Group #22558708 — subgroups

Subgroups — #22558708

Order 8576.

240 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16863449	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863133	C₂	1	4	yes	maximal
4	#533×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₆₇	—	2	1	no	—
6	C₂×C₄×Dic₆₇	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₆₇⋊C₈	—	2	1	no	—
10	C₆₇⋊#95	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#125×C₆₇	—	2	1	no	—
14	C₂²⋊Q₈×C₆₇	—	2	1	no	—
15	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₆₇	—	2	1	no	—
17	C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
18	#117×C₆₇	—	2	1	no	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₆₇	—	2	2	no	—
21	C₂×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	4	no	—
23	C₄×Dic₆₇	—	4	2	no	—
24	C₂×Q₈×C₆₇	—	2	1	no	—
25	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²⋊C₄×C₆₇	—	2	2	no	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₆₇	—	4	2	no	—
43	Q₈×C₆₇	—	4	2	no	—
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	4	1	no	—
48	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
53	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
54	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	—	4	2	no	—
58	C₂×Dic₆₇	—	4	2	no	—
59	C₆₇⋊C₈	—	4	2	no	—
60	C₂²×C₆₇	—	2	1	no	—
61	C₂²×C₆₇	—	2	1	no	—
62	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	—	4	2	no	—
70	C₄×C₆₇	#103	1	2	yes	—
71	C₄×C₆₇	#103	1	2	yes	—
72	C₄×C₆₇	#104	1	2	yes	—
73	Dic₆₇	—	4	4	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	#410	1	1	yes	characteristic
77	C₂×C₆₇	C₂×#103	1	1	yes	characteristic
78	C₂×C₆₇	C₂×#104	1	1	yes	characteristic
79	#1503	—	67	1	no	maximal, Sylow
80	C₆₇	#1503	1	1	yes	characteristic, complemented, Sylow
81	#533	D₆₇	1	1	yes	characteristic
82	#431	—	67	1	no	—
83	C₄×C₄⋊C₄	—	67	1	no	—
84	#328	—	67	4	no	—
85	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic, complemented
86	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic, complemented
87	#125	—	2	1	no	—
88	C₂²⋊Q₈	—	2	1	no	—
89	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	Dic₆₇	1	2	yes	complemented
92	#117	—	2	1	no	—
93	C₂×C₄⋊C₄	—	134	1	no	—
94	C₂×C₄²	—	134	1	no	—
95	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
96	C₄⋊C₈	—	134	1	no	—
97	C₂²⋊C₈	—	67	2	no	—
98	C₄×C₈	—	134	1	no	—
99	#95	—	67	2	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
102	C₂²×C₄	—	134	2	no	—
103	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
106	C₂×C₈	—	134	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	134	4	no	—
113	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	268	2	no	—
117	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
118	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
119	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
120	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
121	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	268	2	no	—
129	C₂×C₄	—	268	2	no	—
130	C₂×C₄	—	134	4	no	—
131	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
135	C₂×C₄	#11761615	1	2	yes	—
136	C₂×C₄	#11761618	1	2	yes	—
137	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₈	—	268	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
143	C₂²	C₂×#11761615	1	1	yes	characteristic
144	C₂²	C₂×#11761618	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	268	4	no	—
151	C₄	#14523967	1	2	yes	—
152	C₄	#14523967	1	2	yes	—
153	C₄	#14523970	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16863370	1	1	yes	characteristic
157	C₂	C₂×#14523967	1	1	yes	characteristic
158	C₂	C₂×#14523970	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863449

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

#16863133

C₂

yes

maximal

#533×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄×C₆₇⋊C₈

—

C₆₇⋊#95

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₆₇

C₄

yes

characteristic, complemented

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₄

yes

complemented

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×Q₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#103

yes

—

C₄×C₆₇

#103

yes

—

C₄×C₆₇

#104

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#533

D₆₇

yes

characteristic

#431

—

C₄×C₄⋊C₄

—

#328

—

C₄⋊Q₈

Dic₆₇

yes

characteristic, complemented

C₄⋊Q₈

Dic₆₇

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₆₇

yes

complemented

#117

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₄×C₈

—

134

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₆₇

yes

—

102

C₂²×C₄

—

134

—

103

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

104

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

105

C₂²×C₄

C₂²×D₆₇

yes

characteristic

106

C₂×C₈

—

134

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

C₂×Dic₆₇

yes

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

268

—

117

C₄²

C₂×Dic₆₇

yes

characteristic

118

C₄²

C₂×Dic₆₇

yes

characteristic

119

C₄²

C₂²⋊D₆₇

yes

characteristic

120

C₄²

C₂²⋊D₆₇

yes

characteristic

121

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

268

—

129

C₂×C₄

—

268

—

130

C₂×C₄

—

134

—

131

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

132

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

135

C₂×C₄

#11761615

yes

—

136

C₂×C₄

#11761618

yes

—

137

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

139

C₈

—

268

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

143

C₂²

C₂×#11761615

yes

characteristic

144

C₂²

C₂×#11761618

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

268

—

151

C₄

#14523967

yes

—

152

C₄

#14523967

yes

—

153

C₄

#14523970

yes

—

154

C₂

—

155

C₂

—

156

C₂

#16863370

yes

characteristic

157

C₂

C₂×#14523967

yes

characteristic

158

C₂

C₂×#14523970

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#533×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#533	D₆₇
hypocenter	C₆₇	#1503
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#533×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₆₇	C₄	semidirect
12. C₄⋊Q₈×C₆₇	C₄	semidirect
17. C₄×Q₈×C₆₇	C₄	semidirect
80. C₆₇	#1503	semidirect
85. C₄⋊Q₈	Dic₆₇	semidirect
86. C₄⋊Q₈	Dic₆₇	semidirect
91. C₄×Q₈	Dic₆₇	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₆₇

C₄

semidirect

12. C₄⋊Q₈×C₆₇

C₄

semidirect

17. C₄×Q₈×C₆₇

C₄

semidirect

80. C₆₇

#1503

semidirect

85. C₄⋊Q₈

Dic₆₇

semidirect

86. C₄⋊Q₈

Dic₆₇

semidirect

91. C₄×Q₈

Dic₆₇

semidirect

Subgroups — #22558708

Conjugacy classes of subgroups

Complements