Group #22561654 — subgroups

Subgroups — #22561654

Order 8576.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863397	C₂	1	2	yes	maximal
3	#16863164	C₂	1	2	yes	maximal, complemented
4	#16863161	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	#14523975	—	2	2	no	—
9	#14523967	—	2	2	no	—
10	#14523964	—	2	2	no	—
11	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₆₇	—	2	2	no	—
15	C₂²×Dic₆₇	—	2	2	no	—
16	#11761612	—	2	2	no	—
17	#11761612	—	2	2	no	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	#41×C₆₇	—	2	2	no	—
22	#41×C₆₇	—	2	2	no	—
23	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×D₄×C₆₇	—	2	1	no	—
25	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	—	2	1	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	C₂³×C₆₇	—	2	1	no	—
33	C₂³×C₆₇	#41	1	1	yes	characteristic
34	Q₈×C₆₇	—	2	2	no	—
35	D₄×C₆₇	—	2	2	no	—
36	D₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	—	2	1	no	—
38	C₂×C₄×C₆₇	—	4	1	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#41	1	2	yes	—
43	C₂×C₄×C₆₇	#41	1	2	yes	—
44	C₂×Dic₆₇	—	4	2	no	—
45	C₂×Dic₆₇	—	8	2	no	—
46	C₈×C₆₇	—	2	2	no	—
47	C₈×C₆₇	—	4	1	no	—
48	C₆₇⋊C₈	—	4	2	no	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	—	4	1	no	—
51	C₂²×C₆₇	—	4	2	no	—
52	C₂²×C₆₇	#121	1	1	yes	characteristic
53	C₂²×C₆₇	#126	1	2	yes	—
54	C₄×C₆₇	—	4	1	no	—
55	C₄×C₆₇	#123	1	1	yes	characteristic
56	C₄×C₆₇	#123	1	2	yes	—
57	C₄×C₆₇	#124	1	1	yes	characteristic
58	Dic₆₇	—	8	2	no	—
59	C₂×C₆₇	—	2	2	no	—
60	C₂×C₆₇	—	8	1	no	—
61	C₂×C₆₇	#397	1	1	yes	characteristic
62	C₂×C₆₇	#471	1	2	yes	—
63	#2079	—	67	1	no	maximal, Sylow
64	C₆₇	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	—	67	1	no	—
66	#417	—	67	2	no	—
67	#408	D₆₇	1	1	yes	characteristic
68	#343	—	67	2	no	—
69	#341	—	67	1	no	—
70	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
71	C₂×M₁₆	—	67	2	no	—
72	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	67	2	no	—
75	#106	—	134	2	no	—
76	#103	—	134	2	no	—
77	#102	—	134	2	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	#41	—	2	2	no	—
80	#41	—	2	2	no	—
81	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	D₂₆₈	1	1	yes	characteristic
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	134	2	no	—
86	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	134	2	no	—
89	M₁₆	—	134	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	134	2	no	—
93	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
94	C₄⋊C₄	—	134	2	no	—
95	C₄⋊C₄	—	134	2	no	—
96	C₂³	—	2	1	no	—
97	C₂³	#11761628	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	268	2	no	—
105	C₂×C₄	—	536	2	no	—
106	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
108	C₂×C₄	#11761627	1	2	yes	—
109	C₂×C₄	#11761627	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	1	no	—
112	C₈	—	268	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	—	4	2	no	—
116	C₂²	#14524017	1	1	yes	characteristic
117	C₂²	#14524036	1	2	yes	—
118	C₄	—	4	1	no	—
119	C₄	—	536	2	no	—
120	C₄	#14524024	1	1	yes	characteristic
121	C₄	#14524027	1	2	yes	—
122	C₄	#14524030	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	8	1	no	—
125	C₂	#16863321	1	1	yes	characteristic
126	C₂	#16863659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523975

C₂

yes

maximal, characteristic, complemented

#16863397

C₂

yes

maximal

#16863164

C₂

yes

maximal, complemented

#16863161

C₂

yes

maximal, characteristic, complemented

#408×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523975

—

#14523967

—

#14523964

—

C₂×#41×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

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

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#126

yes

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

characteristic

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#397

yes

characteristic

C₂×C₆₇

#471

yes

—

#2079

—

maximal, Sylow

C₆₇

#2079

yes

characteristic, complemented, Sylow

C₂×#106

—

#417

—

#408

D₆₇

yes

characteristic

#343

—

#341

—

C₂×#41

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

134

—

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₂₆₈

yes

characteristic

C₂×D₄

—

C₂×D₄

D₂₆₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂³

—

C₂³

#11761628

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

536

—

106

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

107

C₂×C₄

C₂×D₂₆₈

yes

characteristic

108

C₂×C₄

#11761627

yes

—

109

C₂×C₄

#11761627

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

268

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#14524017

yes

characteristic

117

C₂²

#14524036

yes

—

118

C₄

—

119

C₄

—

536

—

120

C₄

#14524024

yes

characteristic

121

C₄

#14524027

yes

—

122

C₄

#14524030

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16863321

yes

characteristic

126

C₂

#16863659

yes

—

	subgroup	G/H
center	C₂²	#14524017
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#408×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#408	D₆₇
hypocenter	C₆₇	#2079
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#14524017

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#408×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#14523975	C₂	semidirect
3. #16863164	C₂	semidirect
4. #16863161	C₂	semidirect
64. C₆₇	#2079	semidirect

H ≅ G/N

type

1. C₂×#14523975

C₂

semidirect

3. #16863164

C₂

semidirect

4. #16863161

C₂

semidirect

64. C₆₇

#2079

semidirect

Subgroups — #22561654

Conjugacy classes of subgroups

Complements