Group #22561512 — subgroups

Subgroups — #22561512

Order 8576.

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

normal subgroups only

This group has 264 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	#16863424	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863392	C₂	1	2	yes	maximal, complemented
3	#16863370	C₂	1	2	yes	maximal, complemented
4	#16863153	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	#14524002	C₂²	1	2	yes	—
8	#14524002	C₂²	1	1	yes	characteristic
9	Dic₆₇⋊C₈	—	2	2	no	—
10	#14523967	—	2	2	no	—
11	#14523964	—	2	2	no	—
12	#14523958	—	2	4	no	—
13	C₂×#41×C₆₇	C₂²	1	2	yes	—
14	#137×C₆₇	—	2	2	no	—
15	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
16	#135×C₆₇	—	2	2	no	—
17	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₄⋊Dic₆₇	—	2	1	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₂²⋊Dic₆₇	—	2	2	no	—
23	C₂²⋊Dic₆₇	D₄	1	2	yes	—
24	C₄×Dic₆₇	—	2	1	no	—
25	C₄×Dic₆₇	—	2	2	no	—
26	#41×C₆₇	—	2	2	no	—
27	#41×C₆₇	—	2	2	no	—
28	#41×C₆₇	—	2	2	no	—
29	#41×C₆₇	—	2	2	no	—
30	C₂×Q₈×C₆₇	D₄	1	2	yes	—
31	C₂×D₄×C₆₇	—	2	2	no	—
32	C₂×D₄×C₆₇	D₄	1	2	yes	—
33	C₂²×C₄×C₆₇	—	2	2	no	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	Q₁₆×C₆₇	—	2	2	no	—
36	SD₁₆×C₆₇	—	2	4	no	—
37	D₈×C₆₇	—	2	2	no	—
38	M₁₆×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	D₄	1	2	yes	—
40	C₂³×C₆₇	—	2	2	no	—
41	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₆₇	—	2	2	no	—
43	Q₈×C₆₇	—	2	2	no	—
44	D₄×C₆₇	—	2	2	no	—
45	D₄×C₆₇	—	2	2	no	—
46	D₄×C₆₇	—	4	2	no	—
47	D₄×C₆₇	—	4	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	2	yes	—
55	C₂×Dic₆₇	—	2	2	no	—
56	C₂×Dic₆₇	—	4	2	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₂²×C₆₇	—	4	1	no	—
60	C₂²×C₆₇	—	4	2	no	—
61	C₂²×C₆₇	—	8	2	no	—
62	C₂²×C₆₇	#121	1	1	yes	characteristic
63	C₂²×C₆₇	#121	1	2	yes	—
64	C₄×C₆₇	—	4	2	no	—
65	C₄×C₆₇	#120	1	1	yes	characteristic
66	C₄×C₆₇	#123	1	2	yes	—
67	C₄×C₆₇	#124	1	1	yes	characteristic
68	Dic₆₇	—	4	2	no	—
69	Dic₆₇	—	8	2	no	—
70	C₂×C₆₇	—	2	1	no	—
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	—	8	2	no	—
73	C₂×C₆₇	#394	1	1	yes	characteristic
74	#2053	—	67	1	no	maximal, Sylow
75	C₆₇	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	D₆₇	1	1	yes	characteristic
77	#424	—	67	1	no	—
78	#416	—	67	2	no	—
79	#410	—	67	2	no	—
80	H(3,ℤ/4)	—	67	1	no	—
81	C₂×#41	C₂×D₆₇	1	2	yes	—
82	#137	—	2	2	no	—
83	AGL(1,ℤ/8)	—	2	2	no	—
84	#135	—	2	2	no	—
85	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂²×C₈	—	67	1	no	—
87	#117	—	67	1	no	—
88	#117	—	67	2	no	—
89	C₄⋊C₈	—	134	2	no	—
90	#103	—	134	2	no	—
91	#102	—	134	2	no	—
92	#99	—	134	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂²⋊D₆₇	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₂²⋊D₆₇	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	SD₁₆	—	2	4	no	—
104	D₈	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	134	2	no	—
107	C₂×C₈	—	134	1	no	—
108	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
109	C₄⋊C₄	—	134	2	no	—
110	C₄⋊C₄	—	134	1	no	—
111	C₂²⋊C₄	—	67	2	no	—
112	C₂²⋊C₄	—	134	2	no	—
113	C₄²	—	134	2	no	—
114	C₄²	—	134	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	D₄×D₆₇	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	268	2	no	—
127	C₂×C₄	—	134	2	no	—
128	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
130	C₂×C₄	#11761627	1	1	yes	characteristic
131	C₂×C₄	#11761627	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	268	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#14524014	1	1	yes	characteristic
138	C₂²	#14524015	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	268	2	no	—
141	C₄	—	536	2	no	—
142	C₄	#14524012	1	1	yes	characteristic
143	C₄	#14524024	1	2	yes	—
144	C₄	#14524030	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	8	2	no	—
148	C₂	#16863310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863424

C₂

yes

maximal, characteristic, complemented

#16863392

C₂

yes

maximal, complemented

#16863370

C₂

yes

maximal, complemented

#16863153

C₂

yes

maximal, characteristic, complemented

#575×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

#14524002

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523967

—

#14523964

—

#14523958

—

C₂×#41×C₆₇

C₂²

yes

—

#137×C₆₇

—

AGL(1,ℤ/8)×C₆₇

—

#135×C₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

D₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#121

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#575

D₆₇

yes

—

—

—

—

C₂×#41

C₂×D₆₇

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

C₄⋊C₈

—

134

—

#103

—

134

—

#102

—

134

—

#99

—

134

—

—

—

—

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₆₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₆₇

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

C₂²⋊D₆₇

yes

—

109

C₄⋊C₄

—

134

—

110

C₄⋊C₄

—

134

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

134

—

113

C₄²

—

134

—

114

C₄²

—

134

—

115

C₂³

—

116

C₂³

D₄×D₆₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

134

—

128

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

130

C₂×C₄

#11761627

yes

characteristic

131

C₂×C₄

#11761627

yes

—

132

C₈

—

133

C₈

—

268

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#14524014

yes

characteristic

138

C₂²

#14524015

yes

—

139

C₄

—

140

C₄

—

268

—

141

C₄

—

536

—

142

C₄

#14524012

yes

characteristic

143

C₄

#14524024

yes

—

144

C₄

#14524030

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#16863310

yes

characteristic

	subgroup	G/H
center	C₄	#14524030
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#575×C₆₇	C₂
socle	C₂×C₆₇	#394
hypercenter	#575	D₆₇
hypocenter	C₆₇	#2053
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#14524030

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#575×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #16863424	C₂	semidirect
2. #16863392	C₂	semidirect
3. #16863370	C₂	semidirect
4. #16863153	C₂	semidirect
75. C₆₇	#2053	semidirect

H ≅ G/N

type

1. #16863424

C₂

semidirect

2. #16863392

C₂

semidirect

3. #16863370

C₂

semidirect

4. #16863153

C₂

semidirect

75. C₆₇

#2053

semidirect

Subgroups — #22561512

Conjugacy classes of subgroups

Complements