Group #22559635 — subgroups

Subgroups — #22559635

Order 8576.

166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863748	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863677	C₂	1	1	yes	maximal, characteristic
3	#16863598	C₂	1	1	yes	maximal, characteristic
4	#16863461	C₂	1	1	yes	maximal, characteristic
5	#16863146	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863143	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761615	—	2	1	no	—
9	C₂×C₈×D₆₇	—	2	1	no	—
10	#14524043	C₂²	1	1	yes	characteristic
11	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
12	#14523977	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#14523967	—	2	1	no	—
15	#14523966	—	2	1	no	—
16	#14523966	—	2	1	no	—
17	#14523965	—	2	1	no	—
18	(C₂×D₆₇)⋊C₈	—	2	1	no	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×D₂₆₈	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₇	—	2	1	no	—
26	#11761615	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₆₇	—	4	2	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	(C₂×D₆₇)⋊C₄	—	2	1	no	—
35	C₂×Q₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₆₇	—	2	1	no	—
42	Q₈×C₆₇	—	4	1	no	—
43	Q₈×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×Dic₆₇	—	2	1	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	D₂₆₈	—	2	1	no	—
51	D₂₆₈	SD₁₆	1	2	yes	—
52	C₄×D₆₇	—	4	1	no	—
53	C₄×D₆₇	—	4	1	no	—
54	Dic₁₃₄	—	4	1	no	—
55	Dic₁₃₄	—	4	1	no	—
56	C₈×C₆₇	—	4	1	no	—
57	C₈×C₆₇	—	4	1	no	—
58	C₆₇⋊C₈	—	2	1	no	—
59	C₆₇⋊C₈	—	2	1	no	—
60	C₆₇⋊C₈	—	4	1	no	—
61	C₂²×C₆₇	#121	1	1	yes	characteristic
62	C₂×D₆₇	—	2	2	no	—
63	C₂×D₆₇	—	4	1	no	—
64	C₂×D₆₇	—	4	1	no	—
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	8	1	no	—
67	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₆₇	#135	1	1	yes	characteristic
69	C₄×C₆₇	#135	1	1	yes	characteristic
70	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	8	1	no	—
73	C₂×C₆₇	#463	1	1	yes	characteristic
74	C₂×C₆₇	#465	1	1	yes	characteristic
75	C₂×C₆₇	#473	1	1	yes	characteristic
76	D₆₇	—	4	2	no	—
77	#1679	—	67	1	no	maximal, Sylow
78	C₆₇	#1679	1	1	yes	characteristic, complemented, Sylow
79	#487	—	67	1	no	—
80	#474	—	67	1	no	—
81	#461	—	67	1	no	—
82	C₈×D₄	—	67	1	no	—
83	#334	—	67	1	no	—
84	#332	—	67	1	no	—
85	#332	D₆₇	1	1	yes	characteristic, complemented
86	C₂×SD₁₆	—	134	1	no	—
87	C₂²×C₈	—	134	1	no	—
88	C₄⋊Q₈	—	67	1	no	—
89	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
90	C₄×D₄	—	67	1	no	—
91	#107	—	134	1	no	—
92	C₄⋊C₈	—	67	1	no	—
93	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
95	#103	—	134	1	no	—
96	#103	—	134	1	no	—
97	#103	—	134	1	no	—
98	#102	—	134	1	no	—
99	C₂²⋊C₈	—	134	1	no	—
100	C₄×C₈	—	67	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	134	1	no	—
103	C₂×D₄	—	67	1	no	—
104	C₂²×C₄	—	134	1	no	—
105	SD₁₆	—	268	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	268	2	no	—
109	C₂×C₈	—	67	1	no	—
110	C₂×C₈	—	67	1	no	—
111	C₂×C₈	—	134	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	67	1	no	—
115	C₄⋊C₄	—	134	1	no	—
116	C₄⋊C₄	—	134	1	no	—
117	C₂²⋊C₄	—	134	1	no	—
118	C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₂³	—	134	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	268	1	no	—
123	Q₈	—	268	1	no	—
124	D₄	—	67	2	no	—
125	D₄	—	134	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	—	268	1	no	—
129	C₂×C₄	—	268	1	no	—
130	C₂×C₄	—	134	1	no	—
131	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
133	C₂×C₄	#11761628	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	268	1	no	—
137	C₈	—	134	1	no	—
138	C₈	—	134	1	no	—
139	C₂²	—	268	1	no	—
140	C₂²	—	268	1	no	—
141	C₂²	—	134	2	no	—
142	C₂²	#14524016	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	8	1	no	—
145	C₄	—	268	1	no	—
146	C₄	—	536	1	no	—
147	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
148	C₄	#14524062	1	1	yes	characteristic
149	C₄	#14524066	1	1	yes	characteristic
150	C₄	#14524070	1	1	yes	characteristic
151	C₂	—	268	2	no	—
152	C₂	#16863614	1	1	yes	characteristic
153	C₂	#16863624	1	1	yes	characteristic
154	C₂	#16863670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863748

