Group #22561367 — subgroups

Subgroups — #22561367

Order 8576.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16863404	C₂	1	2	yes	maximal
2	#16863301	C₂	1	1	yes	maximal, characteristic
3	#16863165	C₂	1	1	yes	maximal, characteristic
4	#16863155	C₂	1	2	yes	maximal
5	#544×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×#11761612	C₂²	1	2	yes	—
8	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	#14523966	—	2	2	no	—
10	#14523966	—	2	2	no	—
11	C₆₇⋊#95	—	2	2	no	—
12	C₄⋊Q₈×C₆₇	—	2	2	no	—
13	#125×C₆₇	—	2	2	no	—
14	C₂²⋊Q₈×C₆₇	—	2	2	no	—
15	#117×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₁₃₄	—	2	1	no	—
20	C₂×Dic₁₃₄	Q₈	1	2	yes	complemented
21	C₂²×Dic₆₇	—	2	1	no	—
22	C₂²×Dic₆₇	—	2	1	no	—
23	#11761612	—	2	2	no	—
24	#11761612	—	2	2	no	—
25	C₂×C₆₇⋊C₈	—	2	2	no	—
26	C₄×Dic₆₇	—	2	2	no	—
27	C₂×Q₈×C₆₇	—	2	2	no	—
28	C₂²×C₄×C₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
34	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
35	C₂²⋊C₄×C₆₇	—	2	2	no	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₇	—	8	2	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	4	2	no	—
43	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#41	1	2	yes	—
46	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	2	no	—
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	C₂×Dic₆₇	—	4	1	no	—
51	C₂×Dic₆₇	—	4	1	no	—
52	C₂×Dic₆₇	—	4	1	no	—
53	Dic₁₃₄	—	2	4	no	—
54	Dic₁₃₄	—	4	2	no	—
55	C₆₇⋊C₈	—	4	2	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	#120	1	1	yes	characteristic
59	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₆₇	#124	1	1	yes	characteristic
61	C₄×C₆₇	—	4	2	no	—
62	C₄×C₆₇	—	4	2	no	—
63	C₄×C₆₇	—	4	2	no	—
64	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₆₇	#124	1	2	yes	—
66	Dic₆₇	—	4	2	no	—
67	Dic₆₇	—	4	2	no	—
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	#393	1	1	yes	characteristic
71	C₂×C₆₇	#455	1	1	yes	characteristic
72	C₂×C₆₇	#456	1	1	yes	characteristic
73	#2028	—	67	1	no	maximal, Sylow
74	C₆₇	#2028	1	1	yes	characteristic, complemented, Sylow
75	#544	D₆₇	1	1	yes	characteristic
76	#419	—	67	2	no	—
77	#391	—	67	1	no	—
78	#343	—	67	1	no	—
79	#339	—	67	2	no	—
80	C₂²×Q₈	—	67	1	no	—
81	C₂×M₁₆	—	67	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	#125	—	2	2	no	—
84	C₂²⋊Q₈	—	2	2	no	—
85	#117	C₂×D₆₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
87	C₂×C₄²	—	67	1	no	—
88	#103	—	134	2	no	—
89	#103	—	134	2	no	—
90	#95	—	134	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	67	2	no	—
93	C₂×Q₈	—	134	2	no	—
94	C₂×Q₈	—	134	2	no	—
95	C₂×Q₈	—	134	1	no	—
96	C₂²×C₄	—	134	1	no	—
97	C₂²×C₄	—	134	1	no	—
98	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	M₁₆	—	134	2	no	—
101	M₁₆	—	134	2	no	—
102	C₂×C₈	—	134	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
107	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	134	2	no	—
111	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
112	Q₈	—	8	2	no	—
113	Q₈	—	268	2	no	—
114	Q₈	—	134	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	268	1	no	—
120	C₂×C₄	—	268	1	no	—
121	C₂×C₄	—	268	1	no	—
122	C₂×C₄	—	268	1	no	—
123	C₂×C₄	—	134	2	no	—
124	C₂×C₄	—	134	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
126	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
127	C₂×C₄	#11761629	1	2	yes	—
128	C₂×C₄	#11761629	1	1	yes	characteristic
129	C₈	—	268	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#14524012	1	1	yes	characteristic
133	C₂²	#14524019	1	1	yes	characteristic
134	C₂²	#14524032	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	268	2	no	—
139	C₄	—	268	2	no	—
140	C₄	#14524019	1	2	yes	—
141	C₄	#14524032	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16863307	1	1	yes	characteristic
145	C₂	#16863574	1	1	yes	characteristic
146	C₂	#16863579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863404

C₂

yes

maximal

#16863301

C₂

yes

maximal, characteristic

#16863165

C₂

yes

maximal, characteristic

#16863155

C₂

yes

maximal

#544×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523966

—

#14523966

—

C₆₇⋊#95

—

C₄⋊Q₈×C₆₇

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

#117×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₈

yes

complemented

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

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×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#124

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#124

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#544

D₆₇

yes

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

#103

—

134

—

#103

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

M₁₆

—

134

—

101

M₁₆

—

134

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₆₇

yes

—

107

C₄⋊C₄

C₂²⋊D₆₇

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

134

—

111

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

112

Q₈

—

113

Q₈

—

268

—

114

Q₈

—

134

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

268

—

120

C₂×C₄

—

268

—

121

C₂×C₄

—

268

—

122

C₂×C₄

—

268

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

126

C₂×C₄

Q₈×D₆₇

yes

characteristic

127

C₂×C₄

#11761629

yes

—

128

C₂×C₄

#11761629

yes

characteristic

129

C₈

—

268

—

130

C₂²

—

131

C₂²

—

132

C₂²

#14524012

yes

characteristic

133

C₂²

#14524019

yes

characteristic

134

C₂²

#14524032

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

268

—

139

C₄

—

268

—

140

C₄

#14524019

yes

—

141

C₄

#14524032

yes

—

142

C₂

—

143

C₂

—

144

C₂

#16863307

yes

characteristic

145

C₂

#16863574

yes

characteristic

146

C₂

#16863579

yes

characteristic

	subgroup	G/H
center	C₂²	#14524012
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#544×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#544	D₆₇
hypocenter	C₆₇	#2028
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#14524012

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#544×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
20. C₂×Dic₁₃₄	Q₈	semidirect
74. C₆₇	#2028	semidirect

H ≅ G/N

type

20. C₂×Dic₁₃₄

Q₈

semidirect

74. C₆₇

#2028

semidirect

Subgroups — #22561367

Conjugacy classes of subgroups

Complements