Subgroups — C₇×#3123 #11742432

Order 896.

394 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#518	C₂	1	1	yes	maximal, characteristic
3	C₇×#485	C₂	1	2	yes	maximal, complemented
4	C₇×#478	C₂	1	2	yes	maximal, complemented
5	C₇×#461	C₂	1	2	yes	maximal, complemented
6	C₇×#450	C₂	1	2	yes	maximal
7	C₇×#440	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
9	C₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
13	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₇×#126	—	2	2	no	—
15	C₇×#125	—	2	2	no	—
16	C₇×#125	C₂²	1	1	yes	characteristic, complemented
17	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₄×C₇×Q₈	C₂²	1	2	yes	complemented
19	C₄×C₇×D₄	—	2	2	no	—
20	C₄×C₇×D₄	C₂²	1	2	yes	—
21	C₄×C₇×D₄	C₂²	1	2	yes	—
22	C₇×#117	C₂²	1	1	yes	characteristic
23	C₇×#117	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
25	C₇×#107	C₂²	1	2	yes	complemented
26	C₇×#106	C₂²	1	2	yes	complemented
27	C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₇×#103	C₂²	1	2	yes	complemented
29	C₇×#103	C₂²	1	2	yes	complemented
30	C₇×#102	C₂²	1	2	yes	—
31	C₇×#102	C₂²	1	2	yes	—
32	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₇×#97	C₂²	1	1	yes	characteristic, complemented
34	C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
35	#3123	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂⁴×C₇	—	2	1	no	—
37	C₂×C₇×Q₈	C₂³	1	2	yes	—
38	C₂×C₇×D₄	—	2	4	no	—
39	C₂×C₇×D₄	—	2	1	no	—
40	C₂×C₇×D₄	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	—	2	2	no	—
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₇×SD₁₆	—	2	4	no	—
45	C₇×SD₁₆	—	2	4	no	—
46	C₂×C₇×C₈	C₂³	1	2	yes	—
47	C₂×C₇×C₈	C₂³	1	2	yes	—
48	C₇×C₄⋊C₄	—	2	2	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	D₄	1	2	yes	—
53	C₇×C₄⋊C₄	C₂³	1	2	yes	—
54	C₇×C₄⋊C₄	C₂³	1	2	yes	—
55	C₇×C₂²⋊C₄	—	2	1	no	—
56	C₇×C₂²⋊C₄	—	2	2	no	—
57	C₇×C₂²⋊C₄	—	2	2	no	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	D₄	1	2	yes	—
60	C₄²×C₇	—	2	1	no	—
61	C₄²×C₇	—	2	2	no	—
62	C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₄²×C₇	C₂³	1	1	yes	characteristic
64	#541	C₂×C₇	1	1	yes	characteristic, complemented
65	#518	C₂×C₇	1	1	yes	characteristic
66	#485	C₂×C₇	1	2	yes	complemented
67	#478	C₂×C₇	1	2	yes	complemented
68	#461	C₂×C₇	1	2	yes	complemented
69	#450	C₂×C₇	1	2	yes	—
70	#440	C₂×C₇	1	2	yes	complemented
71	C₄×SD₁₆	C₂×C₇	1	2	yes	complemented
72	#433	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂³×C₇	—	2	4	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	8	2	no	—
76	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
77	C₇×Q₈	—	2	2	no	—
78	C₇×Q₈	—	2	2	no	—
79	C₇×D₄	—	2	4	no	—
80	C₇×D₄	—	2	2	no	—
81	C₇×D₄	#41	1	4	yes	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	8	2	no	—
87	C₂×C₄×C₇	—	8	2	no	—
88	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
92	C₇×C₈	—	2	2	no	—
93	C₇×C₈	—	2	2	no	—
94	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
95	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
96	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
97	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
98	#126	—	2	2	no	—
99	#125	—	2	2	no	—
100	#125	C₂²×C₇	1	1	yes	characteristic, complemented
101	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
102	C₄×Q₈	C₂²×C₇	1	2	yes	complemented
103	C₄×D₄	—	2	2	no	—
104	C₄×D₄	C₂²×C₇	1	2	yes	—
