Subgroups — #13096551

Order 1568.

340 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716131	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716072	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716071	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716062	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716061	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇⋊D₇	—	2	1	no	—
9	C₇²×D₄	C₂²	1	1	yes	characteristic
10	C₇²×D₄	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊(C₇⋊D₇)	—	2	1	no	—
12	C₂²⋊(C₇⋊D₇)	—	2	1	no	—
13	C₂×C₇⋊Dic₇	—	2	1	no	—
14	C₄⋊(C₇⋊D₇)	C₂²	1	1	yes	characteristic
15	C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
16	#87648	C₂²	1	1	yes	characteristic, complemented
17	C₇²×C₈	C₂²	1	1	yes	characteristic, complemented
18	C₇⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
19	#6474	—	7	8	no	maximal
20	C₂²×C₇²	—	2	1	no	—
21	C₂²×C₇²	—	2	1	no	—
22	C₂×C₇⋊D₇	—	2	1	no	—
23	C₂×C₇⋊D₇	—	4	2	no	—
24	C₂×C₇⋊D₇	D₄	1	1	yes	characteristic
25	C₄×C₇²	C₂³	1	1	yes	characteristic
26	C₇⋊Dic₇	—	2	1	no	—
27	C₇⋊Dic₇	D₄	1	1	yes	characteristic
28	#1159	—	7	8	no	—
29	D₄×D₇	—	7	8	no	—
30	C₇×D₈	D₇	1	8	yes	complemented
31	#1142	—	7	8	no	—
32	#1141	—	7	8	no	—
33	#1132	—	7	8	no	—
34	#1131	—	7	8	no	—
35	C₂×C₇²	—	4	1	no	—
36	C₂×C₇²	—	4	1	no	—
37	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
38	C₇⋊D₇	—	2	1	no	—
39	C₇⋊D₇	—	4	1	no	—
40	C₂²×D₇	—	14	8	no	—
41	C₇×D₄	C₂×D₇	1	8	yes	—
42	C₇×D₄	C₂×D₇	1	8	yes	complemented
43	C₂²⋊D₇	—	14	8	no	—
44	C₂²⋊D₇	—	14	8	no	—
45	C₂×Dic₇	—	14	8	no	—
46	D₂₈	—	7	8	no	—
47	C₄×D₇	—	7	8	no	—
48	Dic₁₄	—	7	8	no	—
49	C₇×C₈	C₂×D₇	1	8	yes	complemented
50	C₇⋊C₈	—	7	8	no	—
51	C₇²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
52	AGL(1,ℤ/8)	—	49	1	no	Sylow
53	C₂²×C₇	—	2	8	no	—
54	C₂²×C₇	—	2	8	no	—
55	C₂×D₇	—	7	8	no	—
56	C₂×D₇	—	14	8	no	—
57	C₂×D₇	—	28	16	no	—
58	C₄×C₇	C₂²×D₇	1	8	yes	—
59	Dic₇	—	7	8	no	—
60	Dic₇	—	14	8	no	—
61	#41	—	49	1	no	—
62	C₂×D₄	—	49	1	no	—
63	SD₁₆	—	49	1	no	—
64	SD₁₆	—	49	1	no	—
65	D₈	—	49	1	no	—
66	D₈	C₇⋊D₇	1	1	yes	characteristic, complemented
67	M₁₆	—	49	1	no	—
68	C₂×C₇	—	4	8	no	—
69	C₂×C₇	—	4	8	no	—
70	C₂×C₇	D₄×D₇	1	8	yes	—
71	D₇	—	14	8	no	—
72	D₇	—	28	8	no	—
73	C₂³	—	98	1	no	—
74	Q₈	—	49	1	no	—
75	D₄	—	49	1	no	—
76	D₄	—	98	1	no	—
77	D₄	—	98	1	no	—
78	D₄	C₂×C₇⋊D₇	1	1	yes	characteristic
79	D₄	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented
80	C₂×C₄	—	49	1	no	—
81	C₂×C₄	—	98	1	no	—
82	C₈	—	49	1	no	—
83	C₈	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented
84	C₇	#6474	1	8	yes	complemented
85	C₂²	—	2	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	49	1	no	—
88	C₂²	—	98	1	no	—
89	C₂²	—	196	2	no	—
90	C₄	—	49	1	no	—
91	C₄	—	98	1	no	—
92	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
93	C₂	—	4	1	no	—
94	C₂	—	4	1	no	—
95	C₂	—	98	1	no	—
96	C₂	—	196	1	no	—
97	C₂	D₄×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₇⋊D₇
commutator, cosocle	C₄×C₇²	C₂³
Frattini	C₄	C₂²×C₇⋊D₇
Fitting	C₇²×D₈	C₂
socle	C₂×C₇²	C₂×D₄
hypercenter	D₈	C₇⋊D₇
hypocenter	C₇²	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #11716131	C₂	semidirect
2. D₄×C₇⋊D₇	C₂	semidirect
3. C₇²×D₈	C₂	semidirect
4. #11716072	C₂	semidirect
5. #11716071	C₂	semidirect
6. #11716062	C₂	semidirect
7. #11716061	C₂	semidirect
10. C₇²×D₄	C₂²	semidirect
16. #87648	C₂²	semidirect
17. C₇²×C₈	C₂²	semidirect
18. C₇⋊(C₇⋊C₈)	C₂²	semidirect
30. C₇×D₈	D₇	semidirect
42. C₇×D₄	C₂×D₇	semidirect
49. C₇×C₈	C₂×D₇	semidirect
51. C₇²	AGL(1,ℤ/8)	semidirect
66. D₈	C₇⋊D₇	semidirect
79. D₄	C₂×C₇⋊D₇	semidirect
83. C₈	C₂×C₇⋊D₇	semidirect
84. C₇	#6474	semidirect

Maximal subgroups

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

subgroup	index
#11716131	2
D₄×C₇⋊D₇	2
C₇²×D₈	2
#11716072	2
#11716071	2
#11716062	2
#11716061	2
#6474	7