Group #22561466 — subgroups

Subgroups — #22561466

Order 8576.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#16863422	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863402	C₂	1	2	yes	maximal, complemented
4	C₂×#14523959	C₂	1	2	yes	maximal, complemented
5	#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	2	yes	—
7	C₂×D₅₃₆	—	2	4	no	—
8	C₂×D₅₃₆	—	2	2	no	—
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	#14524002	C₂²	1	1	yes	characteristic
11	Dic₆₇⋊C₈	—	2	2	no	—
12	#14523965	—	2	4	no	—
13	#14523959	—	2	4	no	—
14	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₆₇	—	2	2	no	—
17	C₂×D₂₆₈	—	2	2	no	—
18	C₂×D₂₆₈	—	2	2	no	—
19	C₂×D₂₆₈	—	2	2	no	—
20	C₂×D₂₆₈	—	2	2	no	—
21	C₂×D₂₆₈	—	2	2	no	—
22	C₂×D₂₆₈	D₄	1	4	yes	—
23	D₅₃₆	—	2	8	no	—
24	#11761612	—	2	2	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₄⋊Dic₆₇	D₄	1	2	yes	—
27	C₂²⋊Dic₆₇	—	2	1	no	—
28	C₄×Dic₆₇	—	2	1	no	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₆₇	—	2	2	no	—
31	M₁₆×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	D₄	1	2	yes	—
35	C₂³×C₆₇	#41	1	1	yes	characteristic
36	C₂²×D₆₇	—	2	4	no	—
37	C₂²×D₆₇	—	4	2	no	—
38	C₂²×D₆₇	—	4	2	no	—
39	C₂²×D₆₇	—	8	4	no	—
40	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#41	1	2	yes	—
44	C₂×Dic₆₇	—	2	2	no	—
45	D₂₆₈	—	2	4	no	—
46	D₂₆₈	—	2	4	no	—
47	D₂₆₈	—	4	2	no	—
48	D₂₆₈	—	4	2	no	—
49	C₈×C₆₇	—	2	2	no	—
50	C₈×C₆₇	—	4	2	no	—
51	C₆₇⋊C₈	—	4	1	no	—
52	C₂²×C₆₇	—	2	2	no	—
53	C₂²×C₆₇	#121	1	1	yes	characteristic
54	C₂²×C₆₇	#123	1	2	yes	—
55	C₂×D₆₇	—	4	4	no	—
56	C₂×D₆₇	—	8	4	no	—
57	C₂×D₆₇	—	8	4	no	—
58	C₂×D₆₇	—	8	4	no	—
59	C₄×C₆₇	#120	1	1	yes	characteristic
60	C₄×C₆₇	#121	1	2	yes	—
61	C₄×C₆₇	#124	1	1	yes	characteristic
62	Dic₆₇	—	4	2	no	—
63	C₂×C₆₇	—	2	2	no	—
64	C₂×C₆₇	#394	1	1	yes	characteristic
65	C₂×C₆₇	#472	1	2	yes	—
66	D₆₇	—	8	4	no	—
67	#2045	—	67	1	no	maximal, Sylow
68	C₆₇	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	67	1	no	—
70	#424	—	67	1	no	—
71	#418	—	67	2	no	—
72	C₂×#100	—	67	2	no	—
73	#341	D₆₇	1	1	yes	characteristic, complemented
74	C₂²×D₄	—	67	2	no	—
75	C₂×D₈	—	134	4	no	—
76	C₂×D₈	—	134	2	no	—
77	C₂×M₁₆	—	67	1	no	—
78	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
79	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
80	#117	—	67	1	no	—
81	C₄⋊C₈	—	134	2	no	—
82	#102	—	134	4	no	—
83	#100	—	134	4	no	—
84	C₂⁴	—	134	2	no	—
85	C₂×D₄	—	67	4	no	—
86	C₂×D₄	—	134	2	no	—
87	C₂×D₄	—	134	2	no	—
88	C₂×D₄	—	134	2	no	—
89	C₂×D₄	—	134	2	no	—
90	C₂×D₄	—	134	2	no	—
91	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
92	D₈	—	134	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	134	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	134	1	no	—
99	C₂×C₈	D₂₆₈	1	2	yes	—
100	C₄⋊C₄	—	67	2	no	—
101	C₂²⋊C₄	—	134	1	no	—
102	C₄²	—	134	1	no	—
103	C₂³	—	268	2	no	—
104	C₂³	—	268	2	no	—
105	C₂³	—	536	4	no	—
106	C₂³	—	134	4	no	—
107	C₂³	#11761629	1	1	yes	characteristic
108	D₄	—	268	2	no	—
109	D₄	—	268	2	no	—
110	D₄	—	134	4	no	—
111	D₄	—	134	4	no	—
112	C₂×C₄	—	134	2	no	—
113	C₂×C₄	D₄×D₆₇	1	2	yes	—
114	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
115	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
116	C₂×C₄	#11761628	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	268	1	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	268	4	no	—
122	C₂²	—	536	4	no	—
123	C₂²	—	536	4	no	—
124	C₂²	—	536	4	no	—
125	C₂²	#14524018	1	1	yes	characteristic
126	C₂²	#14524028	1	2	yes	—
127	C₄	—	268	2	no	—
128	C₄	#14524013	1	1	yes	characteristic
129	C₄	#14524016	1	2	yes	—
130	C₄	#14524031	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	536	4	no	—
133	C₂	#16863314	1	1	yes	characteristic
134	C₂	#16863667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₃₆

