Group #22562327 — subgroups

Subgroups — #22562327

Order 8576.

148 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 47 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×#14523984	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863813	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863666	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863225	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863209	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863208	C₂	1	1	yes	maximal, characteristic, complemented
7	#472×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524068	—	2	1	no	—
9	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
10	#14523984	—	2	1	no	—
11	#14523984	—	2	1	no	—
12	#14523983	—	2	1	no	—
13	#14523983	C₂²	1	1	yes	characteristic
14	C₂×C₆₇⋊C₁₆	—	2	1	no	—
15	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14523979	C₂²	1	1	yes	characteristic
17	#14523959	—	2	1	no	—
18	AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
19	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
20	#131×C₆₇	C₂²	1	1	yes	characteristic
21	#108×C₆₇	C₂²	1	1	yes	characteristic
22	#100×C₆₇	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	#11761615	—	4	1	no	—
25	D₅₃₆	—	2	1	no	—
26	D₅₃₆	—	2	1	no	—
27	#11761613	—	4	1	no	—
28	#11761612	—	2	1	no	—
29	C₆₇⋊C₁₆	—	2	1	no	—
30	C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
31	C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
32	#41×C₆₇	D₄	1	1	yes	characteristic
33	C₂×D₄×C₆₇	—	2	1	no	—
34	SD₁₆×C₆₇	—	4	1	no	—
35	D₈×C₆₇	—	2	1	no	—
36	D₈×C₆₇	—	2	1	no	—
37	D₈×C₆₇	—	4	1	no	—
38	M₁₆×C₆₇	—	2	1	no	—
39	M₁₆×C₆₇	—	2	1	no	—
40	M₁₆×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₆₇	—	4	1	no	—
44	C₂²×D₆₇	—	4	1	no	—
45	Q₈×C₆₇	D₈	1	1	yes	characteristic
46	D₄×C₆₇	—	2	1	no	—
47	D₄×C₆₇	—	4	1	no	—
48	D₄×C₆₇	—	4	1	no	—
49	D₄×C₆₇	D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	D₂₆₈	—	4	1	no	—
53	D₂₆₈	—	4	1	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₆₇	#41	1	1	yes	characteristic
58	C₆₇⋊C₈	—	4	1	no	—
59	C₂²×C₆₇	—	2	1	no	—
60	C₂²×C₆₇	—	8	1	no	—
61	C₂²×C₆₇	—	16	1	no	—
62	C₂²×C₆₇	#135	1	1	yes	characteristic
63	C₂×D₆₇	—	8	1	no	—
64	C₂×D₆₇	—	16	1	no	—
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	#121	1	1	yes	characteristic
67	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	—	4	1	no	—
70	C₂×C₆₇	—	16	1	no	—
71	C₂×C₆₇	#466	1	1	yes	characteristic
72	D₆₇	—	16	1	no	—
73	#2221	—	67	1	no	maximal, Sylow
74	C₆₇	#2221	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	67	1	no	—
76	#504	—	67	1	no	—
77	#472	—	67	1	no	—
78	#472	D₆₇	1	1	yes	characteristic, complemented
79	#368	—	67	1	no	—
80	#361	—	67	1	no	—
81	#361	—	67	1	no	—
82	AGL(1,ℤ/8)	—	2	1	no	—
83	AGL(1,ℤ/8)	—	134	1	no	—
84	C₂×D₈	—	67	1	no	—
85	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
86	#131	C₂×D₆₇	1	1	yes	characteristic
87	D₁₆	—	134	1	no	—
88	D₁₆	—	134	1	no	—
89	M₃₂	—	67	1	no	—
90	M₃₂	—	134	1	no	—
91	C₂×C₁₆	—	67	1	no	—
92	C₂×C₁₆	—	134	1	no	—
93	#108	—	67	1	no	—
94	#108	C₂×D₆₇	1	1	yes	characteristic
95	#100	—	2	1	no	—
96	#100	—	134	1	no	—
97	#41	C₂²⋊D₆₇	1	1	yes	characteristic
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	134	1	no	—
100	SD₁₆	—	4	1	no	—
101	SD₁₆	—	268	1	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	4	1	no	—
105	D₈	—	268	1	no	—
106	D₈	—	134	1	no	—
107	D₈	—	134	1	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	134	1	no	—
111	M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₁₆	—	67	1	no	—
115	C₁₆	—	67	1	no	—
116	C₁₆	—	134	1	no	—
117	C₂³	—	4	1	no	—
118	C₂³	—	268	1	no	—
119	Q₈	#11761613	1	1	yes	characteristic
120	D₄	—	2	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	268	1	no	—
124	D₄	—	268	1	no	—
125	D₄	#11761613	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	268	1	no	—
131	C₈	D₄×D₆₇	1	1	yes	characteristic
132	C₈	#11761627	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	16	1	no	—
136	C₂²	—	536	1	no	—
137	C₂²	—	1072	1	no	—
138	C₂²	#14524063	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	#14524015	1	1	yes	characteristic
141	C₄	C₂×#11761613	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	4	1	no	—
144	C₂	—	16	1	no	—
145	C₂	—	1072	1	no	—
146	C₂	#16863630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523984

