Subgroups — C₇×#23842 #13201058

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2331	C₂	1	2	yes	maximal
2	C₇×#2085	C₂	1	2	yes	maximal
3	C₇×#2085	C₂	1	2	yes	maximal
4	C₇×#2058	C₂	1	2	yes	maximal
5	C₇×#2057	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×C₄⋊Q₈	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×#391	—	2	2	no	—
18	C₇×#391	—	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×#382	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	#23842	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₇×Q₈	—	2	2	no	—
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₇×C₄⋊Q₈	—	2	4	no	—
41	C₇×C₄⋊Q₈	—	2	4	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	C₂³	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₄²×C₇	—	2	2	no	—
50	C₂×C₄²×C₇	—	2	2	no	—
51	C₂×C₄²×C₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	4	no	—
54	C₇×#107	—	2	4	no	—
55	C₇×#106	—	2	4	no	—
56	C₇×#106	—	2	4	no	—
57	C₇×#97	Q₈	1	4	yes	complemented
58	C₄×C₇×C₈	Q₈	1	4	yes	complemented
59	C₇×#95	—	2	2	no	—
60	C₇×#95	—	2	2	no	—
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	#2331	C₂×C₇	1	2	yes	—
69	#2085	C₂×C₇	1	2	yes	—
70	#2085	C₂×C₇	1	2	yes	—
71	#2058	C₂×C₇	1	2	yes	—
72	#2057	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	C₂×C₇×Q₈	—	2	8	no	—
78	C₂×C₇×Q₈	—	8	2	no	—
79	C₂×C₇×Q₈	—	8	2	no	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
96	C₇×C₄⋊C₄	—	2	8	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	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	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×C₄⋊Q₈	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	#391	—	2	2	no	—
121	#391	—	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	#382	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	2	yes	—
132	#340	C₂²×C₇	1	1	yes	characteristic
133	#340	C₂²×C₇	1	1	yes	characteristic
134	#340	C₂²×C₇	1	1	yes	characteristic
135	#340	C₂²×C₇	1	2	yes	—
136	#336	C₂²×C₇	1	2	yes	—
137	#336	C₂²×C₇	1	2	yes	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₇×Q₈	—	8	8	no	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	1	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	4	no	—
146	C₂×C₄×C₇	—	2	4	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	—	2	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	—	4	2	no	—
161	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
162	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
163	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×#41	1	2	yes	—
166	C₂×C₄×C₇	C₂×#41	1	2	yes	—
167	C₇×C₈	—	2	2	no	—
168	C₇×C₈	—	2	4	no	—
169	C₇×C₈	—	2	2	no	—
170	C₂²×Q₈	—	2	2	no	—
171	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₇	1	2	yes	—
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	2	4	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	C₂³×C₇	1	2	yes	—
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×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#107	—	2	4	no	—
189	#106	—	2	4	no	—
190	#106	—	2	4	no	—
191	#97	C₇×Q₈	1	4	yes	complemented
192	C₄×C₈	C₇×Q₈	1	4	yes	complemented
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	C₇×D₄	1	1	yes	characteristic
200	#95	C₇×D₄	1	2	yes	—
201	#95	C₇×D₄	1	1	yes	characteristic
202	C₂²×C₇	#525	1	2	yes	—
203	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
204	C₂²×C₇	#576	1	1	yes	characteristic
205	C₂²×C₇	#577	1	2	yes	—
206	C₂²×C₇	#578	1	1	yes	characteristic
207	C₄×C₇	—	2	2	no	—
208	C₄×C₇	—	2	2	no	—
209	C₄×C₇	—	4	2	no	—
210	C₄×C₇	—	4	4	no	—
211	C₄×C₇	—	4	4	no	—
212	C₄×C₇	—	4	2	no	—
213	C₄×C₇	—	8	2	no	—
214	C₄×C₇	—	8	2	no	—
215	C₄×C₇	#533	1	1	yes	characteristic
216	C₄×C₇	#533	1	2	yes	—
217	C₄×C₇	#533	1	1	yes	characteristic
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	8	2	no	—
220	C₂×Q₈	—	8	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
227	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
235	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
236	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	8	2	no	—
245	C₄⋊C₄	—	8	2	no	—
246	C₄⋊C₄	C₇×#41	1	4	yes	—
247	C₄⋊C₄	C₇×#41	1	4	yes	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	8	2	no	—
251	C₄²	—	8	2	no	—
252	C₄²	C₂×C₇×Q₈	1	2	yes	—
253	C₄²	C₂×C₇×Q₈	1	2	yes	—
254	C₂×C₇	#2399	1	1	yes	characteristic
255	C₂×C₇	#3118	1	2	yes	—
256	C₂×C₇	#3121	1	2	yes	—
257	C₂×C₇	#3163	1	1	yes	characteristic
258	C₂×C₇	#3164	1	1	yes	characteristic
259	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
260	Q₈	—	8	8	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	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	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
283	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
284	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₇×#41	1	2	yes	—
287	C₂×C₄	C₂×C₇×#41	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₇	#23842	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	C₇×#525	1	2	yes	—
293	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
294	C₂²	C₇×#576	1	1	yes	characteristic
295	C₂²	C₇×#577	1	2	yes	—
296	C₂²	C₇×#578	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₇×#533	1	1	yes	characteristic
306	C₄	C₇×#533	1	2	yes	—
307	C₄	C₇×#533	1	1	yes	characteristic
308	C₂	C₇×#2399	1	1	yes	characteristic
309	C₂	C₇×#3118	1	2	yes	—
310	C₂	C₇×#3121	1	2	yes	—
311	C₂	C₇×#3163	1	1	yes	characteristic
312	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. #23842	C₇	direct (N × H)
57. C₇×#97	Q₈	semidirect
58. C₄×C₇×C₈	Q₈	semidirect
191. #97	C₇×Q₈	semidirect
192. C₄×C₈	C₇×Q₈	semidirect
291. C₇	#23842	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×#2331	2
C₇×#2085	2
C₇×#2058	2
C₇×#2057	2
C₇×#1776	2
C₄×C₇×#107	2
C₄×C₇×#106	2
C₇×#1746	2
#23842	7

Subgroups — C7×#23842 #13201058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23842 #13201058