Group #22562296 — subgroups

Subgroups — #22562296

Order 8576.

174 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 51 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×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863633	C₂	1	2	yes	maximal, complemented
4	#16863196	C₂	1	2	yes	maximal, complemented
5	#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₃₆	C₂²	1	2	yes	—
7	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	#14524018	—	2	2	no	—
9	D₁₀₇₂	—	2	2	no	—
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₆×D₆₇	—	2	2	no	—
12	#14523963	—	2	2	no	—
13	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#107×C₆₇	C₂²	1	2	yes	—
15	C₂×D₂₆₈	—	2	2	no	—
16	C₂×D₂₆₈	—	4	2	no	—
17	C₂×C₄×D₆₇	D₄	1	1	yes	characteristic
18	D₅₃₆	—	2	2	no	—
19	D₅₃₆	—	2	2	no	—
20	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
21	C₈×D₆₇	—	2	1	no	—
22	C₈×D₆₇	—	2	1	no	—
23	(C₂×D₆₇)⋊C₄	—	4	2	no	—
24	C₆₇⋊C₁₆	—	2	1	no	—
25	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₁₆×C₆₇	D₄	1	2	yes	complemented
28	C₂²×D₆₇	—	4	2	no	—
29	C₂²×D₆₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	4	2	no	—
31	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₆₇	D₈	1	1	yes	characteristic
33	D₂₆₈	—	4	2	no	—
34	D₂₆₈	—	4	2	no	—
35	D₂₆₈	—	8	4	no	—
36	C₄×D₆₇	—	2	1	no	—
37	C₄×D₆₇	—	2	1	no	—
38	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₆₇	#41	1	1	yes	characteristic
40	C₆₇⋊C₈	—	2	1	no	—
41	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
42	C₂×D₆₇	—	2	1	no	—
43	C₂×D₆₇	—	2	1	no	—
44	C₂×D₆₇	—	8	2	no	—
45	C₂×D₆₇	—	8	2	no	—
46	C₂×D₆₇	—	8	2	no	—
47	C₂×D₆₇	D₁₆	1	2	yes	complemented
48	C₄×C₆₇	—	8	2	no	—
49	C₄×C₆₇	#121	1	1	yes	characteristic
50	C₄×C₆₇	#135	1	1	yes	characteristic
51	Dic₆₇	—	2	1	no	—
52	C₂×C₆₇	#466	1	1	yes	characteristic
53	C₂×C₆₇	C₂×D₁₆	1	1	yes	characteristic
54	C₂×C₆₇	#508	1	1	yes	characteristic
55	D₆₇	—	2	2	no	—
56	D₆₇	—	16	2	no	—
57	#2215	—	67	1	no	maximal, Sylow
58	C₆₇	#2215	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₁₆	—	67	1	no	—
60	C₂²×C₁₆	—	67	1	no	—
61	#466	—	67	2	no	—
62	#366	D₆₇	1	1	yes	characteristic, complemented
63	#357	—	67	2	no	—
64	C₂×D₈	—	67	2	no	—
65	C₂²×C₈	—	67	1	no	—
66	#121	—	134	2	no	—
67	D₁₆	—	134	2	no	—
68	C₂×C₁₆	—	67	1	no	—
69	C₂×C₁₆	—	134	2	no	—
70	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
71	#107	C₂×D₆₇	1	2	yes	—
72	#102	—	134	2	no	—
73	C₂×D₄	—	268	2	no	—
74	C₂×D₄	—	134	2	no	—
75	C₂²×C₄	—	67	1	no	—
76	D₈	—	134	2	no	—
77	D₈	—	134	2	no	—
78	C₂×C₈	—	67	1	no	—
79	C₂×C₈	—	134	1	no	—
80	C₂×C₈	—	134	1	no	—
81	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
82	C₄⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	268	2	no	—
84	C₁₆	—	134	1	no	—
85	C₁₆	D₂₆₈	1	2	yes	complemented
86	C₂³	—	268	2	no	—
87	C₂³	—	67	1	no	—
88	D₄	—	268	2	no	—
89	D₄	—	268	2	no	—
90	D₄	—	536	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	67	1	no	—
93	C₂×C₄	—	134	1	no	—
94	C₂×C₄	—	134	1	no	—
95	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
96	C₈	—	134	1	no	—
97	C₈	C₂×D₂₆₈	1	1	yes	characteristic
98	C₈	#11761629	1	1	yes	characteristic
99	C₂²	—	536	2	no	—
100	C₂²	—	536	2	no	—
101	C₂²	—	536	2	no	—
102	C₂²	—	67	2	no	—
103	C₂²	—	134	1	no	—
104	C₂²	—	134	1	no	—
105	C₂²	D₈×D₆₇	1	1	yes	characteristic
106	C₄	—	8	2	no	—
107	C₄	—	134	1	no	—
108	C₄	#14524018	1	1	yes	characteristic
109	C₄	#14524066	1	1	yes	characteristic
110	C₂	—	1072	2	no	—
111	C₂	—	134	2	no	—
112	C₂	#16863633	1	1	yes	characteristic
113	C₂	D₁₆×D₆₇	1	1	yes	characteristic
114	C₂	#16863829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₇₂

