Group #22562425 — subgroups

Subgroups — #22562425

Order 8576.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16863650	C₂	1	1	yes	maximal, characteristic
2	#16863443	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863218	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863210	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863208	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863177	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11761611	C₂²	1	1	yes	characteristic
10	M₁₆×D₆₇	—	2	1	no	—
11	#14524018	—	2	1	no	—
12	#14523983	C₂²	1	1	yes	characteristic
13	#14523979	C₂²	1	1	yes	characteristic
14	#14523978	—	2	1	no	—
15	#14523963	—	2	1	no	—
16	M₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	#108×C₆₇	C₂²	1	1	yes	characteristic
18	#106×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×D₂₆₈	—	2	1	no	—
20	C₂×D₂₆₈	—	4	1	no	—
21	C₂×C₄×D₆₇	D₄	1	1	yes	characteristic
22	D₅₃₆	—	2	1	no	—
23	D₅₃₆	—	2	1	no	—
24	#11761612	—	2	1	no	—
25	#11761611	—	2	1	no	—
26	#11761611	—	2	1	no	—
27	#11761611	—	4	1	no	—
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₈×D₆₇	—	4	1	no	—
30	(C₂×D₆₇)⋊C₄	—	4	1	no	—
31	C₆₇⋊C₁₆	—	2	1	no	—
32	M₁₆×C₆₇	—	2	1	no	—
33	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₁₆×C₆₇	—	2	1	no	—
36	C₂²×D₆₇	—	4	1	no	—
37	C₂²×D₆₇	SD₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	4	1	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₆₇	SD₁₆	1	1	yes	characteristic
41	D₂₆₈	—	4	1	no	—
42	D₂₆₈	—	4	1	no	—
43	D₂₆₈	—	8	2	no	—
44	C₄×D₆₇	—	2	1	no	—
45	C₄×D₆₇	—	2	1	no	—
46	C₈×C₆₇	—	4	1	no	—
47	C₈×C₆₇	#41	1	1	yes	characteristic
48	C₈×C₆₇	#41	1	1	yes	characteristic
49	C₆₇⋊C₈	—	2	1	no	—
50	C₆₇⋊C₈	—	4	1	no	—
51	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×D₆₇	—	2	1	no	—
53	C₂×D₆₇	—	4	1	no	—
54	C₂×D₆₇	—	8	1	no	—
55	C₂×D₆₇	—	8	1	no	—
56	C₂×D₆₇	—	8	1	no	—
57	C₄×C₆₇	—	8	1	no	—
58	C₄×C₆₇	#123	1	1	yes	characteristic
59	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₆₇	—	2	1	no	—
61	C₂×C₆₇	—	2	1	no	—
62	C₂×C₆₇	#481	1	1	yes	characteristic
63	D₆₇	—	4	1	no	—
64	D₆₇	—	16	1	no	—
65	#2238	—	67	1	no	maximal, Sylow
66	C₆₇	#2238	1	1	yes	characteristic, complemented, Sylow
67	#469	—	67	1	no	—
68	#430	—	67	1	no	—
69	#365	—	67	1	no	—
70	#365	D₆₇	1	1	yes	characteristic, complemented
71	#361	—	67	1	no	—
72	#361	—	67	1	no	—
73	#349	—	67	1	no	—
74	C₂×D₈	—	67	1	no	—
75	C₂×M₁₆	—	67	1	no	—
76	C₂×M₁₆	—	134	1	no	—
77	#121	—	134	1	no	—
78	M₃₂	—	67	1	no	—
79	M₃₂	C₂×D₆₇	1	1	yes	characteristic
80	#108	—	67	1	no	—
81	#108	—	134	1	no	—
82	#108	C₂×D₆₇	1	1	yes	characteristic
83	#106	C₂×D₆₇	1	1	yes	characteristic
84	#102	—	134	1	no	—
85	C₂×D₄	—	268	1	no	—
86	C₂×D₄	—	134	1	no	—
87	C₂²×C₄	—	67	1	no	—
88	D₈	—	134	1	no	—
89	D₈	—	134	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	268	1	no	—
92	M₁₆	—	134	1	no	—
93	M₁₆	—	134	1	no	—
94	M₁₆	—	134	1	no	—
95	C₂×C₈	—	268	1	no	—
96	C₂×C₈	—	67	1	no	—
97	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	268	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	134	1	no	—
102	C₂³	—	268	1	no	—
103	C₂³	—	67	1	no	—
104	D₄	—	268	1	no	—
105	D₄	—	268	1	no	—
106	D₄	—	536	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	67	1	no	—
109	C₂×C₄	—	134	1	no	—
110	C₂×C₄	—	134	1	no	—
111	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	268	1	no	—
114	C₈	—	134	1	no	—
115	C₈	#11761628	1	1	yes	characteristic
116	C₈	#11761629	1	1	yes	characteristic
117	C₂²	—	268	1	no	—
118	C₂²	—	536	1	no	—
119	C₂²	—	536	1	no	—
120	C₂²	—	536	1	no	—
121	C₂²	—	134	1	no	—
122	C₂²	SD₁₆×D₆₇	1	1	yes	characteristic
123	C₄	—	8	1	no	—
124	C₄	—	134	1	no	—
125	C₄	#14524028	1	1	yes	characteristic
126	C₄	#14524070	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	268	1	no	—
129	C₂	—	1072	1	no	—
130	C₂	#16863716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863650

