Group #22562365 — subgroups

Subgroups — #22562365

Order 8576.

136 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16863797	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863478	C₂	1	1	yes	maximal, characteristic
3	#16863221	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863198	C₂	1	1	yes	maximal, characteristic
6	#16863196	C₂	1	1	yes	maximal, characteristic
7	#367×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
9	#14524034	—	2	1	no	—
10	#14524005	—	2	1	no	—
11	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14523977	C₂²	1	1	yes	characteristic
13	#14523974	—	2	1	no	—
14	#14523963	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#107×C₆₇	C₂²	1	1	yes	characteristic
18	#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×D₂₆₈	—	2	1	no	—
20	C₂×C₄×D₆₇	—	4	1	no	—
21	D₅₃₆	—	2	1	no	—
22	D₅₃₆	Q₈	1	2	yes	—
23	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₆₇	—	2	1	no	—
25	C₄⋊Dic₆₇	—	4	1	no	—
26	Dic₆₇⋊C₄	—	4	1	no	—
27	(C₂×D₆₇)⋊C₄	—	4	1	no	—
28	C₄×Dic₆₇	D₄	1	1	yes	characteristic
29	C₆₇⋊C₁₆	—	2	1	no	—
30	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₁₆×C₆₇	—	2	1	no	—
34	C₂²×D₆₇	—	4	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×Dic₆₇	—	4	1	no	—
39	C₂×Dic₆₇	D₈	1	1	yes	characteristic
40	C₂×Dic₆₇	D₈	1	1	yes	characteristic
41	D₂₆₈	—	2	2	no	—
42	D₂₆₈	—	4	1	no	—
43	C₄×D₆₇	—	8	2	no	—
44	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₇	#41	1	1	yes	characteristic
46	C₆₇⋊C₈	—	2	1	no	—
47	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
48	C₂×D₆₇	—	4	2	no	—
49	C₂×D₆₇	—	8	1	no	—
50	C₂×D₆₇	—	8	1	no	—
51	C₄×C₆₇	—	8	1	no	—
52	C₄×C₆₇	—	8	1	no	—
53	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₆₇	#137	1	1	yes	characteristic
55	Dic₆₇	—	2	1	no	—
56	Dic₆₇	—	2	1	no	—
57	Dic₆₇	—	8	1	no	—
58	C₂×C₆₇	#474	1	1	yes	characteristic
59	C₂×C₆₇	#508	1	1	yes	characteristic
60	C₂×C₆₇	#509	1	1	yes	characteristic
61	D₆₇	—	8	2	no	—
62	#2228	—	67	1	no	maximal, Sylow
63	C₆₇	#2228	1	1	yes	characteristic, complemented, Sylow
64	#499	—	67	1	no	—
65	C₄×D₈	—	67	1	no	—
66	#367	D₆₇	1	1	yes	characteristic, complemented
67	#366	—	67	1	no	—
68	C₄⋊C₁₆	—	67	1	no	—
69	#357	—	67	1	no	—
70	#357	—	67	1	no	—
71	C₂×D₈	—	67	1	no	—
72	#125	—	134	1	no	—
73	C₄×D₄	—	134	1	no	—
74	C₂×C₁₆	—	67	1	no	—
75	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
76	#107	—	67	1	no	—
77	#107	C₂×D₆₇	1	1	yes	characteristic
78	#107	C₂×D₆₇	1	1	yes	characteristic
79	#106	—	134	1	no	—
80	#102	—	134	1	no	—
81	C₄×C₈	—	67	1	no	—
82	C₂×D₄	—	134	1	no	—
83	C₂²×C₄	—	268	1	no	—
84	D₈	—	67	2	no	—
85	D₈	—	134	1	no	—
86	C₂×C₈	—	67	1	no	—
87	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	268	1	no	—
91	C₄⋊C₄	—	268	1	no	—
92	C₄⋊C₄	—	134	1	no	—
93	C₂²⋊C₄	—	268	1	no	—
94	C₄²	—	67	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	134	1	no	—
97	C₂³	—	268	1	no	—
98	D₄	—	268	1	no	—
99	D₄	—	134	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	536	2	no	—
104	C₂×C₄	—	67	1	no	—
105	C₂×C₄	—	67	1	no	—
106	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
107	C₈	—	134	1	no	—
108	C₈	Q₈×D₆₇	1	1	yes	characteristic
109	C₈	#11761628	1	1	yes	characteristic
110	C₂²	—	268	2	no	—
111	C₂²	—	536	1	no	—
112	C₂²	—	536	1	no	—
113	C₂²	D₈×D₆₇	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	8	1	no	—
116	C₄	—	536	1	no	—
117	C₄	—	134	1	no	—
118	C₄	—	134	1	no	—
119	C₄	#14524021	1	1	yes	characteristic
120	C₄	#14524075	1	1	yes	characteristic
121	C₂	—	536	2	no	—
122	C₂	#16863675	1	1	yes	characteristic
123	C₂	#16863827	1	1	yes	characteristic
124	C₂	#16863833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863797

C₂

yes

maximal, characteristic, complemented

#16863478

C₂

yes

maximal, characteristic

#16863221

C₂

yes

maximal, characteristic, complemented

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16863198

C₂

yes

maximal, characteristic

#16863196

C₂

yes

maximal, characteristic

#367×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

C₂²

yes

characteristic

#14524034

—

#14524005

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523974

—

#14523963

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

—

C₂×C₄×D₆₇

—

D₅₃₆

—

D₅₃₆

Q₈

yes

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#137

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#2228

—

maximal, Sylow

C₆₇

#2228

yes

characteristic, complemented, Sylow

#499

—

C₄×D₈

—

#367

D₆₇

yes

characteristic, complemented

#366

—

C₄⋊C₁₆

—

#357

—

#357

—

C₂×D₈

—

#125

—

134

—

C₄×D₄

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

—

#107

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

characteristic

#106

—

134

—

#102

—

134

—

C₄×C₈

—

C₂×D₄

—

134

—

C₂²×C₄

—

268

—

D₈

—

D₈

—

134

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

C₂³

—

268

—

D₄

—

268

—

D₄

—

134

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

536

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

D₄×D₆₇

yes

characteristic

107

C₈

—

134

—

108

C₈

Q₈×D₆₇

yes

characteristic

109

C₈

#11761628

yes

characteristic

110

C₂²

—

268

—

111

C₂²

—

536

—

112

C₂²

—

536

—

113

C₂²

D₈×D₆₇

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

536

—

117

C₄

—

134

—

118

C₄

—

134

—

119

C₄

#14524021

yes

characteristic

120

C₄

#14524075

yes

characteristic

121

C₂

—

536

—

122

C₂

#16863675

yes

characteristic

123

C₂

#16863827

yes

characteristic

124

C₂

#16863833

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#367×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#367	D₆₇
hypocenter	C₆₇	#2228
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#367×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #16863797	C₂	semidirect
3. #16863221	C₂	semidirect
4. Dic₆₇⋊C₁₆	C₂	semidirect
7. #367×C₆₇	C₂	semidirect
63. C₆₇	#2228	semidirect
66. #367	D₆₇	semidirect

H ≅ G/N

type

1. #16863797

C₂

semidirect

3. #16863221

C₂

semidirect

4. Dic₆₇⋊C₁₆

C₂

semidirect

7. #367×C₆₇

C₂

semidirect

63. C₆₇

#2228

semidirect

66. #367

D₆₇

semidirect

Subgroups — #22562365

Conjugacy classes of subgroups

Complements