Group #22562494 — subgroups

Subgroups — #22562494

Order 8576.

120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863800	C₂	1	1	yes	maximal, characteristic
2	#16863797	C₂	1	1	yes	maximal, characteristic
3	#16863223	C₂	1	1	yes	maximal, characteristic
4	#16863222	C₂	1	1	yes	maximal, characteristic
5	#16863221	C₂	1	1	yes	maximal, characteristic
6	#16863171	C₂	1	1	yes	maximal, characteristic
7	#367×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524042	—	2	1	no	—
9	#14524034	—	2	1	no	—
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14523977	C₂²	1	1	yes	characteristic
12	#14523977	C₂²	1	1	yes	characteristic
13	#14523976	—	2	1	no	—
14	#14523974	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#107×C₆₇	C₂²	1	1	yes	characteristic
18	#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₄	—	4	1	no	—
20	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₆₇	—	2	1	no	—
22	C₄⋊Dic₆₇	—	2	1	no	—
23	C₄⋊Dic₆₇	—	4	1	no	—
24	Dic₆₇⋊C₄	—	4	1	no	—
25	Dic₆₇⋊C₄	—	4	1	no	—
26	C₄×Dic₆₇	D₄	1	1	yes	characteristic
27	C₆₇⋊C₁₆	Q₈	1	2	yes	complemented
28	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₁₆×C₆₇	Q₈	1	2	yes	complemented
32	C₂×C₄×C₆₇	—	4	1	no	—
33	C₂×C₄×C₆₇	—	4	1	no	—
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₆₇	—	4	1	no	—
36	C₂×Dic₆₇	—	4	1	no	—
37	C₂×Dic₆₇	D₈	1	1	yes	characteristic
38	C₂×Dic₆₇	D₈	1	1	yes	characteristic
39	Dic₁₃₄	—	8	2	no	—
40	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₆₇⋊C₈	Q₁₆	1	2	yes	—
43	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
44	C₄×C₆₇	—	8	1	no	—
45	C₄×C₆₇	—	8	1	no	—
46	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₆₇	—	2	1	no	—
49	Dic₆₇	—	2	1	no	—
50	Dic₆₇	—	8	1	no	—
51	Dic₆₇	—	8	1	no	—
52	C₂×C₆₇	#500	1	1	yes	characteristic
53	C₂×C₆₇	#509	1	1	yes	characteristic
54	C₂×C₆₇	#510	1	1	yes	characteristic
55	#2255	—	67	1	no	maximal, Sylow
56	C₆₇	#2255	1	1	yes	characteristic, complemented, Sylow
57	#500	—	67	1	no	—
58	#499	—	67	1	no	—
59	#367	—	67	1	no	—
60	#367	D₆₇	1	1	yes	characteristic
61	#366	—	67	1	no	—
62	#366	—	67	1	no	—
63	#346	—	67	1	no	—
64	C₄⋊Q₈	—	134	1	no	—
65	#125	—	134	1	no	—
66	C₂×C₁₆	—	67	1	no	—
67	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
68	#107	—	67	1	no	—
69	#107	—	67	1	no	—
70	#107	—	134	1	no	—
71	#107	C₂×D₆₇	1	1	yes	characteristic
72	#107	C₂×D₆₇	1	1	yes	characteristic
73	#106	—	134	1	no	—
74	C₄×C₈	—	67	1	no	—
75	C₂×Q₈	—	268	1	no	—
76	C₂×C₈	—	67	1	no	—
77	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	268	1	no	—
81	C₄⋊C₄	—	268	1	no	—
82	C₄⋊C₄	—	268	1	no	—
83	C₄⋊C₄	—	134	1	no	—
84	C₄⋊C₄	—	134	1	no	—
85	C₄²	—	67	1	no	—
86	C₁₆	—	67	2	no	—
87	C₁₆	Dic₁₃₄	1	2	yes	complemented
88	Q₈	—	536	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	268	1	no	—
92	C₂×C₄	—	268	1	no	—
93	C₂×C₄	—	67	1	no	—
94	C₂×C₄	—	67	1	no	—
95	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
96	C₈	—	67	2	no	—
97	C₈	Q₈×D₆₇	1	1	yes	characteristic
98	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
99	C₂²	D₈×D₆₇	1	1	yes	characteristic
100	C₄	—	8	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	536	1	no	—
103	C₄	—	536	1	no	—
104	C₄	—	134	1	no	—
105	C₄	—	134	1	no	—
106	C₄	#14524042	1	1	yes	characteristic
107	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
108	C₂	#16863800	1	1	yes	characteristic
109	C₂	#16863833	1	1	yes	characteristic
110	C₂	#16863835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863800

C₂

yes

maximal, characteristic

#16863797

C₂

yes

maximal, characteristic

#16863223

C₂

yes

maximal, characteristic

#16863222

C₂

yes

maximal, characteristic

#16863221

C₂

yes

maximal, characteristic

#16863171

C₂

yes

maximal, characteristic

#367×C₆₇

C₂

yes

maximal, characteristic

#14524042

—

#14524034

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523976

—

#14523974

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

Q₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₆₇⋊C₈

Q₁₆

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

—

—

—

C₄⋊Q₈

—

134

—

#125

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

—

—

134

—

#107

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

characteristic

#106

—

134

—

C₄×C₈

—

C₂×Q₈

—

268

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

C₁₆

—

C₁₆

Dic₁₃₄

yes

complemented

Q₈

—

536

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₈

—

C₈

Q₈×D₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

D₈×D₆₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

536

—

103

C₄

—

536

—

104

C₄

—

134

—

105

C₄

—

134

—

106

C₄

#14524042

yes

characteristic

107

C₄

Q₁₆×D₆₇

yes

characteristic

108

C₂

#16863800

yes

characteristic

109

C₂

#16863833

yes

characteristic

110

C₂

#16863835

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#367×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#367	D₆₇
hypocenter	C₆₇	#2255
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#367×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
27. C₆₇⋊C₁₆	Q₈	semidirect
31. C₁₆×C₆₇	Q₈	semidirect
56. C₆₇	#2255	semidirect
87. C₁₆	Dic₁₃₄	semidirect

H ≅ G/N

type

27. C₆₇⋊C₁₆

Q₈

semidirect

31. C₁₆×C₆₇

Q₈

semidirect

56. C₆₇

#2255

semidirect

87. C₁₆

Dic₁₃₄

semidirect

Subgroups — #22562494

Conjugacy classes of subgroups

Complements