Group #22566329 — subgroups

Subgroups — #22566329

Order 8576.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14524035	C₂	1	1	yes	maximal, characteristic
2	#16863327	C₂	1	4	yes	maximal
3	#16863315	C₂	1	4	yes	maximal
4	#16863265	C₂	1	2	yes	maximal
5	#16863261	C₂	1	2	yes	maximal
6	#16863242	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524035	—	2	4	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
12	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
15	C₆₇⋊#95	C₂²	1	2	yes	—
16	C₆₇⋊#95	C₂²	1	2	yes	—
17	C₆₇⋊#95	C₂²	1	8	yes	—
18	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₆₇	C₂³	1	2	yes	—
24	C₂²×Dic₆₇	C₂³	1	2	yes	—
25	C₂²×Dic₆₇	C₂³	1	4	yes	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	2	no	—
28	Dic₆₇⋊C₄	—	2	4	no	—
29	Dic₆₇⋊C₄	—	2	4	no	—
30	Dic₆₇⋊C₄	—	4	8	no	—
31	C₄×Dic₆₇	Q₈	1	8	yes	complemented
32	C₂×Q₈×C₆₇	—	2	4	no	—
33	C₂×Q₈×C₆₇	—	4	4	no	—
34	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₄²×C₆₇	D₄	1	4	yes	—
41	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₆₇	—	4	8	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	—	2	8	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
51	C₂×Dic₆₇	—	2	4	no	—
52	C₂×Dic₆₇	—	2	4	no	—
53	C₂×Dic₆₇	—	2	2	no	—
54	C₂×Dic₆₇	—	2	2	no	—
55	C₂×Dic₆₇	—	2	4	no	—
56	C₂×Dic₆₇	—	2	4	no	—
57	C₂×Dic₆₇	C₂×Q₈	1	8	yes	—
58	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
60	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₆₇	—	2	2	no	—
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	—	4	4	no	—
65	C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
66	Dic₆₇	—	2	8	no	—
67	Dic₆₇	—	4	2	no	—
68	Dic₆₇	—	4	2	no	—
69	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₆₇	#535	1	1	yes	characteristic
71	C₂×C₆₇	#536	1	1	yes	characteristic
72	C₂×C₆₇	#544	1	4	yes	—
73	#2660	—	67	1	no	maximal, Sylow
74	C₆₇	#2660	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
76	C₂×#125	—	67	1	no	—
77	#398	—	67	4	no	—
78	#395	—	67	4	no	—
79	#384	—	67	2	no	—
80	#382	—	67	2	no	—
81	#376	—	67	1	no	—
82	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
83	C₄⋊Q₈	—	2	4	no	—
84	#125	—	134	4	no	—
85	C₂×C₄⋊C₄	—	67	8	no	—
86	C₂×C₄⋊C₄	—	67	2	no	—
87	C₂×C₄⋊C₄	—	67	2	no	—
88	C₂×C₄⋊C₄	—	67	1	no	—
89	C₂×C₄⋊C₄	—	67	1	no	—
90	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
92	C₂×C₄²	—	67	2	no	—
93	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
94	#95	—	67	2	no	—
95	#95	—	67	2	no	—
96	#95	—	67	8	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	4	4	no	—
99	C₂²×C₄	—	67	4	no	—
100	C₂²×C₄	—	67	2	no	—
101	C₂²×C₄	—	67	2	no	—
102	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
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	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	268	8	no	—
109	C₄⋊C₄	—	134	4	no	—
110	C₄⋊C₄	—	134	4	no	—
111	C₄⋊C₄	—	134	2	no	—
112	C₄⋊C₄	—	134	2	no	—
113	C₄²	—	67	8	no	—
114	C₄²	C₂²⋊D₆₇	1	4	yes	—
115	C₂³	C₂³×D₆₇	1	1	yes	characteristic
116	Q₈	—	4	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	67	8	no	—
122	C₂×C₄	—	134	2	no	—
123	C₂×C₄	—	134	2	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	—	134	4	no	—
126	C₂×C₄	—	134	4	no	—
127	C₂×C₄	—	134	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
131	C₂×C₄	Q₈×D₆₇	1	4	yes	—
132	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₆₇	1	2	yes	—
134	C₂²	#14524088	1	2	yes	—
135	C₂²	#14524091	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	268	2	no	—
140	C₄	—	268	2	no	—
141	C₄	—	134	8	no	—
142	C₄	#14524041	1	4	yes	—
143	C₂	C₂×#14524041	1	1	yes	characteristic
144	C₂	#16863963	1	1	yes	characteristic
145	C₂	#16863970	1	1	yes	characteristic
146	C₂	#16864023	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

#16863327

C₂

yes

maximal

#16863315

C₂

yes

maximal

#16863265

C₂

yes

maximal

#16863261

C₂

yes

maximal

#16863242

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

Q₈

yes

complemented

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×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

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×Q₈

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2660

—

maximal, Sylow

C₆₇

#2660

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

—

—

—

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₆₇

yes

—

103

C₂²×C₄

C₂²×D₆₇

yes

characteristic

104

C₂²×C₄

C₂²×D₆₇

yes

characteristic

105

C₂²×C₄

C₂²×D₆₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

268

—

109

C₄⋊C₄

—

134

—

110

C₄⋊C₄

—

134

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₄²

—

114

C₄²

C₂²⋊D₆₇

yes

—

115

C₂³

C₂³×D₆₇

yes

characteristic

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

—

134

—

126

C₂×C₄

—

134

—

127

C₂×C₄

—

134

—

128

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

131

C₂×C₄

Q₈×D₆₇

yes

—

132

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

133

C₂²

C₂×Q₈×D₆₇

yes

—

134

C₂²

#14524088

yes

—

135

C₂²

#14524091

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

268

—

140

C₄

—

268

—

141

C₄

—

134

—

142

C₄

#14524041

yes

—

143

C₂

C₂×#14524041

yes

characteristic

144

C₂

#16863963

yes

characteristic

145

C₂

#16863970

yes

characteristic

146

C₂

#16864023

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2660
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄×Dic₆₇	Q₈	semidirect
74. C₆₇	#2660	semidirect

H ≅ G/N

type

31. C₄×Dic₆₇

Q₈

semidirect

74. C₆₇

#2660

semidirect

Subgroups — #22566329

Conjugacy classes of subgroups

Complements