Subgroups — C₇×#2645 #11741954

Order 896.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#401	C₂	1	2	yes	maximal
2	C₇×#400	C₂	1	6	yes	maximal
3	C₇×#391	C₂	1	3	yes	maximal
4	C₇×#389	C₂	1	3	yes	maximal
5	C₇×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂×C₄²×C₇	C₂²	1	3	yes	—
10	C₇×#95	C₂²	1	3	yes	—
11	C₇×#95	C₂²	1	1	yes	characteristic
12	C₇×#95	C₂²	1	6	yes	—
13	C₇×#95	C₂²	1	6	yes	—
14	C₇×#95	C₂²	1	6	yes	—
15	#2645	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×Q₈	—	2	6	no	—
17	C₂²×C₄×C₇	C₂³	1	3	yes	—
18	C₂²×C₄×C₇	C₂³	1	2	yes	—
19	C₂²×C₄×C₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₇	C₂³	1	6	yes	—
21	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₇×C₄⋊C₄	—	2	6	no	—
23	C₇×C₄⋊C₄	—	4	6	no	—
24	C₄²×C₇	—	2	6	no	—
25	#401	C₂×C₇	1	2	yes	—
26	#400	C₂×C₇	1	6	yes	—
27	#391	C₂×C₇	1	3	yes	—
28	#389	C₂×C₇	1	3	yes	—
29	#376	C₂×C₇	1	1	yes	characteristic
30	C₂³×C₇	C₂⁴	1	1	yes	characteristic
31	C₇×Q₈	—	4	4	no	—
32	C₂×C₄×C₇	—	2	3	no	—
33	C₂×C₄×C₇	—	2	3	no	—
34	C₂×C₄×C₇	—	2	6	no	—
35	C₂×C₄×C₇	—	2	6	no	—
36	C₂×C₄×C₇	—	2	3	no	—
37	C₂×C₄×C₇	—	2	6	no	—
38	C₂×C₄×C₇	—	2	6	no	—
39	C₂×C₄×C₇	—	2	3	no	—
40	C₂×C₄×C₇	—	2	6	no	—
41	C₂×C₄×C₇	#41	1	6	yes	—
42	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
43	C₂×C₄⋊C₄	C₂²×C₇	1	3	yes	—
44	C₂×C₄⋊C₄	C₂²×C₇	1	6	yes	—
45	C₂×C₄²	C₂²×C₇	1	3	yes	—
46	#95	C₂²×C₇	1	3	yes	—
47	#95	C₂²×C₇	1	1	yes	characteristic
48	#95	C₂²×C₇	1	6	yes	—
49	#95	C₂²×C₇	1	6	yes	—
50	#95	C₂²×C₇	1	6	yes	—
51	C₂²×C₇	C₂×#41	1	3	yes	—
52	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
54	C₄×C₇	—	4	2	no	—
55	C₄×C₇	—	4	3	no	—
56	C₄×C₇	—	4	3	no	—
57	C₄×C₇	—	4	6	no	—
58	C₄×C₇	#126	1	4	yes	—
59	C₂×Q₈	—	2	6	no	—
60	C₂²×C₄	C₂³×C₇	1	3	yes	—
61	C₂²×C₄	C₂³×C₇	1	2	yes	—
62	C₂²×C₄	C₂³×C₇	1	3	yes	—
63	C₂²×C₄	C₂³×C₇	1	6	yes	—
64	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	6	no	—
66	C₄⋊C₄	—	4	6	no	—
67	C₄²	—	2	6	no	—
68	C₂×C₇	C₂×#126	1	1	yes	characteristic
69	C₂×C₇	#541	1	3	yes	—
70	C₂×C₇	#542	1	3	yes	—
71	C₂³	C₂⁴×C₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	C₂×C₄	—	2	3	no	—
74	C₂×C₄	—	2	3	no	—
75	C₂×C₄	—	2	6	no	—
76	C₂×C₄	—	2	6	no	—
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	6	no	—
79	C₂×C₄	—	2	6	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₇×#41	1	6	yes	—
83	C₇	#2645	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²	C₂×C₇×#41	1	3	yes	—
85	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
86	C₂²	C₇×2₋¹⁺⁴	1	3	yes	—
87	C₄	—	4	2	no	—
88	C₄	—	4	3	no	—
89	C₄	—	4	3	no	—
90	C₄	—	4	6	no	—
91	C₄	C₇×#126	1	4	yes	—
92	C₂	C₂×C₇×#126	1	1	yes	characteristic
93	C₂	C₇×#541	1	3	yes	—
94	C₂	C₇×#542	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #2645	C₇	direct (N × H)
83. C₇	#2645	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×#401	2
C₇×#400	2
C₇×#391	2
C₇×#389	2
C₇×#376	2
#2645	7

Subgroups — C7×#2645 #11741954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2645 #11741954