Subgroups — C₇×#3116 #11742425

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇×#41	C₂	1	1	yes	maximal, characteristic
3	C₇×#485	C₂	1	2	yes	maximal, complemented
4	C₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#452	C₂	1	2	yes	maximal
9	C₇×#441	C₂	1	2	yes	maximal, complemented
10	C₇×#440	C₂	1	2	yes	maximal, complemented
11	C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#41	C₂²	1	1	yes	characteristic
13	C₂×C₇×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
15	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#125	—	2	1	no	—
18	C₇×C₂²⋊Q₈	—	2	1	no	—
19	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×C₇×Q₈	—	2	1	no	—
21	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
22	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
23	C₄×C₇×Q₈	C₂²	1	2	yes	—
24	C₄×C₇×D₄	—	2	1	no	—
25	C₄×C₇×D₄	—	2	1	no	—
26	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
28	C₇×#117	—	2	1	no	—
29	C₇×#117	—	2	1	no	—
30	C₇×#117	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₇	—	2	1	no	—
32	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₇×#107	C₂²	1	2	yes	—
34	C₇×#106	C₂²	1	2	yes	—
35	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₇×#103	C₂²	1	2	yes	—
38	C₇×#103	C₂²	1	2	yes	—
39	C₇×#103	C₂²	1	2	yes	—
40	C₇×#102	C₂²	1	2	yes	—
41	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₇×#97	C₂²	1	2	yes	—
43	#3116	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₇×#41	—	2	2	no	—
45	C₇×#41	—	2	2	no	—
46	C₂×C₇×Q₈	—	2	1	no	—
47	C₂×C₇×Q₈	C₂³	1	2	yes	—
48	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
49	C₂×C₇×D₄	—	2	1	no	—
50	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₇×Q₁₆	—	2	4	no	—
58	C₇×SD₁₆	—	2	4	no	—
59	C₂×C₇×C₈	C₂³	1	2	yes	—
60	C₂×C₇×C₈	C₂³	1	2	yes	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	—	2	1	no	—
64	C₇×C₄⋊C₄	—	2	1	no	—
65	C₇×C₄⋊C₄	—	2	1	no	—
66	C₇×C₄⋊C₄	—	2	2	no	—
67	C₇×C₄⋊C₄	—	2	2	no	—
68	C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₇×C₂²⋊C₄	—	2	2	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₇×C₂²⋊C₄	—	2	1	no	—
74	C₇×C₂²⋊C₄	—	2	1	no	—
75	C₄²×C₇	—	2	1	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	1	no	—
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	D₄	1	1	yes	characteristic
80	C₄²×C₇	D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂³	1	1	yes	characteristic
82	C₄²×C₇	C₂³	1	1	yes	characteristic
83	#533	C₂×C₇	1	1	yes	characteristic, complemented
84	C₄×#41	C₂×C₇	1	1	yes	characteristic
85	#485	C₂×C₇	1	2	yes	complemented
86	#477	C₂×C₇	1	1	yes	characteristic, complemented
87	#476	C₂×C₇	1	1	yes	characteristic, complemented
88	#462	C₂×C₇	1	1	yes	characteristic, complemented
89	#461	C₂×C₇	1	1	yes	characteristic, complemented
90	#452	C₂×C₇	1	2	yes	—
91	#441	C₂×C₇	1	2	yes	complemented
92	#440	C₂×C₇	1	2	yes	complemented
93	#432	C₂×C₇	1	1	yes	characteristic, complemented
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
96	C₇×Q₈	—	2	1	no	—
97	C₇×Q₈	—	2	2	no	—
98	C₇×Q₈	—	2	2	no	—
99	C₇×Q₈	—	4	2	no	—
100	C₇×Q₈	#41	1	2	yes	—
101	C₇×D₄	—	2	4	no	—
102	C₇×D₄	—	2	1	no	—
103	C₇×D₄	#41	1	2	yes	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	4	1	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
122	C₇×C₈	—	2	2	no	—
123	C₇×C₈	—	2	2	no	—
