Group #22559598 — subgroups

Subgroups — #22559598

Order 8576.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16863801	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863601	C₂	1	2	yes	maximal
3	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863141	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761616	—	2	2	no	—
7	#14524043	C₂²	1	2	yes	—
8	#14523977	—	2	1	no	—
9	#14523977	—	2	1	no	—
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#14523966	—	2	2	no	—
12	C₂²×C₈×C₆₇	—	2	1	no	—
13	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	#11761616	—	4	4	no	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	—	2	1	no	—
24	C₂×C₈×C₆₇	—	2	1	no	—
25	C₂×C₈×C₆₇	—	4	2	no	—
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂²⋊C₄×C₆₇	—	2	1	no	—
30	C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₂³×C₆₇	—	2	1	no	—
32	D₄×C₆₇	—	2	1	no	—
33	D₄×C₆₇	D₈	1	2	yes	—
34	C₂×C₄×C₆₇	—	2	1	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	—	4	1	no	—
37	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
40	C₂×Dic₆₇	—	4	2	no	—
41	Dic₁₃₄	—	4	2	no	—
42	Dic₁₃₄	—	4	2	no	—
43	C₈×C₆₇	—	2	1	no	—
44	C₈×C₆₇	—	4	1	no	—
45	C₈×C₆₇	SD₁₆	1	2	yes	complemented
46	C₆₇⋊C₈	—	4	2	no	—
47	C₂²×C₆₇	—	2	2	no	—
48	C₂²×C₆₇	—	4	1	no	—
49	C₂²×C₆₇	—	4	1	no	—
50	C₂²×C₆₇	#121	1	1	yes	characteristic
51	C₄×C₆₇	—	2	1	no	—
52	C₄×C₆₇	—	4	1	no	—
53	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₆₇	#135	1	1	yes	characteristic
56	C₄×C₆₇	#137	1	1	yes	characteristic
57	Dic₆₇	—	8	2	no	—
58	C₂×C₆₇	—	4	2	no	—
59	C₂×C₆₇	#461	1	1	yes	characteristic
60	C₂×C₆₇	#466	1	1	yes	characteristic
61	C₂×C₆₇	#473	1	1	yes	characteristic
62	#1673	—	67	1	no	maximal, Sylow
63	C₆₇	#1673	1	1	yes	characteristic, complemented, Sylow
64	#500	—	67	1	no	—
65	#461	—	67	2	no	—
66	C₈×D₄	D₆₇	1	1	yes	characteristic
67	#335	—	67	1	no	—
68	#332	—	67	2	no	—
69	C₂×SD₁₆	—	134	2	no	—
70	C₂²×C₈	—	2	1	no	—
71	C₄⋊Q₈	—	67	2	no	—
72	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
73	#107	—	134	1	no	—
74	#107	—	134	1	no	—
75	C₄⋊C₈	—	67	2	no	—
76	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
77	#103	—	134	2	no	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
80	C₂×Q₈	—	134	2	no	—
81	C₂×D₄	D₂₆₈	1	1	yes	characteristic
82	C₂²×C₄	—	2	1	no	—
83	SD₁₆	—	268	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	134	2	no	—
87	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
89	C₄⋊C₄	—	134	2	no	—
90	C₄⋊C₄	—	134	2	no	—
91	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
92	C₂²⋊C₄	—	2	1	no	—
93	C₄²	C₂²×D₆₇	1	1	yes	characteristic
94	C₂³	—	2	1	no	—
95	Q₈	—	268	2	no	—
96	Q₈	—	268	2	no	—
97	D₄	—	2	1	no	—
98	D₄	D₅₃₆	1	2	yes	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	268	2	no	—
103	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
105	C₂×C₄	#11761628	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	4	1	no	—
108	C₈	—	268	2	no	—
109	C₈	#11761616	1	2	yes	complemented
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	1	no	—
113	C₂²	#14524017	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	4	1	no	—
116	C₄	—	536	2	no	—
117	C₄	C₂×D₅₃₆	1	1	yes	characteristic
118	C₄	C₂×#11761616	1	1	yes	characteristic
119	C₄	#14524065	1	1	yes	characteristic
120	C₄	#14524073	1	1	yes	characteristic
121	C₂	—	4	2	no	—
122	C₂	#16863601	1	1	yes	characteristic
123	C₂	#16863632	1	1	yes	characteristic
124	C₂	#16863671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863801

C₂

yes

maximal, characteristic, complemented

#16863601

C₂

yes

maximal

#16863149

C₂

yes

maximal, characteristic, complemented

#16863141

C₂

yes

maximal, complemented

C₈×D₄×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

—

#14524043

C₂²

yes

—

#14523977

—

#14523977

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523966

—

C₂²×C₈×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

#11761616

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#500

—

#461

—

C₈×D₄

D₆₇

yes

characteristic

#335

—

#332

—

C₂×SD₁₆

—

134

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₆₇

yes

characteristic

#107

—

134

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

C₂×D₄

D₂₆₈

yes

characteristic

C₂²×C₄

—

SD₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

D₂₆₈

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

Q₈

—

268

—

Q₈

—

268

—

D₄

—

D₄

D₅₃₆

yes

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

268

—

103

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

104

C₂×C₄

C₂×D₂₆₈

yes

characteristic

105

C₂×C₄

#11761628

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

268

—

109

C₈

#11761616

yes

complemented

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14524017

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

536

—

117

C₄

C₂×D₅₃₆

yes

characteristic

118

C₄

C₂×#11761616

yes

characteristic

119

C₄

#14524065

yes

characteristic

120

C₄

#14524073

yes

characteristic

121

C₂

—

122

C₂

#16863601

yes

characteristic

123

C₂

#16863632

yes

characteristic

124

C₂

#16863671

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈×D₄×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#1673
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14524017

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

C₈×D₄×C₆₇

C₂

socle

C₂²×C₆₇

#121

hypercenter

C₈×D₄

D₆₇

hypocenter

C₆₇

#1673

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #16863801	C₂	semidirect
3. #16863149	C₂	semidirect
4. #16863141	C₂	semidirect
45. C₈×C₆₇	SD₁₆	semidirect
63. C₆₇	#1673	semidirect
109. C₈	#11761616	semidirect

H ≅ G/N

type

1. #16863801

C₂

semidirect

3. #16863149

C₂

semidirect

4. #16863141

C₂

semidirect

45. C₈×C₆₇

SD₁₆

semidirect

63. C₆₇

#1673

semidirect

109. C₈

#11761616

semidirect

Subgroups — #22559598

Conjugacy classes of subgroups

Complements