²	1	2	yes	—
34	C₇×#336	C₂²	1	2	yes	—
35	#23843	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	2	yes	—
39	C₇×#125	—	2	4	no	—
40	C₇×#125	—	2	4	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	—	2	2	no	—
52	C₂×C₄²×C₇	—	2	2	no	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#107	—	2	4	no	—
56	C₇×#107	—	2	4	no	—
57	C₇×#106	—	2	4	no	—
58	C₇×#106	—	2	4	no	—
59	C₇×#97	Q₈	1	4	yes	—
60	C₄×C₇×C₈	Q₈	1	4	yes	—
61	C₇×#95	—	2	2	no	—
62	C₇×#95	—	2	2	no	—
63	C₇×#95	—	2	2	no	—
64	C₇×#95	—	2	2	no	—
65	C₇×#95	D₄	1	1	yes	characteristic
66	C₇×#95	D₄	1	2	yes	—
67	C₇×#95	D₄	1	1	yes	characteristic
68	#2330	C₂×C₇	1	2	yes	—
69	#2084	C₂×C₇	1	2	yes	—
70	#2083	C₂×C₇	1	2	yes	—
71	#2059	C₂×C₇	1	2	yes	—
72	#2059	C₂×C₇	1	2	yes	—
73	#1776	C₂×C₇	1	2	yes	—
74	C₄×#107	C₂×C₇	1	1	yes	characteristic
75	C₄×#106	C₂×C₇	1	1	yes	characteristic
76	#1746	C₂×C₇	1	1	yes	characteristic
77	C₂²×C₄×C₇	—	2	2	no	—
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	2	no	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
91	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
93	C₇×C₄⋊C₄	—	2	8	no	—
94	C₇×C₄⋊C₄	—	2	8	no	—
95	C₇×C₄⋊C₄	—	2	2	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	8	2	no	—
102	C₇×C₄⋊C₄	—	8	2	no	—
103	C₇×C₄⋊C₄	—	8	2	no	—
104	C₇×C₄⋊C₄	—	8	2	no	—
105	C₇×C₄⋊C₄	#41	1	4	yes	—
106	C₇×C₄⋊C₄	#41	1	4	yes	—
107	C₄²×C₇	—	2	2	no	—
108	C₄²×C₇	—	2	2	no	—
109	C₄²×C₇	—	8	2	no	—
110	C₄²×C₇	—	8	2	no	—
111	C₄²×C₇	C₂×Q₈	1	2	yes	—
112	C₄²×C₇	C₂×Q₈	1	2	yes	—
113	C₂×#125	C₂²×C₇	1	2	yes	—
114	C₂×#107	C₂²×C₇	1	2	yes	—
115	C₂×#107	C₂²×C₇	1	2	yes	—
116	C₂×#106	C₂²×C₇	1	2	yes	—
117	C₂×#106	C₂²×C₇	1	2	yes	—
118	C₂×#97	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	#389	—	2	2	no	—
121	#389	—	2	2	no	—
122	#389	C₂²×C₇	1	2	yes	—
123	#389	C₂²×C₇	1	2	yes	—
124	#387	—	2	2	no	—
125	#387	—	2	2	no	—
126	#384	C₂²×C₇	1	2	yes	—
127	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
128	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
129	#376	C₂²×C₇	1	1	yes	characteristic
130	#340	C₂²×C₇	1	1	yes	characteristic
131	#340	C₂²×C₇	1	1	yes	characteristic
132	#340	C₂²×C₇	1	2	yes	—
133	#340	C₂²×C₇	1	1	yes	characteristic
134	#340	C₂²×C₇	1	2	yes	—
135	#340	C₂²×C₇	1	1	yes	characteristic
136	#336	C₂²×C₇	1	2	yes	—
137	#336	C₂²×C₇	1	2	yes	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	1	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	—	2	4	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	2	no	—
153	C₂×C₄×C₇	—	4	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
161	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
162	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×#41	1	2	yes	—
165	C₂×C₄×C₇	C₂×#41	1	2	yes	—
166	C₇×C₈	—	2	2	no	—
167	C₇×C₈	—	2	4	no	—
168	C₇×C₈	—	2	2	no	—
169	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₇	1	2	yes	—
172	#125	—	2	4	no	—
173	#125	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₇	1	2	yes	—
187	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	—	2	4	no	—
190	#106	—	2	4	no	—
191	#106	—	2	4	no	—
192	#97	C₇×Q₈	1	4	yes	—
193	C₄×C₈	C₇×Q₈	1	4	yes	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	C₇×D₄	1	1	yes	characteristic
199	#95	C₇×D₄	1	2	yes	—
200	#95	C₇×D₄	1	1	yes	characteristic
201	C₂²×C₇	#525	1	2	yes	—
202	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
203	C₂²×C₇	#576	1	1	yes	characteristic
204	C₂²×C₇	#577	1	2	yes	—
205	C₂²×C₇	#578	1	1	yes	characteristic
206	C₄×C₇	—	2	2	no	—
207	C₄×C₇	—	2	2	no	—
208	C₄×C₇	—	4	2	no	—
209	C₄×C₇	—	4	4	no	—
210	C₄×C₇	—	4	2	no	—
211	C₄×C₇	—	4	4	no	—
212	C₄×C₇	—	8	2	no	—
213	C₄×C₇	—	8	2	no	—
214	C₄×C₇	#533	1	1	yes	characteristic
215	C₄×C₇	#533	1	2	yes	—
216	C₄×C₇	#533	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
223	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
231	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
232	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	—	8	2	no	—
245	C₄⋊C₄	C₇×#41	1	4	yes	—
246	C₄⋊C₄	C₇×#41	1	4	yes	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	8	2	no	—
250	C₄²	—	8	2	no	—
251	C₄²	C₂×C₇×Q₈	1	2	yes	—
252	C₄²	C₂×C₇×Q₈	1	2	yes	—
253	C₂×C₇	#2399	1	1	yes	characteristic
254	C₂×C₇	#3119	1	2	yes	—
255	C₂×C₇	#3120	1	2	yes	—
256	C₂×C₇	#3163	1	1	yes	characteristic
257	C₂×C₇	#3164	1	1	yes	characteristic
258	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
281	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
282	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₇×#41	1	2	yes	—
285	C₂×C₄	C₂×C₇×#41	1	2	yes	—
286	C₈	—	2	2	no	—
287	C₈	—	2	4	no	—
288	C₈	—	2	2	no	—
289	C₇	#23843	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	C₇×#525	1	2	yes	—
291	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
292	C₂²	C₇×#576	1	1	yes	characteristic
293	C₂²	C₇×#577	1	2	yes	—
294	C₂²	C₇×#578	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	C₇×#533	1	1	yes	characteristic
304	C₄	C₇×#533	1	2	yes	—
305	C₄	C₇×#533	1	1	yes	characteristic
306	C₂	C₇×#2399	1	1	yes	characteristic
307	C₂	C₇×#3119	1	2	yes	—
308	C₂	C₇×#3120	1	2	yes	—
309	C₂	C₇×#3163	1	1	yes	characteristic
310	C₂	C₇×#3164	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
35. #23843	C₇	direct (N × H)
289. C₇	#23843	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×#2330	2
C₇×#2084	2
C₇×#2083	2
C₇×#2059	2
C₇×#1776	2
C₄×C₇×#107	2
C₄×C₇×#106	2
C₇×#1746	2
#23843	7

Subgroups — C7×#23843 #13201059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23843 #13201059