Group #22559575 — subgroups

Subgroups — #22559575

Order 8576.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863677	C₂	1	2	yes	maximal
3	C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
4	#16863141	C₂	1	2	yes	maximal, complemented
5	#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₆₈	—	2	1	no	—
7	C₂×Dic₂₆₈	—	2	1	no	—
8	C₂×C₈×D₆₇	—	2	1	no	—
9	#14524043	C₂²	1	2	yes	—
10	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
11	#14523977	—	2	2	no	—
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#14523965	—	2	2	no	—
14	(C₂×D₆₇)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂×D₂₆₈	D₄	1	1	yes	characteristic
19	C₂×C₄×D₆₇	—	2	1	no	—
20	Dic₂₆₈	—	4	2	no	—
21	Dic₂₆₈	—	4	2	no	—
22	C₂×C₆₇⋊C₈	—	2	1	no	—
23	C₈×D₆₇	—	4	2	no	—
24	C₄⋊Dic₆₇	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
27	(C₂×D₆₇)⋊C₄	—	2	1	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×D₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
36	C₂×Dic₆₇	—	2	1	no	—
37	C₂×Dic₆₇	—	4	2	no	—
38	D₂₆₈	—	2	1	no	—
39	D₂₆₈	Q₁₆	1	2	yes	complemented
40	C₄×D₆₇	—	4	1	no	—
41	C₄×D₆₇	—	4	1	no	—
42	Dic₁₃₄	—	4	2	no	—
43	Dic₁₃₄	—	4	2	no	—
44	C₈×C₆₇	—	2	1	no	—
45	C₈×C₆₇	—	4	2	no	—
46	C₈×C₆₇	D₈	1	2	yes	—
47	C₆₇⋊C₈	—	4	1	no	—
48	C₂²×C₆₇	#121	1	1	yes	characteristic
49	C₂×D₆₇	—	2	2	no	—
50	C₂×D₆₇	—	4	1	no	—
51	C₂×D₆₇	—	4	1	no	—
52	C₄×C₆₇	—	2	1	no	—
53	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₆₇	#135	1	1	yes	characteristic
56	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	Dic₆₇	—	4	1	no	—
58	Dic₆₇	—	8	2	no	—
59	C₂×C₆₇	#459	1	1	yes	characteristic
60	C₂×C₆₇	#467	1	1	yes	characteristic
61	C₂×C₆₇	#473	1	1	yes	characteristic
62	D₆₇	—	4	2	no	—
63	#1669	—	67	1	no	maximal, Sylow
64	C₆₇	#1669	1	1	yes	characteristic, complemented, Sylow
65	#494	—	67	1	no	—
66	#474	—	67	2	no	—
67	C₈×D₄	—	67	1	no	—
68	#335	D₆₇	1	1	yes	characteristic, complemented
69	#332	—	67	2	no	—
70	C₂×Q₁₆	—	134	1	no	—
71	C₂×Q₁₆	—	134	1	no	—
72	C₂²×C₈	—	134	1	no	—
73	C₄⋊Q₈	—	67	2	no	—
74	C₄×D₄	—	67	1	no	—
75	#107	—	134	2	no	—
76	C₄⋊C₈	—	67	1	no	—
77	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
78	#102	—	134	2	no	—
79	C₂²⋊C₈	—	134	1	no	—
80	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
81	C₂×Q₈	—	134	2	no	—
82	C₂×D₄	—	67	1	no	—
83	C₂²×C₄	—	134	1	no	—
84	Q₁₆	—	268	2	no	—
85	Q₁₆	—	268	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	268	2	no	—
88	C₂×C₈	—	134	1	no	—
89	C₂×C₈	D₂₆₈	1	1	yes	characteristic
90	C₂×C₈	D₂₆₈	1	1	yes	characteristic
91	C₄⋊C₄	—	67	1	no	—
92	C₄⋊C₄	—	134	2	no	—
93	C₄⋊C₄	—	134	2	no	—
94	C₂²⋊C₄	—	134	1	no	—
95	C₄²	C₂²×D₆₇	1	1	yes	characteristic
96	C₂³	—	134	1	no	—
97	Q₈	—	268	2	no	—
98	Q₈	—	268	2	no	—
99	D₄	—	67	2	no	—
100	D₄	—	134	1	no	—
101	C₂×C₄	—	268	1	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	268	2	no	—
104	C₂×C₄	—	134	1	no	—
105	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
107	C₂×C₄	#11761629	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	4	2	no	—
110	C₈	—	268	1	no	—
111	C₈	D₅₃₆	1	2	yes	—
112	C₂²	—	268	1	no	—
113	C₂²	—	268	1	no	—
114	C₂²	—	134	2	no	—
115	C₂²	#14524018	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	268	1	no	—
118	C₄	—	536	2	no	—
119	C₄	C₂×D₅₃₆	1	1	yes	characteristic
120	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
121	C₄	#14524062	1	1	yes	characteristic
122	C₄	#14524071	1	1	yes	characteristic
123	C₂	—	268	2	no	—
124	C₂	#16863592	1	1	yes	characteristic
125	C₂	#16863638	1	1	yes	characteristic
126	C₂	#16863672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863776

