Subgroups — C₇×#11288 #13188504

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#344	C₂	1	2	yes	maximal
3	C₇×#1740	C₂	1	1	yes	maximal, characteristic
4	C₇×#1729	C₂	1	2	yes	maximal
5	C₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	2	yes	—
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₇×#424	—	2	4	no	—
12	C₇×#423	—	2	2	no	—
13	C₇×#344	—	2	4	no	—
14	C₇×#342	C₄	1	4	yes	—
15	C₇×#336	—	2	4	no	—
16	C₇×#336	—	2	2	no	—
17	#11288	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	1	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	D₄	1	2	yes	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	1	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	—	2	1	no	—
34	C₂²×C₇×C₈	D₄	1	4	yes	—
35	C₂²×C₇×C₈	Q₈	1	2	yes	—
36	C₇×#117	—	2	2	no	—
37	C₇×#117	—	2	2	no	—
38	C₇×#117	—	2	2	no	—
39	C₇×#117	—	2	2	no	—
40	C₇×#117	—	2	2	no	—
41	C₇×#117	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
44	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	D₄	1	2	yes	—
48	C₇×C₄⋊C₈	—	2	8	no	—
49	#2905	C₂×C₇	1	1	yes	characteristic
50	C₂×#344	C₂×C₇	1	2	yes	—
51	#1740	C₂×C₇	1	1	yes	characteristic
52	#1729	C₂×C₇	1	2	yes	—
53	#1729	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₇	#41	1	2	yes	—
64	C₂²×C₄×C₇	#41	1	2	yes	—
65	C₂²×C₄×C₇	#41	1	2	yes	—
66	C₂²×C₄×C₇	#41	1	1	yes	characteristic
67	C₇×M₁₆	—	2	4	no	—
68	C₇×M₁₆	—	4	2	no	—
69	C₇×M₁₆	—	4	2	no	—
70	C₇×M₁₆	—	4	2	no	—
71	C₇×M₁₆	—	4	2	no	—
72	C₇×M₁₆	—	4	2	no	—
73	C₂×C₇×C₈	—	2	4	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	4	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	4	1	no	—
80	C₂×C₇×C₈	—	4	1	no	—
81	C₂×C₇×C₈	—	4	1	no	—
82	C₂×C₇×C₈	—	4	2	no	—
83	C₂×C₇×C₈	—	4	1	no	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	4	2	no	—
87	C₇×C₂²⋊C₄	—	2	4	no	—
88	C₇×C₂²⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	2	4	no	—
90	C₄²×C₇	—	2	2	no	—
91	C₄²×C₇	—	2	2	no	—
92	C₄²×C₇	—	4	2	no	—
93	C₄²×C₇	—	4	2	no	—
94	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
95	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
96	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
97	C₂×#117	C₂²×C₇	1	2	yes	—
98	C₂×#117	C₂²×C₇	1	1	yes	characteristic
99	#424	—	2	4	no	—
100	#423	—	2	2	no	—
101	#344	—	2	4	no	—
102	#342	C₄×C₇	1	4	yes	—
103	#336	—	2	4	no	—
104	#336	—	2	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₇	#117	1	2	yes	—
108	C₂³×C₇	C₄×D₄	1	2	yes	—
109	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇	—	2	4	no	—
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	—	4	8	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₇	#121	1	2	yes	—
118	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇	#123	1	2	yes	—
122	C₂×C₄×C₇	#123	1	2	yes	—
123	C₂×C₄×C₇	#123	1	1	yes	characteristic
124	C₂×C₄×C₇	#124	1	1	yes	characteristic
125	C₂×C₄×C₇	#124	1	2	yes	—
126	C₂×C₄×C₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₇	#125	1	2	yes	—
128	C₂×C₄×C₇	#125	1	2	yes	—
129	C₂×C₄×C₇	#125	1	1	yes	characteristic
130	C₂×C₄×C₇	#126	1	2	yes	—
131	C₂×C₄×C₇	#126	1	2	yes	—
132	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
133	C₇×C₈	—	4	2	no	—
134	C₇×C₈	—	4	4	no	—
135	C₇×C₈	—	4	2	no	—
136	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	C₇×D₄	1	2	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₇×D₄	1	4	yes	—