C₂

yes

maximal, characteristic, complemented

#16863813

C₂

yes

maximal, characteristic, complemented

#16863666

C₂

yes

maximal, characteristic, complemented

#16863225

C₂

yes

maximal, characteristic, complemented

#16863209

C₂

yes

maximal, characteristic, complemented

#16863208

C₂

yes

maximal, characteristic, complemented

#472×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524068

—

C₂×D₅₃₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523979

C₂²

yes

characteristic

#14523959

—

AGL(1,ℤ/8)×C₆₇

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

#131×C₆₇

C₂²

yes

characteristic

#108×C₆₇

C₂²

yes

characteristic

#100×C₆₇

—

C₂×D₂₆₈

—

#11761615

—

D₅₃₆

—

D₅₃₆

—

#11761613

—

#11761612

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

D₄

yes

characteristic

C₆₇⋊C₁₆

D₄

yes

characteristic

#41×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

D₈

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#135

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#466

yes

characteristic

D₆₇

—

#2221

—

maximal, Sylow

C₆₇

#2221

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#504

—

#472

—

#472

D₆₇

yes

characteristic, complemented

—

—

—

—

—

134

—

C₂×D₈

—

C₂×D₈

C₂×D₆₇

yes

characteristic

#131

C₂×D₆₇

yes

characteristic

D₁₆

—

134

—

D₁₆

—

134

—

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

#108

—

#108

C₂×D₆₇

yes

characteristic

#100

—

#100

—

134

—

#41

C₂²⋊D₆₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

134

—

100

SD₁₆

—

101

SD₁₆

—

268

—

102

D₈

—

103

D₈

—

104

D₈

—

105

D₈

—

268

—

106

D₈

—

134

—

107

D₈

—

134

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

134

—

111

M₁₆

C₂²⋊D₆₇

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₆₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

134

—

117

C₂³

—

118

C₂³

—

268

—

119

Q₈

#11761613

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

268

—

124

D₄

—

268

—

125

D₄

#11761613

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

268

—

131

C₈

D₄×D₆₇

yes

characteristic

132

C₈

#11761627

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

536

—

137

C₂²

—

1072

—

138

C₂²

#14524063

yes

characteristic

139

C₄

—

140

C₄

#14524015

yes

characteristic

141

C₄

C₂×#11761613

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

—

1072

—

146

C₂

#16863630

yes

characteristic

	subgroup	G/H
center	C₂	#16863630
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#472×C₆₇	C₂
socle	C₂×C₆₇	#466
hypercenter	#472	D₆₇
hypocenter	C₆₇	#2221
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

subgroup

G/H

center

C₂

#16863630

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#472×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

N	H ≅ G/N	type
1. C₂×#14523984	C₂	semidirect
2. #16863813	C₂	semidirect
3. #16863666	C₂	semidirect
4. #16863225	C₂	semidirect
5. #16863209	C₂	semidirect
6. #16863208	C₂	semidirect
7. #472×C₆₇	C₂	semidirect
74. C₆₇	#2221	semidirect
78. #472	D₆₇	semidirect

H ≅ G/N

type

1. C₂×#14523984

C₂

semidirect

2. #16863813

C₂

semidirect

3. #16863666

C₂

semidirect

4. #16863225

C₂

semidirect

5. #16863209

C₂

semidirect

6. #16863208

C₂

semidirect

7. #472×C₆₇

C₂

semidirect

74. C₆₇

#2221

semidirect

78. #472

D₆₇

semidirect

Subgroups — #22562327

Conjugacy classes of subgroups

Complements