124	C₂×#41	C₂²×C₇	1	1	yes	characteristic
125	C₂×Q₁₆	C₂²×C₇	1	2	yes	—
126	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
127	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
128	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
129	#125	—	2	1	no	—
130	C₂²⋊Q₈	—	2	1	no	—
131	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
134	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
135	C₄×Q₈	C₂²×C₇	1	2	yes	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
139	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
140	#117	—	2	1	no	—
141	#117	—	2	1	no	—
142	#117	C₂²×C₇	1	1	yes	characteristic
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
145	#107	C₂²×C₇	1	2	yes	—
146	#106	C₂²×C₇	1	2	yes	—
147	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
149	#103	C₂²×C₇	1	2	yes	—
150	#103	C₂²×C₇	1	2	yes	—
151	#103	C₂²×C₇	1	2	yes	—
152	#102	C₂²×C₇	1	2	yes	—
153	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
154	#97	C₂²×C₇	1	2	yes	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	4	1	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	1	no	—
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	4	2	no	—
168	C₄×C₇	—	4	1	no	—
169	C₄×C₇	—	4	1	no	—
170	C₄×C₇	—	4	2	no	—
171	C₄×C₇	#137	1	2	yes	—
172	C₄×C₇	C₂×#41	1	1	yes	characteristic
173	C₄×C₇	C₂×#41	1	1	yes	characteristic
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂³×C₇	1	2	yes	—
178	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
187	Q₁₆	—	2	4	no	—
188	SD₁₆	—	2	4	no	—
189	C₂×C₈	C₂³×C₇	1	2	yes	—
190	C₂×C₈	C₂³×C₇	1	2	yes	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₂³×C₇	1	2	yes	—
199	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₇×D₄	1	1	yes	characteristic
210	C₄²	C₇×D₄	1	1	yes	characteristic
211	C₄²	C₂³×C₇	1	1	yes	characteristic
212	C₄²	C₂³×C₇	1	1	yes	characteristic
213	C₂×C₇	—	4	1	no	—
214	C₂×C₇	—	4	2	no	—
215	C₂×C₇	#525	1	1	yes	characteristic
216	C₂×C₇	C₂×#137	1	1	yes	characteristic
217	C₂×C₇	#575	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	4	2	no	—
224	Q₈	C₇×#41	1	2	yes	—
225	D₄	—	2	4	no	—
226	D₄	—	2	1	no	—
227	D₄	C₇×#41	1	2	yes	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₇	#3116	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	C₇×#137	1	2	yes	—
266	C₄	C₂×C₇×#41	1	1	yes	characteristic
267	C₄	C₂×C₇×#41	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	C₇×#525	1	1	yes	characteristic
271	C₂	C₂×C₇×#137	1	1	yes	characteristic
272	C₂	C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#533	C₂	semidirect
3. C₇×#485	C₂	semidirect
4. C₇×#477	C₂	semidirect
5. C₇×#476	C₂	semidirect
6. C₇×#462	C₂	semidirect
7. C₇×#461	C₂	semidirect
9. C₇×#441	C₂	semidirect
10. C₇×#440	C₂	semidirect
11. C₇×#432	C₂	semidirect
43. #3116	C₇	direct (N × H)
83. #533	C₂×C₇	semidirect
85. #485	C₂×C₇	semidirect
86. #477	C₂×C₇	semidirect
87. #476	C₂×C₇	semidirect
88. #462	C₂×C₇	semidirect
89. #461	C₂×C₇	semidirect
91. #441	C₂×C₇	semidirect
92. #440	C₂×C₇	semidirect
93. #432	C₂×C₇	semidirect
248. C₇	#3116	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×#533	2
C₄×C₇×#41	2
C₇×#485	2
C₇×#477	2
C₇×#476	2
C₇×#462	2
C₇×#461	2
C₇×#452	2
C₇×#441	2
C₇×#440	2
C₇×#432	2
#3116	7

Subgroups — C7×#3116 #11742425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3116 #11742425