Subgroups — #13096570

Order 1568.

470 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₇⋊D₇)	C₂	1	2	yes	maximal, complemented
3	C₂×C₄⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716070	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₇⋊Dic₇)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇²	C₂²	1	1	yes	characteristic
7	C₂²×C₇⋊D₇	C₂²	1	2	yes	—
8	C₂×C₄×C₇²	—	2	2	no	—
9	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₇⋊D₇)	—	2	4	no	—
12	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
13	C₄⋊(C₇⋊D₇)	—	2	2	no	—
14	#6493	—	7	8	no	maximal
15	C₂²×C₇²	—	2	1	no	—
16	C₂²×C₇²	—	2	1	no	—
17	C₂²×C₇²	D₄	1	2	yes	complemented
18	C₂²×C₇²	C₂³	1	1	yes	characteristic
19	C₂×C₇⋊D₇	—	2	2	no	—
20	C₂×C₇⋊D₇	—	2	2	no	—
21	C₂×C₇⋊D₇	—	2	2	no	—
22	C₄×C₇²	—	2	1	no	—
23	C₄×C₇²	D₄	1	2	yes	complemented
24	C₇⋊Dic₇	—	2	2	no	—
25	C₂²×C₄×C₇	D₇	1	8	yes	complemented
26	C₂×C₂²⋊D₇	—	7	16	no	—
27	C₂×D₂₈	—	7	8	no	—
28	(C₂×D₇)⋊C₄	—	7	16	no	—
29	C₄⋊Dic₇	—	7	8	no	—
30	C₂×C₇²	—	2	2	no	—
31	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
32	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
33	C₂×C₇²	#41	1	1	yes	characteristic
34	C₇⋊D₇	—	4	2	no	—
35	C₂³×C₇	C₂×D₇	1	8	yes	—
36	C₂²×D₇	—	7	16	no	—
37	C₂×C₄×C₇	—	2	16	no	—
38	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
39	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
40	C₂²⋊D₇	—	14	32	no	—
41	C₂×Dic₇	—	7	16	no	—
42	D₂₈	—	14	16	no	—
43	C₇²	#121	1	1	yes	characteristic, complemented, Sylow
44	#121	—	49	1	no	Sylow
45	C₂²×C₇	—	2	8	no	—
46	C₂²×C₇	—	2	8	no	—
47	C₂²×C₇	D₂₈	1	16	yes	complemented
48	C₂²×C₇	C₂²×D₇	1	8	yes	—
49	C₂×D₇	—	14	16	no	—
50	C₂×D₇	—	14	16	no	—
51	C₂×D₇	—	14	16	no	—
52	C₄×C₇	—	2	8	no	—
53	C₄×C₇	C₂²⋊D₇	1	16	yes	complemented
54	Dic₇	—	14	16	no	—
55	C₂×D₄	—	49	2	no	—
56	C₂×D₄	—	49	1	no	—
57	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic, complemented
58	C₄⋊C₄	—	49	1	no	—
59	C₂²⋊C₄	—	49	2	no	—
60	C₂×C₇	—	2	16	no	—
61	C₂×C₇	C₂×D₂₈	1	8	yes	—
62	C₂×C₇	#1157	1	8	yes	—
63	C₂×C₇	C₂×C₂²⋊D₇	1	8	yes	—
64	D₇	—	28	16	no	—
65	C₂³	—	49	2	no	—
66	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
67	D₄	—	98	4	no	—
68	D₄	—	98	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	49	2	no	—
71	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
72	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
73	C₇	#6493	1	8	yes	complemented
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	98	2	no	—
77	C₂²	—	98	2	no	—
78	C₂²	—	98	2	no	—
79	C₂²	C₄⋊(C₇⋊D₇)	1	2	yes	complemented
80	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	98	2	no	—
83	C₄	C₂²⋊(C₇⋊D₇)	1	2	yes	complemented
84	C₂	—	2	2	no	—
85	C₂	—	196	2	no	—
86	C₂	C₂×C₄⋊(C₇⋊D₇)	1	1	yes	characteristic
87	C₂	#11716129	1	1	yes	characteristic
88	C₂	C₂×C₂²⋊(C₇⋊D₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₇⋊D₇
commutator, socle, cosocle	C₂²×C₇²	C₂³
Fitting	C₂²×C₄×C₇²	C₂
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	#121
LCS	G ▷ C₂²×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×C₇²	C₂	semidirect
2. C₂×C₂²⋊(C₇⋊D₇)	C₂	semidirect
3. C₂×C₄⋊(C₇⋊D₇)	C₂	semidirect
4. #11716070	C₂	semidirect
5. C₄⋊(C₇⋊Dic₇)	C₂	semidirect
17. C₂²×C₇²	D₄	semidirect
23. C₄×C₇²	D₄	semidirect
25. C₂²×C₄×C₇	D₇	semidirect
43. C₇²	#121	semidirect
47. C₂²×C₇	D₂₈	semidirect
53. C₄×C₇	C₂²⋊D₇	semidirect
57. C₂²×C₄	C₇⋊D₇	semidirect
73. C₇	#6493	semidirect
79. C₂²	C₄⋊(C₇⋊D₇)	semidirect
83. C₄	C₂²⋊(C₇⋊D₇)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇²	2
C₂×C₂²⋊(C₇⋊D₇)	2
C₂×C₄⋊(C₇⋊D₇)	2
#11716070	2
C₄⋊(C₇⋊Dic₇)	2
#6493	7