C₂

yes

maximal, characteristic, complemented

#16863677

C₂

yes

maximal

C₈×D₂₆₈

C₂

yes

maximal, characteristic

#16863141

C₂

yes

maximal, complemented

#335×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×C₈×D₆₇

—

#14524043

C₂²

yes

—

C₄×D₂₆₈

C₂²

yes

characteristic

#14523977

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523965

—

(C₂×D₆₇)⋊C₈

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₄×D₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

(C₂×D₆₇)⋊C₄

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

Q₁₆

yes

complemented

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

D₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1669

—

maximal, Sylow

C₆₇

#1669

yes

characteristic, complemented, Sylow

#494

—

#474

—

C₈×D₄

—

#335

D₆₇

yes

characteristic, complemented

#332

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂²×C₈

—

134

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

—

#102

—

134

—

C₂²⋊C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

C₂×D₄

—

C₂²×C₄

—

134

—

Q₁₆

—

268

—

Q₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

134

—

Q₈

—

268

—

Q₈

—

268

—

D₄

—

100

D₄

—

134

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

268

—

104

C₂×C₄

—

134

—

105

C₂×C₄

D₄×D₆₇

yes

characteristic

106

C₂×C₄

C₂×D₂₆₈

yes

characteristic

107

C₂×C₄

#11761629

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

268

—

111

C₈

D₅₃₆

yes

—

112

C₂²

—

268

—

113

C₂²

—

268

—

114

C₂²

—

134

—

115

C₂²

#14524018

yes

characteristic

116

C₄

—

117

C₄

—

268

—

118

C₄

—

536

—

119

C₄

C₂×D₅₃₆

yes

characteristic

120

C₄

Q₁₆×D₆₇

yes

characteristic

121

C₄

#14524062

yes

characteristic

122

C₄

#14524071

yes

characteristic

123

C₂

—

268

—

124

C₂

#16863592

yes

characteristic

125

C₂

#16863638

yes

characteristic

126

C₂

#16863672

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#335×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#335	D₆₇
hypocenter	C₆₇	#1669
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14524018

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#335×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16863776	C₂	semidirect
4. #16863141	C₂	semidirect
5. #335×C₆₇	C₂	semidirect
39. D₂₆₈	Q₁₆	semidirect
64. C₆₇	#1669	semidirect
68. #335	D₆₇	semidirect

H ≅ G/N

type

1. #16863776

C₂

semidirect

4. #16863141

C₂

semidirect

5. #335×C₆₇

C₂

semidirect

39. D₂₆₈

Q₁₆

semidirect

64. C₆₇

#1669

semidirect

68. #335

D₆₇

semidirect

Subgroups — #22559575

Conjugacy classes of subgroups

Complements