105	C₄×D₄	C₂²×C₇	1	2	yes	—
106	#117	C₂²×C₇	1	1	yes	characteristic
107	#117	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	—	2	2	no	—
109	#107	C₂²×C₇	1	2	yes	complemented
110	#106	C₂²×C₇	1	2	yes	complemented
111	C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
112	#103	C₂²×C₇	1	2	yes	complemented
113	#103	C₂²×C₇	1	2	yes	complemented
114	#102	C₂²×C₇	1	2	yes	—
115	#102	C₂²×C₇	1	2	yes	—
116	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
117	#97	C₂²×C₇	1	1	yes	characteristic, complemented
118	C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	—	2	4	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	—	8	4	no	—
125	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	2	no	—
128	C₄×C₇	—	2	2	no	—
129	C₄×C₇	—	4	2	no	—
130	C₄×C₇	—	4	2	no	—
131	C₄×C₇	—	4	2	no	—
132	C₄×C₇	C₂×#41	1	2	yes	—
133	C₂⁴	—	2	1	no	—
134	C₂×Q₈	C₂³×C₇	1	2	yes	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	C₂³×C₇	1	2	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
141	SD₁₆	—	2	4	no	—
142	SD₁₆	—	2	4	no	—
143	C₂×C₈	C₂³×C₇	1	2	yes	—
144	C₂×C₈	C₂³×C₇	1	2	yes	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	C₇×D₄	1	2	yes	—
150	C₄⋊C₄	C₂³×C₇	1	2	yes	—
151	C₄⋊C₄	C₂³×C₇	1	2	yes	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	C₇×D₄	1	2	yes	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	2	no	—
159	C₄²	C₂³×C₇	1	1	yes	characteristic
160	C₄²	C₂³×C₇	1	1	yes	characteristic
161	C₂×C₇	—	4	1	no	—
162	C₂×C₇	—	4	4	no	—
163	C₂×C₇	#525	1	1	yes	characteristic
164	C₂×C₇	#577	1	2	yes	—
165	C₂³	—	2	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	D₄	C₇×#41	1	4	yes	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
181	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
183	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₇	#3123	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²	—	2	1	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×C₇×#41	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	4	4	no	—
203	C₂	C₇×#525	1	1	yes	characteristic
204	C₂	C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#541	C₂	semidirect
3. C₇×#485	C₂	semidirect
4. C₇×#478	C₂	semidirect
5. C₇×#461	C₂	semidirect
7. C₇×#440	C₂	semidirect
8. C₄×C₇×SD₁₆	C₂	semidirect
9. C₇×#433	C₂	semidirect
13. C₇×C₄⋊Q₈	C₂²	semidirect
16. C₇×#125	C₂²	semidirect
18. C₄×C₇×Q₈	C₂²	semidirect
25. C₇×#107	C₂²	semidirect
26. C₇×#106	C₂²	semidirect
27. C₇×C₄⋊C₈	C₂²	semidirect
28. C₇×#103	C₂²	semidirect
29. C₇×#103	C₂²	semidirect
33. C₇×#97	C₂²	semidirect
34. C₄×C₇×C₈	C₂²	semidirect
35. #3123	C₇	direct (N × H)
64. #541	C₂×C₇	semidirect
66. #485	C₂×C₇	semidirect
67. #478	C₂×C₇	semidirect
68. #461	C₂×C₇	semidirect
70. #440	C₂×C₇	semidirect
71. C₄×SD₁₆	C₂×C₇	semidirect
72. #433	C₂×C₇	semidirect
97. C₄⋊Q₈	C₂²×C₇	semidirect
100. #125	C₂²×C₇	semidirect
102. C₄×Q₈	C₂²×C₇	semidirect
109. #107	C₂²×C₇	semidirect
110. #106	C₂²×C₇	semidirect
111. C₄⋊C₈	C₂²×C₇	semidirect
112. #103	C₂²×C₇	semidirect
113. #103	C₂²×C₇	semidirect
117. #97	C₂²×C₇	semidirect
118. C₄×C₈	C₂²×C₇	semidirect
186. C₇	#3123	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×#541	2
C₇×#518	2
C₇×#485	2
C₇×#478	2
C₇×#461	2
C₇×#450	2
C₇×#440	2
C₄×C₇×SD₁₆	2
C₇×#433	2
#3123	7

Subgroups — C7×#3123 #11742432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3123 #11742432