Subgroups — C₇×#3164 #11742473

Order 896.

340 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#544	C₂	1	2	yes	maximal
2	C₇×#501	C₂	1	4	yes	maximal, complemented
3	C₇×#500	C₂	1	2	yes	maximal, complemented
4	C₇×#499	C₂	1	2	yes	maximal, complemented
5	C₇×#428	C₂	1	2	yes	maximal
6	C₇×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#107	C₂	1	1	yes	maximal, characteristic
8	C₇×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊Q₈	—	2	2	no	—
12	C₇×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₇×#125	—	2	2	no	—
14	C₇×#125	C₂²	1	4	yes	—
15	C₇×C₂²⋊Q₈	—	2	4	no	—
16	C₇×#117	C₂²	1	2	yes	—
17	C₇×#117	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₇×#107	C₂²	1	1	yes	characteristic
20	C₇×#107	C₂²	1	2	yes	—
21	C₇×#107	C₂²	1	1	yes	characteristic
22	C₇×#107	C₂²	1	4	yes	—
23	C₇×#107	C₂²	1	2	yes	—
24	C₇×#106	C₂²	1	4	yes	—
25	C₇×#106	C₂²	1	2	yes	—
26	C₇×#97	C₂²	1	2	yes	—
27	C₄×C₇×C₈	C₂²	1	2	yes	—
28	#3164	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₇×Q₈	—	2	4	no	—
30	C₂²×C₄×C₇	—	2	2	no	—
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₇×M₁₆	Q₈	1	4	yes	complemented
33	C₂×C₇×C₈	Q₈	1	4	yes	complemented
34	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₇×C₈	C₂³	1	2	yes	—
36	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊C₄	—	2	2	no	—
38	C₇×C₄⋊C₄	—	2	4	no	—
39	C₇×C₄⋊C₄	—	2	4	no	—
40	C₇×C₄⋊C₄	—	2	4	no	—
41	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₇×C₄⋊C₄	C₂³	1	2	yes	—
44	C₇×C₄⋊C₄	C₂³	1	4	yes	—
45	C₇×C₄⋊C₄	C₂³	1	2	yes	—
46	C₇×C₂²⋊C₄	—	2	2	no	—
47	C₇×C₂²⋊C₄	D₄	1	2	yes	—
48	C₄²×C₇	—	2	2	no	—
49	C₄²×C₇	C₂³	1	2	yes	—
50	#544	C₂×C₇	1	2	yes	—
51	#501	C₂×C₇	1	4	yes	complemented
52	#500	C₂×C₇	1	2	yes	complemented
53	#499	C₂×C₇	1	2	yes	complemented
54	#428	C₂×C₇	1	2	yes	—
55	#427	C₂×C₇	1	1	yes	characteristic
56	C₂×#107	C₂×C₇	1	1	yes	characteristic
57	#405	C₂×C₇	1	1	yes	characteristic
58	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
59	C₇×Q₈	—	8	4	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	—	2	2	no	—
62	C₂×C₄×C₇	—	2	4	no	—
63	C₂×C₄×C₇	—	8	2	no	—
64	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
70	C₇×C₈	C₂×Q₈	1	2	yes	—
71	C₇×C₈	C₂×Q₈	1	2	yes	—
72	C₇×C₈	C₂×Q₈	1	4	yes	—
73	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	C₂²×C₇	1	4	yes	—
77	#125	—	2	2	no	—
78	#125	C₂²×C₇	1	4	yes	—
79	C₂²⋊Q₈	—	2	4	no	—
80	#117	C₂²×C₇	1	2	yes	—
81	#117	C₂²×C₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
83	#107	C₂²×C₇	1	2	yes	—
84	#107	C₂²×C₇	1	2	yes	—
85	#107	C₂²×C₇	1	1	yes	characteristic
86	#107	C₂²×C₇	1	1	yes	characteristic
87	#107	C₂²×C₇	1	4	yes	—
88	#106	C₂²×C₇	1	2	yes	—
89	#106	C₂²×C₇	1	4	yes	—
90	#97	C₂²×C₇	1	2	yes	—
91	C₄×C₈	C₂²×C₇	1	2	yes	—
92	C₂²×C₇	—	2	1	no	—
93	C₂²×C₇	—	2	1	no	—
94	C₂²×C₇	C₄⋊Q₈	1	2	yes	—
95	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₇	—	2	2	no	—
97	C₄×C₇	—	2	2	no	—
98	C₄×C₇	—	4	4	no	—
99	C₄×C₇	—	4	2	no	—
100	C₄×C₇	—	4	2	no	—
101	C₄×C₇	C₄⋊Q₈	1	2	yes	—
102	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×Q₈	—	2	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
107	M₁₆	C₇×Q₈	1	4	yes	complemented
108	C₂×C₈	C₇×Q₈	1	4	yes	complemented
109	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
110	C₂×C₈	C₂³×C₇	1	2	yes	—
111	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
117	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
118	C₄⋊C₄	C₂³×C₇	1	2	yes	—
119	C₄⋊C₄	C₂³×C₇	1	4	yes	—
120	C₄⋊C₄	C₂³×C₇	1	2	yes	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	C₇×D₄	1	2	yes	—
123	C₄²	—	2	2	no	—
124	C₄²	C₂³×C₇	1	2	yes	—
125	C₂×C₇	—	2	2	no	—
126	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₇	#576	1	1	yes	characteristic
128	C₂×C₇	#578	1	1	yes	characteristic
129	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
130	Q₈	—	8	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
136	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
137	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
139	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
140	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
141	C₈	C₂×C₇×Q₈	1	2	yes	—
142	C₈	C₂×C₇×Q₈	1	2	yes	—
143	C₈	C₂×C₇×Q₈	1	4	yes	—
144	C₇	#3164	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	C₇×C₄⋊Q₈	1	2	yes	—
148	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	C₇×C₄⋊Q₈	1	2	yes	—
155	C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
156	C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
159	C₂	C₇×#576	1	1	yes	characteristic
160	C₂	C₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂³×C₇	C₂×D₄
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
2. C₇×#501	C₂	semidirect
3. C₇×#500	C₂	semidirect
4. C₇×#499	C₂	semidirect
28. #3164	C₇	direct (N × H)
32. C₇×M₁₆	Q₈	semidirect
33. C₂×C₇×C₈	Q₈	semidirect
51. #501	C₂×C₇	semidirect
52. #500	C₂×C₇	semidirect
53. #499	C₂×C₇	semidirect
107. M₁₆	C₇×Q₈	semidirect
108. C₂×C₈	C₇×Q₈	semidirect
144. C₇	#3164	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×#544	2
C₇×#501	2
C₇×#500	2
C₇×#499	2
C₇×#428	2
C₇×#427	2
C₂×C₇×#107	2
C₇×#405	2
#3164	7

Subgroups — C7×#3164 #11742473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3164 #11742473