C₂

yes

maximal, characteristic, complemented

#16863677

C₂

yes

maximal, characteristic

#16863598

C₂

yes

maximal, characteristic

#16863461

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic, complemented

#16863143

C₂

yes

maximal, characteristic, complemented

#332×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761615

—

C₂×C₈×D₆₇

—

#14524043

C₂²

yes

characteristic

C₄×D₂₆₈

C₂²

yes

characteristic

#14523977

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₆₇)⋊C₈

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₄×D₆₇

—

#11761615

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

(C₂×D₆₇)⋊C₄

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

SD₁₆

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1679

—

maximal, Sylow

C₆₇

#1679

yes

characteristic, complemented, Sylow

#487

—

#474

—

#461

—

C₈×D₄

—

#334

—

#332

—

#332

D₆₇

yes

characteristic, complemented

C₂×SD₁₆

—

134

—

C₂²×C₈

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

134

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

134

—

103

C₂×D₄

—

104

C₂²×C₄

—

134

—

105

SD₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

268

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

134

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

134

—

116

C₄⋊C₄

—

134

—

117

C₂²⋊C₄

—

134

—

118

C₄²

C₂²×D₆₇

yes

characteristic

119

C₂³

—

134

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

268

—

123

Q₈

—

268

—

124

D₄

—

125

D₄

—

134

—

126

C₂×C₄

—

127

C₂×C₄

—

268

—

128

C₂×C₄

—

268

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

134

—

131

C₂×C₄

D₄×D₆₇

yes

characteristic

132

C₂×C₄

D₄×D₆₇

yes

characteristic

133

C₂×C₄

#11761628

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

268

—

137

C₈

—

134

—

138

C₈

—

134

—

139

C₂²

—

268

—

140

C₂²

—

268

—

141

C₂²

—

134

—

142

C₂²

#14524016

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

268

—

146

C₄

—

536

—

147

C₄

SD₁₆×D₆₇

yes

characteristic

148

C₄

#14524062

yes

characteristic

149

C₄

#14524066

yes

characteristic

150

C₄

#14524070

yes

characteristic

151

C₂

—

268

—

152

C₂

#16863614

yes

characteristic

153

C₂

#16863624

yes

characteristic

154

C₂

#16863670

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#332×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#332	D₆₇
hypocenter	C₆₇	#1679
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#332×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #16863748	C₂	semidirect
5. #16863146	C₂	semidirect
6. #16863143	C₂	semidirect
7. #332×C₆₇	C₂	semidirect
78. C₆₇	#1679	semidirect
85. #332	D₆₇	semidirect

H ≅ G/N

type

1. #16863748

C₂

semidirect

5. #16863146

C₂

semidirect

6. #16863143

C₂

semidirect

7. #332×C₆₇

C₂

semidirect

78. C₆₇

#1679

semidirect

85. #332

D₆₇

semidirect

Subgroups — #22559635

Conjugacy classes of subgroups

Complements