Group #22561555 — subgroups

Subgroups — #22561555

Order 8576.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16864027	C₂	1	1	yes	maximal, characteristic
2	#16863438	C₂	1	2	yes	maximal
3	#16863265	C₂	1	1	yes	maximal, characteristic
4	#16863157	C₂	1	2	yes	maximal
5	#343×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524042	—	2	2	no	—
7	#14524034	—	2	2	no	—
8	#14524022	—	2	2	no	—
9	#14524002	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₇⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	#14523977	—	2	2	no	—
14	#14523975	—	2	2	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂²×Dic₆₇	—	2	1	no	—
19	C₂²×Dic₆₇	—	2	1	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	D₄	1	2	yes	—
23	C₄⋊Dic₆₇	D₄	1	2	yes	—
24	Dic₆₇⋊C₄	—	2	4	no	—
25	Dic₆₇⋊C₄	—	4	4	no	—
26	C₂²⋊Dic₆₇	—	2	2	no	—
27	C₄×Dic₆₇	—	2	2	no	—
28	C₄×Dic₆₇	—	2	1	no	—
29	C₄×Dic₆₇	Q₈	1	2	yes	complemented
30	C₂²×C₄×C₆₇	D₄	1	2	yes	—
31	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₆₇	—	2	2	no	—
33	M₁₆×C₆₇	Q₈	1	4	yes	complemented
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	4	2	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
43	C₂×Dic₆₇	—	2	2	no	—
44	C₂×Dic₆₇	—	2	2	no	—
45	C₂×Dic₆₇	—	2	2	no	—
46	C₂×Dic₆₇	—	2	2	no	—
47	C₂×Dic₆₇	—	2	2	no	—
48	C₂×Dic₆₇	—	4	2	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	Dic₁₃₄	—	8	2	no	—
51	C₈×C₆₇	—	2	4	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	—	2	1	no	—
54	C₂²×C₆₇	#120	1	1	yes	characteristic
55	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₆₇	—	4	2	no	—
58	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
60	Dic₆₇	—	2	4	no	—
61	Dic₆₇	—	4	2	no	—
62	Dic₆₇	—	4	2	no	—
63	Dic₆₇	—	8	1	no	—
64	C₂×C₆₇	—	2	1	no	—
65	C₂×C₆₇	—	2	1	no	—
66	C₂×C₆₇	#395	1	1	yes	characteristic
67	C₂×C₆₇	#453	1	1	yes	characteristic
68	C₂×C₆₇	#456	1	1	yes	characteristic
69	#2060	—	67	1	no	maximal, Sylow
70	C₆₇	#2060	1	1	yes	characteristic, complemented, Sylow
71	#544	—	67	1	no	—
72	#428	—	67	2	no	—
73	#384	—	67	1	no	—
74	#343	D₆₇	1	1	yes	characteristic
75	#339	—	67	2	no	—
76	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
77	C₄⋊Q₈	—	134	2	no	—
78	#125	—	134	2	no	—
79	C₂²⋊Q₈	—	134	2	no	—
80	#117	—	67	2	no	—
81	C₂×C₄⋊C₄	—	67	1	no	—
82	C₂×C₄⋊C₄	—	134	2	no	—
83	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
84	C₂×C₄²	—	67	1	no	—
85	#107	—	134	2	no	—
86	#106	—	134	2	no	—
87	C₂×Q₈	—	134	2	no	—
88	C₂²×C₄	—	134	1	no	—
89	C₂²×C₄	—	134	1	no	—
90	C₂²×C₄	D₂₆₈	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	Dic₁₃₄	1	4	yes	complemented
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	268	4	no	—
97	C₄⋊C₄	—	67	2	no	—
98	C₄⋊C₄	—	67	2	no	—
99	C₄⋊C₄	—	134	2	no	—
100	C₄⋊C₄	—	134	4	no	—
101	C₄⋊C₄	—	134	2	no	—
102	C₂²⋊C₄	—	134	2	no	—
103	C₄²	—	67	2	no	—
104	C₄²	—	134	2	no	—
105	C₄²	—	134	1	no	—
106	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
107	Q₈	—	536	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	268	2	no	—
111	C₂×C₄	—	268	1	no	—
112	C₂×C₄	—	134	2	no	—
113	C₂×C₄	—	134	2	no	—
114	C₂×C₄	—	134	2	no	—
115	C₂×C₄	—	134	2	no	—
116	C₂×C₄	—	134	2	no	—
117	C₂×C₄	D₄×D₆₇	1	2	yes	—
118	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
120	C₂×C₄	#11761627	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#14524013	1	1	yes	characteristic
125	C₂²	#14524023	1	1	yes	characteristic
126	C₂²	#14524043	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	268	2	no	—
129	C₄	—	268	2	no	—
130	C₄	—	536	1	no	—
131	C₄	—	134	4	no	—
132	C₄	#14524020	1	2	yes	—
133	C₄	#14524042	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#16863317	1	1	yes	characteristic
137	C₂	#16863566	1	1	yes	characteristic
138	C₂	#16863581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864027

C₂

yes

maximal, characteristic

#16863438

C₂

yes

maximal

#16863265

C₂

yes

maximal, characteristic

#16863157

C₂

yes

maximal

#343×C₆₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

—

#14523975

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

Q₈

yes

complemented

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

Q₈

yes

complemented

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

#339

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

134

—

#125

—

134

—

C₂²⋊Q₈

—

134

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

#107

—

134

—

#106

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

Dic₁₃₄

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

100

C₄⋊C₄

—

134

—

101

C₄⋊C₄

—

134

—

102

C₂²⋊C₄

—

134

—

103

C₄²

—

104

C₄²

—

134

—

105

C₄²

—

134

—

106

C₂³

C₂×D₂₆₈

yes

characteristic

107

Q₈

—

536

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

268

—

111

C₂×C₄

—

268

—

112

C₂×C₄

—

134

—

113

C₂×C₄

—

134

—

114

C₂×C₄

—

134

—

115

C₂×C₄

—

134

—

116

C₂×C₄

—

134

—

117

C₂×C₄

D₄×D₆₇

yes

—

118

C₂×C₄

Q₈×D₆₇

yes

characteristic

119

C₂×C₄

C₂×Dic₁₃₄

yes

—

120

C₂×C₄

#11761627

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14524013

yes

characteristic

125

C₂²

#14524023

yes

characteristic

126

C₂²

#14524043

yes

characteristic

127

C₄

—

128

C₄

—

268

—

129

C₄

—

268

—

130

C₄

—

536

—

131

C₄

—

134

—

132

C₄

#14524020

yes

—

133

C₄

#14524042

yes

—

134

C₂

—

135

C₂

—

136

C₂

#16863317

yes

characteristic

137

C₂

#16863566

yes

characteristic

138

C₂

#16863581

yes

characteristic

	subgroup	G/H
center	C₂²	#14524013
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#343×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#343	D₆₇
hypocenter	C₆₇	#2060
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#14524013

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#343×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
29. C₄×Dic₆₇	Q₈	semidirect
33. M₁₆×C₆₇	Q₈	semidirect
70. C₆₇	#2060	semidirect
93. M₁₆	Dic₁₃₄	semidirect

H ≅ G/N

type

29. C₄×Dic₆₇

Q₈

semidirect

33. M₁₆×C₆₇

Q₈

semidirect

70. C₆₇

#2060

semidirect

93. M₁₆

Dic₁₃₄

semidirect

Subgroups — #22561555

Conjugacy classes of subgroups

Complements