Subgroups — C₇×#1744 #11730213

Order 896.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₈	C₂	1	2	yes	maximal
2	C₄³×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#336	C₂	1	4	yes	maximal
4	C₂²×C₇×C₈	C₂²	1	4	yes	—
5	C₂×C₄²×C₇	C₄	1	2	yes	—
6	C₂×C₄²×C₇	C₄	1	2	yes	—
7	C₂×C₄²×C₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₈	C₄	1	8	yes	complemented
10	#1744	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₄×C₇	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₇×C₈	—	2	8	no	—
16	C₂×C₇×C₈	C₂×C₄	1	8	yes	—
17	C₄²×C₇	—	2	4	no	—
18	C₄²×C₇	—	2	4	no	—
19	C₄²×C₇	C₈	1	8	yes	complemented
20	C₄²×C₇	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₈	C₂×C₇	1	2	yes	—
22	C₄³	C₂×C₇	1	1	yes	characteristic
23	#336	C₂×C₇	1	4	yes	—
24	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₇	—	2	4	no	—
26	C₂×C₄×C₇	—	2	4	no	—
27	C₂×C₄×C₇	C₄²	1	2	yes	—
28	C₂×C₄×C₇	C₄²	1	2	yes	—
29	C₂×C₄×C₇	C₂×C₈	1	8	yes	—
30	C₂×C₄×C₇	C₂×C₈	1	4	yes	—
31	C₂×C₄×C₇	M₁₆	1	4	yes	—
32	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₇	#41	1	4	yes	—
34	C₇×C₈	—	2	8	no	—
35	C₂²×C₈	C₂²×C₇	1	4	yes	—
36	C₂×C₄²	C₄×C₇	1	2	yes	—
37	C₂×C₄²	C₄×C₇	1	2	yes	—
38	C₂×C₄²	C₂²×C₇	1	2	yes	—
39	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
40	C₄×C₈	C₄×C₇	1	8	yes	complemented
41	C₂²×C₇	C₂×C₄²	1	1	yes	characteristic
42	C₂²×C₇	#117	1	2	yes	—
43	C₂²×C₇	C₂²×C₈	1	2	yes	—
44	C₂²×C₇	C₂×M₁₆	1	2	yes	—
45	C₄×C₇	—	2	8	no	—
46	C₄×C₇	C₄×C₈	1	4	yes	—
47	C₄×C₇	#97	1	4	yes	—
48	C₄×C₇	#117	1	4	yes	—
49	C₂²×C₄	C₂×C₄×C₇	1	4	yes	—
50	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
51	C₂²×C₄	C₂×C₄×C₇	1	1	yes	characteristic
52	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
53	C₂×C₈	—	2	8	no	—
54	C₂×C₈	C₂×C₄×C₇	1	8	yes	—
55	C₄²	—	2	4	no	—
56	C₄²	—	2	4	no	—
57	C₄²	C₇×C₈	1	8	yes	complemented
58	C₄²	C₂×C₄×C₇	1	4	yes	—
59	C₂×C₇	#376	1	1	yes	characteristic
60	C₂×C₇	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₇	C₂×#97	1	1	yes	characteristic
62	C₂×C₇	#431	1	4	yes	—
63	C₂³	C₂²×C₄×C₇	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄²×C₇	1	2	yes	—
67	C₂×C₄	C₄²×C₇	1	2	yes	—
68	C₂×C₄	C₂×C₇×C₈	1	8	yes	—
69	C₂×C₄	C₂×C₇×C₈	1	4	yes	—
70	C₂×C₄	C₇×M₁₆	1	4	yes	—
71	C₂×C₄	C₂²×C₄×C₇	1	2	yes	—
72	C₂×C₄	C₇×#41	1	4	yes	—
73	C₈	—	2	8	no	—
74	C₇	#1744	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	C₂×C₄²×C₇	1	1	yes	characteristic
76	C₂²	C₇×#117	1	2	yes	—
77	C₂²	C₂²×C₇×C₈	1	2	yes	—
78	C₂²	C₂×C₇×M₁₆	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×C₇×C₈	1	4	yes	—
81	C₄	C₇×#97	1	4	yes	—
82	C₄	C₇×#117	1	4	yes	—
83	C₂	C₇×#376	1	1	yes	characteristic
84	C₂	C₂×C₄×C₇×C₈	1	1	yes	characteristic
85	C₂	C₂×C₇×#97	1	1	yes	characteristic
86	C₂	C₇×#431	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₇	C₂²
commutator	C₂	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄×C₇×C₈	C₄	semidirect
10. #1744	C₇	direct (N × H)
19. C₄²×C₇	C₈	semidirect
40. C₄×C₈	C₄×C₇	semidirect
57. C₄²	C₇×C₈	semidirect
74. C₇	#1744	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₇×C₈	2
C₄³×C₇	2
C₇×#336	2
#1744	7

Subgroups — C7×#1744 #11730213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1744 #11730213