153	C₂²×C₈	C₇×Q₈	1	2	yes	—
154	#117	—	2	2	no	—
155	#117	—	2	2	no	—
156	#117	—	2	2	no	—
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₇×Q₈	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₇×D₄	1	2	yes	—
166	C₄⋊C₈	—	2	8	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	4	no	—
169	C₂²×C₇	—	4	2	no	—
170	C₂²×C₇	—	4	1	no	—
171	C₂²×C₇	—	4	1	no	—
172	C₂²×C₇	#382	1	1	yes	characteristic
173	C₂²×C₇	#385	1	1	yes	characteristic
174	C₂²×C₇	#387	1	2	yes	—
175	C₂²×C₇	#388	1	2	yes	—
176	C₂²×C₇	#389	1	1	yes	characteristic
177	C₄×C₇	—	4	4	no	—
178	C₄×C₇	—	8	4	no	—
179	C₄×C₇	#393	1	1	yes	characteristic
180	C₄×C₇	#394	1	1	yes	characteristic
181	C₄×C₇	#397	1	1	yes	characteristic
182	C₄×C₇	#398	1	2	yes	—
183	C₄×C₇	#400	1	1	yes	characteristic
184	C₄×C₇	#400	1	2	yes	—
185	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
188	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
189	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₇×#41	1	1	yes	characteristic
194	C₂²×C₄	C₇×#41	1	2	yes	—
195	C₂²×C₄	C₇×#41	1	2	yes	—
196	C₂²×C₄	C₇×#41	1	2	yes	—
197	C₂²×C₄	C₇×#41	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	4	2	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	2	1	no	—
227	C₂×C₇	—	2	2	no	—
228	C₂×C₇	—	2	1	no	—
229	C₂×C₇	#1442	1	1	yes	characteristic
230	C₂×C₇	#1868	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
233	C₂³	C₇×#117	1	2	yes	—
234	C₂³	C₄×C₇×D₄	1	2	yes	—
235	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
236	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₇×#120	1	1	yes	characteristic
242	C₂×C₄	C₇×#121	1	1	yes	characteristic
243	C₂×C₄	C₇×#121	1	2	yes	—
244	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
247	C₂×C₄	C₇×#123	1	2	yes	—
248	C₂×C₄	C₇×#123	1	2	yes	—
249	C₂×C₄	C₇×#123	1	1	yes	characteristic
250	C₂×C₄	C₇×#124	1	1	yes	characteristic
251	C₂×C₄	C₇×#124	1	2	yes	—
252	C₂×C₄	C₇×#124	1	1	yes	characteristic
253	C₂×C₄	C₇×#125	1	2	yes	—
254	C₂×C₄	C₇×#125	1	2	yes	—
255	C₂×C₄	C₇×#125	1	1	yes	characteristic
256	C₂×C₄	C₇×#126	1	2	yes	—
257	C₂×C₄	C₇×#126	1	2	yes	—
258	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
259	C₈	—	4	2	no	—
260	C₈	—	4	4	no	—
261	C₈	—	4	2	no	—
262	C₇	#11288	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₇×#382	1	1	yes	characteristic
269	C₂²	C₇×#385	1	1	yes	characteristic
270	C₂²	C₇×#387	1	2	yes	—
271	C₂²	C₇×#388	1	2	yes	—
272	C₂²	C₇×#389	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	8	4	no	—
275	C₄	C₇×#393	1	1	yes	characteristic
276	C₄	C₇×#394	1	1	yes	characteristic
277	C₄	C₇×#397	1	1	yes	characteristic
278	C₄	C₇×#398	1	2	yes	—
279	C₄	C₇×#400	1	1	yes	characteristic
280	C₄	C₇×#400	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	C₇×#1442	1	1	yes	characteristic
286	C₂	C₇×#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#120
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#385
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. #11288	C₇	direct (N × H)
262. C₇	#11288	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×#2905	2
C₂×C₇×#344	2
C₇×#1740	2
C₇×#1729	2
#11288	7

Subgroups — C7×#11288 #13188504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11288 #13188504