Group #22561477 — subgroups

Subgroups — #22561477

Order 8576.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#16863404	C₂	1	2	yes	maximal
3	C₂×#14523961	C₂	1	2	yes	maximal
4	#16863161	C₂	1	1	yes	maximal, characteristic
5	#424×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	C₂×Dic₂₆₈	—	2	4	no	—
8	C₂×Dic₂₆₈	—	2	2	no	—
9	C₂×#11761612	C₂²	1	2	yes	—
10	#14523966	—	2	4	no	—
11	#14523961	—	2	4	no	—
12	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	—	2	2	no	—
16	C₂×Dic₁₃₄	—	2	2	no	—
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	D₄	1	4	yes	—
22	C₂²×Dic₆₇	—	2	2	no	—
23	Dic₂₆₈	—	2	8	no	—
24	#11761612	—	2	2	no	—
25	#11761612	—	2	2	no	—
26	C₂×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	1	no	—
31	C₂×C₈×C₆₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₆₇	—	2	1	no	—
34	C₄²×C₆₇	—	2	1	no	—
35	C₂³×C₆₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	#41	1	2	yes	—
41	C₂×Dic₆₇	—	2	4	no	—
42	C₂×Dic₆₇	—	4	2	no	—
43	C₂×Dic₆₇	—	4	2	no	—
44	Dic₁₃₄	—	2	4	no	—
45	Dic₁₃₄	—	2	4	no	—
46	Dic₁₃₄	—	4	2	no	—
47	Dic₁₃₄	—	4	2	no	—
48	C₈×C₆₇	—	2	2	no	—
49	C₈×C₆₇	—	4	1	no	—
50	C₆₇⋊C₈	—	4	2	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	#121	1	1	yes	characteristic
53	C₂²×C₆₇	#123	1	2	yes	—
54	C₄×C₆₇	—	4	2	no	—
55	C₄×C₆₇	#120	1	1	yes	characteristic
56	C₄×C₆₇	#121	1	2	yes	—
57	C₄×C₆₇	#124	1	1	yes	characteristic
58	Dic₆₇	—	4	4	no	—
59	C₂×C₆₇	—	2	2	no	—
60	C₂×C₆₇	#394	1	1	yes	characteristic
61	C₂×C₆₇	#473	1	2	yes	—
62	#2047	—	67	1	no	maximal, Sylow
63	C₆₇	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	67	1	no	—
65	#424	D₆₇	1	1	yes	characteristic
66	#419	—	67	2	no	—
67	C₂×#101	—	67	2	no	—
68	#341	—	67	1	no	—
69	C₂²×Q₈	—	67	2	no	—
70	C₂×Q₁₆	—	134	4	no	—
71	C₂×Q₁₆	—	134	2	no	—
72	C₂×M₁₆	—	67	2	no	—
73	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
75	#117	C₂×D₆₇	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	134	4	no	—
78	#101	—	134	4	no	—
79	C₂×Q₈	—	67	4	no	—
80	C₂×Q₈	—	134	2	no	—
81	C₂×Q₈	—	134	2	no	—
82	C₂×Q₈	—	134	2	no	—
83	C₂×Q₈	—	134	2	no	—
84	C₂×Q₈	—	134	2	no	—
85	C₂²×C₄	—	134	2	no	—
86	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
87	Q₁₆	—	134	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	134	2	no	—
90	M₁₆	—	134	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	134	2	no	—
94	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
95	C₄⋊C₄	D₂₆₈	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#11761628	1	1	yes	characteristic
99	Q₈	—	268	2	no	—
100	Q₈	—	268	2	no	—
101	Q₈	—	134	4	no	—
102	Q₈	—	134	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	268	2	no	—
105	C₂×C₄	—	268	2	no	—
106	C₂×C₄	—	134	4	no	—
107	C₂×C₄	D₄×D₆₇	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
110	C₂×C₄	#11761629	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	268	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#14524017	1	1	yes	characteristic
116	C₂²	#14524028	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	268	4	no	—
119	C₄	#14524011	1	1	yes	characteristic
120	C₄	#14524018	1	2	yes	—
121	C₄	#14524032	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#16863313	1	1	yes	characteristic
124	C₂	#16863671	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

#16863404

C₂

yes

maximal

C₂×#14523961

C₂

yes

maximal

#16863161

C₂

yes

maximal, characteristic

#424×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×#11761612

C₂²

yes

—

#14523966

—

#14523961

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#123

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#394

yes

characteristic

C₂×C₆₇

#473

yes

—

#2047

—

maximal, Sylow

C₆₇

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#424

D₆₇

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

#103

—

134

—

#101

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

Q₁₆

—

134

—

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

D₂₆₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#11761628

yes

characteristic

Q₈

—

268

—

100

Q₈

—

268

—

101

Q₈

—

134

—

102

Q₈

—

134

—

103

C₂×C₄

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

268

—

106

C₂×C₄

—

134

—

107

C₂×C₄

D₄×D₆₇

yes

—

108

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

109

C₂×C₄

C₂×D₂₆₈

yes

characteristic

110

C₂×C₄

#11761629

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

268

—

114

C₂²

—

115

C₂²

#14524017

yes

characteristic

116

C₂²

#14524028

yes

—

117

C₄

—

118

C₄

—

268

—

119

C₄

#14524011

yes

characteristic

120

C₄

#14524018

yes

—

121

C₄

#14524032

yes

characteristic

122

C₂

—

123

C₂

#16863313

yes

characteristic

124

C₂

#16863671

yes

—

	subgroup	G/H
center	C₂²	#14524017
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#424×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#424	D₆₇
hypocenter	C₆₇	#2047
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#14524017

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#424×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
63. C₆₇	#2047	semidirect

H ≅ G/N

type

63. C₆₇

#2047

semidirect

Subgroups — #22561477

Conjugacy classes of subgroups

Complements