Group #22561925 — subgroups

Subgroups — #22561925

Order 8576.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863434	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863197	C₂	1	4	yes	maximal, complemented
4	C₂²×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524002	—	2	1	no	—
6	#14523983	—	2	2	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
8	#14523977	C₂²	1	2	yes	complemented
9	#14523975	—	2	1	no	—
10	C₂²×D₄×C₆₇	—	2	1	no	—
11	C₂×D₈×C₆₇	—	2	1	no	—
12	C₂×D₈×C₆₇	—	2	2	no	—
13	C₂×D₈×C₆₇	C₄	1	4	yes	complemented
14	C₂×D₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₆₇	—	2	2	no	—
17	C₂²⋊Dic₆₇	—	4	1	no	—
18	C₄×Dic₆₇	—	4	1	no	—
19	C₆₇⋊C₁₆	—	2	2	no	—
20	C₂⁴×C₆₇	—	4	1	no	—
21	C₂×D₄×C₆₇	—	2	4	no	—
22	C₂×D₄×C₆₇	—	2	2	no	—
23	C₂×D₄×C₆₇	—	4	1	no	—
24	C₂×D₄×C₆₇	—	4	2	no	—
25	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
26	D₈×C₆₇	—	2	4	no	—
27	D₈×C₆₇	—	2	2	no	—
28	D₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	D₄	1	2	yes	—
32	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₆₇	—	4	4	no	—
34	C₂³×C₆₇	—	4	2	no	—
35	C₂³×C₆₇	—	8	4	no	—
36	C₂³×C₆₇	D₈	1	1	yes	characteristic
37	D₄×C₆₇	—	2	4	no	—
38	D₄×C₆₇	—	4	4	no	—
39	D₄×C₆₇	—	4	2	no	—
40	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₆₇	—	4	2	no	—
45	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₆₇	—	2	2	no	—
49	C₂²×C₆₇	—	4	4	no	—
50	C₂²×C₆₇	—	8	8	no	—
51	C₂²×C₆₇	—	8	4	no	—
52	C₂²×C₆₇	#102	1	2	yes	—
53	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₆₇	#102	1	2	yes	—
55	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₆₇	—	8	2	no	—
58	C₂×C₆₇	—	2	2	no	—
59	C₂×C₆₇	—	8	4	no	—
60	C₂×C₆₇	C₂×#102	1	1	yes	characteristic
61	C₂×C₆₇	#509	1	2	yes	—
62	#2140	—	67	1	no	maximal, Sylow
63	C₆₇	#2140	1	1	yes	characteristic, complemented, Sylow
64	C₂²×D₈	D₆₇	1	1	yes	characteristic
65	C₂×M₃₂	—	67	1	no	—
66	#427	—	67	1	no	—
67	#357	—	67	4	no	—
68	C₂²×D₄	—	2	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	Dic₆₇	1	4	yes	complemented
72	C₂×D₈	C₂×D₆₇	1	2	yes	—
73	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
74	#117	—	134	1	no	—
75	M₃₂	—	134	2	no	—
76	C₂×C₁₆	—	67	2	no	—
77	#107	—	67	2	no	—
78	#106	—	134	1	no	—
79	C₂⁴	—	4	1	no	—
80	C₂×D₄	—	2	4	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
85	D₈	—	2	4	no	—
86	D₈	—	2	2	no	—
87	D₈	C₂×Dic₆₇	1	4	yes	—
88	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
91	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₄⋊C₄	—	134	2	no	—
93	C₂²⋊C₄	—	268	1	no	—
94	C₄²	—	268	1	no	—
95	C₁₆	—	134	2	no	—
96	C₂³	—	4	4	no	—
97	C₂³	—	4	2	no	—
98	C₂³	—	8	4	no	—
99	C₂³	#11761613	1	1	yes	characteristic
100	D₄	—	2	4	no	—
101	D₄	—	4	4	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	268	2	no	—
104	C₂×C₄	C₂²⋊Dic₆₇	1	2	yes	—
105	C₂×C₄	#11761613	1	1	yes	characteristic
106	C₂×C₄	#11761616	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
108	C₈	C₂²⋊Dic₆₇	1	2	yes	—
109	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
110	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	8	no	—
114	C₂²	—	8	4	no	—
115	C₂²	#14523964	1	2	yes	—
116	C₂²	C₂×#11761613	1	1	yes	characteristic
117	C₄	—	536	2	no	—
118	C₄	#14523964	1	2	yes	—
119	C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
120	C₄	C₂×#11761616	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	8	4	no	—
123	C₂	C₂×#14523964	1	1	yes	characteristic
124	C₂	#16863832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic, complemented

#16863434

C₂

yes

maximal, characteristic, complemented

#16863197

C₂

yes

maximal, complemented

C₂²×D₈×C₆₇

C₂

yes

maximal, characteristic

#14524002

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₂²

yes

complemented

#14523977

C₂²

yes

complemented

#14523975

—

C₂²×D₄×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

C₄

yes

complemented

C₂×D₈×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

D₈

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#102

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

#102

yes

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#102

yes

characteristic

C₂×C₆₇

#509

yes

—

#2140

—

maximal, Sylow

C₆₇

#2140

yes

characteristic, complemented, Sylow

C₂²×D₈

D₆₇

yes

characteristic

C₂×M₃₂

—

#427

—

#357

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₆₇

yes

complemented

C₂×D₈

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

134

—

M₃₂

—

134

—

C₂×C₁₆

—

#107

—

#106

—

134

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

D₈

—

D₈

—

D₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₄²

—

268

—

C₁₆

—

134

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#11761613

yes

characteristic

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

268

—

104

C₂×C₄

C₂²⋊Dic₆₇

yes

—

105

C₂×C₄

#11761613

yes

characteristic

106

C₂×C₄

#11761616

yes

—

107

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

108

C₈

C₂²⋊Dic₆₇

yes

—

109

C₈

C₂²×Dic₆₇

yes

characteristic

110

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#14523964

yes

—

116

C₂²

C₂×#11761613

yes

characteristic

117

C₄

—

536

—

118

C₄

#14523964

yes

—

119

C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

120

C₄

C₂×#11761616

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₂×#14523964

yes

characteristic

124

C₂

#16863832

yes

—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂²×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂²×D₈	D₆₇
hypocenter	C₆₇	#2140
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

C₂×#11761613

commutator

C₈×C₆₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂²×D₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

hypercenter

C₂²×D₈

D₆₇

hypocenter

C₆₇

#2140

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#14523983	C₂	semidirect
2. #16863434	C₂	semidirect
3. #16863197	C₂	semidirect
7. C₂×C₆₇⋊C₁₆	C₂²	semidirect
8. #14523977	C₂²	semidirect
13. C₂×D₈×C₆₇	C₄	semidirect
63. C₆₇	#2140	semidirect
71. C₂×D₈	Dic₆₇	semidirect

H ≅ G/N

type

1. C₂×#14523983

C₂

semidirect

2. #16863434

C₂

semidirect

3. #16863197

C₂

semidirect

7. C₂×C₆₇⋊C₁₆

C₂²

semidirect

8. #14523977

C₂²

semidirect

13. C₂×D₈×C₆₇

C₄

semidirect

63. C₆₇

#2140

semidirect

71. C₂×D₈

Dic₆₇

semidirect

Subgroups — #22561925

Conjugacy classes of subgroups

Complements