Group #22561179 — subgroups

Subgroups — #22561179

Order 8576.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#14523954	C₂	1	1	yes	maximal, characteristic
2	#16863266	C₂	1	1	yes	maximal, characteristic
3	#16863158	C₂	1	4	yes	maximal
4	C₂×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₆₇	—	2	2	no	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	#14523954	—	2	2	no	—
9	#125×C₆₇	—	2	2	no	—
10	#125×C₆₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₆₇	—	2	2	no	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂×C₆₇⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₆₇	—	2	4	no	—
19	C₄⋊Dic₆₇	—	4	4	no	—
20	C₂²×C₄×C₆₇	—	2	2	no	—
21	C₂²×C₄×C₆₇	D₄	1	2	yes	—
22	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₆₇	—	2	8	no	—
24	C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
26	C₄²×C₆₇	C₂×C₄	1	2	yes	—
27	C₄²×C₆₇	D₄	1	2	yes	—
28	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	2	no	—
30	C₂×C₄×C₆₇	—	2	2	no	—
31	C₂×C₄×C₆₇	—	2	4	no	—
32	C₂×C₄×C₆₇	—	4	2	no	—
33	C₂×C₄×C₆₇	—	4	2	no	—
34	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₇	#41	1	2	yes	—
39	C₂×Dic₆₇	—	4	2	no	—
40	C₂×Dic₆₇	—	4	4	no	—
41	C₆₇⋊C₈	—	2	4	no	—
42	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₆₇	#124	1	1	yes	characteristic
44	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₆₇	#137	1	2	yes	—
47	C₄×C₆₇	—	2	4	no	—
48	C₄×C₆₇	—	4	4	no	—
49	C₄×C₆₇	C₄×Q₈	1	2	yes	—
50	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
51	Dic₆₇	—	8	2	no	—
52	C₂×C₆₇	#391	1	1	yes	characteristic
53	C₂×C₆₇	#417	1	2	yes	—
54	C₂×C₆₇	#490	1	1	yes	characteristic
55	C₂×C₆₇	#491	1	1	yes	characteristic
56	C₂×C₆₇	#501	1	2	yes	—
57	#1992	—	67	1	no	maximal, Sylow
58	C₆₇	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	D₆₇	1	1	yes	characteristic
60	C₂×#97	—	67	1	no	—
61	#384	—	67	1	no	—
62	#340	—	67	4	no	—
63	C₂²×C₈	—	67	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₆₇	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	67	2	no	—
68	C₂×C₄⋊C₄	—	134	2	no	—
69	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
70	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
71	#97	—	134	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	134	2	no	—
74	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
75	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
76	C₂×C₈	—	67	4	no	—
77	C₂×C₈	—	134	2	no	—
78	C₂×C₈	—	134	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	268	4	no	—
82	C₄⋊C₄	—	134	4	no	—
83	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
84	C₄²	C₂×Dic₆₇	1	2	yes	—
85	C₄²	C₂²⋊D₆₇	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	268	2	no	—
93	C₂×C₄	—	268	4	no	—
94	C₂×C₄	C₂²⋊Dic₆₇	1	4	yes	—
95	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₆₇	1	2	yes	—
98	C₂×C₄	#11761629	1	2	yes	—
99	C₈	—	134	4	no	—
100	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
101	C₂²	#14524032	1	1	yes	characteristic
102	C₂²	#14524041	1	1	yes	characteristic
103	C₂²	#14524068	1	2	yes	—
104	C₂²	#14524073	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	536	2	no	—
108	C₄	Q₈×Dic₆₇	1	2	yes	—
109	C₄	#14524019	1	2	yes	—
110	C₂	#16863303	1	1	yes	characteristic
111	C₂	#16863397	1	2	yes	—
112	C₂	#16863762	1	1	yes	characteristic
113	C₂	#16863766	1	1	yes	characteristic
114	C₂	#16863806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523954

C₂

yes

maximal, characteristic

#16863266

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal

C₂×#125×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523954

—

#125×C₆₇

—

#125×C₆₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#124

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

Dic₆₇

—

C₂×C₆₇

#391

yes

characteristic

C₂×C₆₇

#417

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#1992

—

maximal, Sylow

C₆₇

#1992

yes

characteristic, complemented, Sylow

C₂×#125

D₆₇

yes

characteristic

C₂×#97

—

#384

—

#340

—

C₂²×C₈

—

#125

—

#125

Dic₆₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#97

—

134

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

C₂²⋊Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

Q₈×D₆₇

yes

—

C₂×C₄

#11761629

yes

—

C₈

—

134

—

100

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

101

C₂²

#14524032

yes

characteristic

102

C₂²

#14524041

yes

characteristic

103

C₂²

#14524068

yes

—

104

C₂²

#14524073

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

536

—

108

C₄

Q₈×Dic₆₇

yes

—

109

C₄

#14524019

yes

—

110

C₂

#16863303

yes

characteristic

111

C₂

#16863397

yes

—

112

C₂

#16863762

yes

characteristic

113

C₂

#16863766

yes

characteristic

114

C₂

#16863806

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#125×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#125	D₆₇
hypocenter	C₆₇	#1992
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#125×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₆₇	C₄	semidirect
58. C₆₇	#1992	semidirect
65. #125	Dic₆₇	semidirect

H ≅ G/N

type

10. #125×C₆₇

C₄

semidirect

58. C₆₇

#1992

semidirect

65. #125

Dic₆₇

semidirect

Subgroups — #22561179

Conjugacy classes of subgroups

Complements