C₂

yes

maximal, characteristic

#16863422

C₂

yes

maximal, characteristic, complemented

#16863402

C₂

yes

maximal, complemented

C₂×#14523959

C₂

yes

maximal, complemented

#341×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523965

—

#14523959

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

D₄

yes

—

D₅₃₆

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#123

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#394

yes

characteristic

C₂×C₆₇

#472

yes

—

D₆₇

—

#2045

—

maximal, Sylow

C₆₇

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂²×D₄

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

C₄⋊C₈

—

134

—

#102

—

134

—

#100

—

134

—

C₂⁴

—

134

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

D₈

—

134

—

M₁₆

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

134

—

102

C₄²

—

134

—

103

C₂³

—

268

—

104

C₂³

—

268

—

105

C₂³

—

536

—

106

C₂³

—

134

—

107

C₂³

#11761629

yes

characteristic

108

D₄

—

268

—

109

D₄

—

268

—

110

D₄

—

134

—

111

D₄

—

134

—

112

C₂×C₄

—

134

—

113

C₂×C₄

D₄×D₆₇

yes

—

114

C₂×C₄

D₄×D₆₇

yes

characteristic

115

C₂×C₄

C₂×D₂₆₈

yes

characteristic

116

C₂×C₄

#11761628

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

268

—

120

C₂²

—

121

C₂²

—

268

—

122

C₂²

—

536

—

123

C₂²

—

536

—

124

C₂²

—

536

—

125

C₂²

#14524018

yes

characteristic

126

C₂²

#14524028

yes

—

127

C₄

—

268

—

128

C₄

#14524013

yes

characteristic

129

C₄

#14524016

yes

—

130

C₄

#14524031

yes

characteristic

131

C₂

—

132

C₂

—

536

—

133

C₂

#16863314

yes

characteristic

134

C₂

#16863667

yes

—

	subgroup	G/H
center	C₂²	#14524018
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#341×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#341	D₆₇
hypocenter	C₆₇	#2045
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14524018

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#341×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. #16863422	C₂	semidirect
3. #16863402	C₂	semidirect
4. C₂×#14523959	C₂	semidirect
5. #341×C₆₇	C₂	semidirect
68. C₆₇	#2045	semidirect
73. #341	D₆₇	semidirect

H ≅ G/N

type

2. #16863422

C₂

semidirect

3. #16863402

C₂

semidirect

4. C₂×#14523959

C₂

semidirect

5. #341×C₆₇

C₂

semidirect

68. C₆₇

#2045

semidirect

73. #341

D₆₇

semidirect

Subgroups — #22561466

Conjugacy classes of subgroups

Complements