C₂

yes

maximal, characteristic

#16863443

C₂

yes

maximal, characteristic, complemented

#16863218

C₂

yes

maximal, characteristic, complemented

#16863210

C₂

yes

maximal, characteristic, complemented

#16863208

C₂

yes

maximal, characteristic, complemented

#16863177

C₂

yes

maximal, characteristic, complemented

#365×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

C₂²

yes

characteristic

C₂×#11761611

C₂²

yes

characteristic

M₁₆×D₆₇

—

#14524018

—

#14523983

C₂²

yes

characteristic

#14523979

C₂²

yes

characteristic

#14523978

—

#14523963

—

M₃₂×C₆₇

C₂²

yes

characteristic

#108×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₄×D₆₇

D₄

yes

characteristic

D₅₃₆

—

D₅₃₆

—

—

—

—

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

(C₂×D₆₇)⋊C₄

—

C₆₇⋊C₁₆

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

SD₁₆

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#481

yes

characteristic

D₆₇

—

D₆₇

—

#2238

—

maximal, Sylow

C₆₇

#2238

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

#361

—

#361

—

#349

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

#121

—

134

—

M₃₂

—

M₃₂

C₂×D₆₇

yes

characteristic

#108

—

#108

—

134

—

#108

C₂×D₆₇

yes

characteristic

#106

C₂×D₆₇

yes

characteristic

#102

—

134

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂²×C₄

—

D₈

—

134

—

D₈

—

134

—

M₁₆

—

M₁₆

—

268

—

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

268

—

100

C₁₆

—

101

C₁₆

—

134

—

102

C₂³

—

268

—

103

C₂³

—

104

D₄

—

268

—

105

D₄

—

268

—

106

D₄

—

536

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

134

—

110

C₂×C₄

—

134

—

111

C₂×C₄

D₄×D₆₇

yes

characteristic

112

C₈

—

113

C₈

—

268

—

114

C₈

—

134

—

115

C₈

#11761628

yes

characteristic

116

C₈

#11761629

yes

characteristic

117

C₂²

—

268

—

118

C₂²

—

536

—

119

C₂²

—

536

—

120

C₂²

—

536

—

121

C₂²

—

134

—

122

C₂²

SD₁₆×D₆₇

yes

characteristic

123

C₄

—

124

C₄

—

134

—

125

C₄

#14524028

yes

characteristic

126

C₄

#14524070

yes

characteristic

127

C₂

—

128

C₂

—

268

—

129

C₂

—

1072

—

130

C₂

#16863716

yes

characteristic

	subgroup	G/H
center	C₂	#16863716
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#365×C₆₇	C₂
socle	C₂×C₆₇	#481
hypercenter	#365	D₆₇
hypocenter	C₆₇	#2238
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16863716

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#365×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
2. #16863443	C₂	semidirect
3. #16863218	C₂	semidirect
4. #16863210	C₂	semidirect
5. #16863208	C₂	semidirect
6. #16863177	C₂	semidirect
7. #365×C₆₇	C₂	semidirect
66. C₆₇	#2238	semidirect
70. #365	D₆₇	semidirect

H ≅ G/N

type

2. #16863443

C₂

semidirect

3. #16863218

C₂

semidirect

4. #16863210

C₂

semidirect

5. #16863208

C₂

semidirect

6. #16863177

C₂

semidirect

7. #365×C₆₇

C₂

semidirect

66. C₆₇

#2238

semidirect

70. #365

D₆₇

semidirect

Subgroups — #22562425

Conjugacy classes of subgroups

Complements