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₆₇

C₂

yes

maximal, characteristic, complemented

#16863633

C₂

yes

maximal, complemented

#16863196

C₂

yes

maximal, complemented

#366×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

C₂²

yes

—

C₂×C₈×D₆₇

C₂²

yes

characteristic

#14524018

—

D₁₀₇₂

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₆₇

—

#14523963

—

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₄×D₆₇

D₄

yes

characteristic

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

—

(C₂×D₆₇)⋊C₄

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

D₄

yes

complemented

C₂²×D₆₇

—

C₂²×D₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

D₁₆

yes

complemented

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

#466

yes

characteristic

C₂×C₆₇

C₂×D₁₆

yes

characteristic

C₂×C₆₇

#508

yes

characteristic

D₆₇

—

D₆₇

—

#2215

—

maximal, Sylow

C₆₇

#2215

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂²×C₁₆

—

#466

—

#366

D₆₇

yes

characteristic, complemented

#357

—

C₂×D₈

—

C₂²×C₈

—

#121

—

134

—

D₁₆

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

C₂×D₆₇

yes

—

#102

—

134

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂²×C₄

—

D₈

—

134

—

D₈

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

268

—

C₁₆

—

134

—

C₁₆

D₂₆₈

yes

complemented

C₂³

—

268

—

C₂³

—

D₄

—

268

—

D₄

—

268

—

D₄

—

536

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₈

—

134

—

C₈

C₂×D₂₆₈

yes

characteristic

C₈

#11761629

yes

characteristic

C₂²

—

536

—

100

C₂²

—

536

—

101

C₂²

—

536

—

102

C₂²

—

103

C₂²

—

134

—

104

C₂²

—

134

—

105

C₂²

D₈×D₆₇

yes

characteristic

106

C₄

—

107

C₄

—

134

—

108

C₄

#14524018

yes

characteristic

109

C₄

#14524066

yes

characteristic

110

C₂

—

1072

—

111

C₂

—

134

—

112

C₂

#16863633

yes

characteristic

113

C₂

D₁₆×D₆₇

yes

characteristic

114

C₂

#16863829

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#366×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#366	D₆₇
hypocenter	C₆₇	#2215
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#366×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 ◁ #366

N	H ≅ G/N	type
1. C₂×D₁₀₇₂	C₂	semidirect
2. C₂×C₁₆×D₆₇	C₂	semidirect
3. #16863633	C₂	semidirect
4. #16863196	C₂	semidirect
5. #366×C₆₇	C₂	semidirect
27. C₁₆×C₆₇	D₄	semidirect
47. C₂×D₆₇	D₁₆	semidirect
58. C₆₇	#2215	semidirect
62. #366	D₆₇	semidirect
85. C₁₆	D₂₆₈	semidirect

H ≅ G/N

type

1. C₂×D₁₀₇₂

C₂

semidirect

2. C₂×C₁₆×D₆₇

C₂

semidirect

3. #16863633

C₂

semidirect

4. #16863196

C₂

semidirect

5. #366×C₆₇

C₂

semidirect

27. C₁₆×C₆₇

D₄

semidirect

47. C₂×D₆₇

D₁₆

semidirect

58. C₆₇

#2215

semidirect

62. #366

D₆₇

semidirect

85. C₁₆

D₂₆₈

semidirect

Subgroups — #22562296

Conjugacy classes of subgroups

Complements