Group #30775850 — subgroups

Subgroups — #30775850

Order 17152.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22561969	C₂	1	1	yes	maximal, characteristic
2	#22560997	C₂	1	2	yes	maximal
3	#22558313	C₂	1	1	yes	maximal, characteristic
4	#22558302	C₂	1	2	yes	maximal
5	C₆₇×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	#16863444	—	2	2	no	—
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863351	C₂²	1	1	yes	characteristic
10	Dic₆₇⋊C₁₆	—	2	2	no	—
11	#16863167	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₇	C₂²	1	2	yes	—
14	#368×C₆₇	—	2	2	no	—
15	C₂×#11761612	—	2	2	no	—
16	C₂×#11761612	D₄	1	1	yes	characteristic
17	#14524002	D₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	#14523979	—	2	2	no	—
21	#14523979	Q₈	1	4	yes	—
22	#14523978	—	4	4	no	—
23	#14523953	—	2	1	no	—
24	C₈×Dic₆₇	—	2	1	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
29	#108×C₆₇	—	2	2	no	—
30	#108×C₆₇	—	2	2	no	—
31	#11761612	—	2	4	no	—
32	#11761612	—	2	2	no	—
33	#11761612	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₄⋊Dic₆₇	—	2	1	no	—
37	C₂²⋊Dic₆₇	Q₁₆	1	2	yes	—
38	C₄×Dic₆₇	—	2	1	no	—
39	C₆₇⋊C₁₆	—	4	1	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₆₇	—	4	2	no	—
42	M₁₆×C₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₁₆×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
52	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
53	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
54	C₂×Dic₆₇	—	2	2	no	—
55	C₈×C₆₇	—	8	2	no	—
56	C₈×C₆₇	#123	1	1	yes	characteristic
57	C₈×C₆₇	#125	1	2	yes	—
58	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₆₇⋊C₈	—	4	1	no	—
60	C₆₇⋊C₈	—	4	4	no	—
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	#467	1	1	yes	characteristic
63	C₂²×C₆₇	#477	1	2	yes	—
64	C₄×C₆₇	#398	1	1	yes	characteristic
65	C₄×C₆₇	#469	1	1	yes	characteristic
66	C₄×C₆₇	#478	1	2	yes	—
67	Dic₆₇	—	4	2	no	—
68	#13520	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	#2087	1	1	yes	characteristic
72	#2161	D₆₇	1	1	yes	characteristic
73	#2151	—	67	1	no	—
74	#1953	—	67	2	no	—
75	#1385	—	67	1	no	—
76	#1381	—	67	2	no	—
77	C₆₇	#13520	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
79	C₂²×C₁₆	—	67	1	no	—
80	#430	—	134	2	no	—
81	C₂×#108	—	67	2	no	—
82	C₂×#108	C₂×D₆₇	1	2	yes	—
83	#405	—	67	1	no	—
84	#368	—	2	2	no	—
85	C₄⋊C₁₆	—	134	2	no	—
86	#344	—	134	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	67	1	no	—
89	C₂×M₁₆	—	134	2	no	—
90	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
91	#117	—	67	1	no	—
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	134	1	no	—
94	C₂×C₁₆	—	134	2	no	—
95	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	268	4	no	—
99	#108	—	67	4	no	—
100	#108	—	134	2	no	—
101	#97	—	134	1	no	—
102	C₄×C₈	—	134	1	no	—
103	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
104	M₁₆	—	4	2	no	—
105	M₁₆	—	4	2	no	—
106	M₁₆	—	268	2	no	—
107	M₁₆	—	134	4	no	—
108	M₁₆	—	134	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	268	2	no	—
111	C₂×C₈	—	134	1	no	—
112	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
113	C₂×C₈	Q₈×D₆₇	1	2	yes	—
114	C₂×C₈	#11761627	1	1	yes	characteristic
115	C₂×C₈	#11761628	1	2	yes	—
116	C₄⋊C₄	—	134	1	no	—
117	C₂²⋊C₄	—	67	2	no	—
118	C₄²	—	134	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	268	1	no	—
121	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
122	C₂×C₄	—	134	2	no	—
123	C₂×C₄	#14524014	1	1	yes	characteristic
124	C₂×C₄	#14524021	1	2	yes	—
125	C₂×C₄	#14524066	1	1	yes	characteristic
126	C₂×C₄	#14524067	1	2	yes	—
127	C₈	—	8	2	no	—
128	C₈	—	268	1	no	—
129	C₈	—	268	4	no	—
130	C₈	#14524026	1	1	yes	characteristic
131	C₈	#14524033	1	2	yes	—
132	C₈	#14524041	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	#16863634	1	1	yes	characteristic
135	C₂²	#16863690	1	2	yes	—
136	C₄	—	268	2	no	—
137	C₄	#16863327	1	1	yes	characteristic
138	C₄	#16863646	1	1	yes	characteristic
139	C₄	#16863699	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	#22561703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561969

C₂

yes

maximal, characteristic

#22560997

C₂

yes

maximal

#22558313

C₂

yes

maximal, characteristic

#22558302

C₂

yes

maximal

C₆₇×#2161

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863444

—

C₂×#14523979

C₂²

yes

—

#16863351

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863167

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#368×C₆₇

—

C₂×#11761612

—

C₂×#11761612

D₄

yes

characteristic

#14524002

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523979

Q₈

yes

—

#14523978

—

#14523953

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

Q₁₆

yes

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#123

yes

characteristic

C₈×C₆₇

#125

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#467

yes

characteristic

C₂²×C₆₇

#477

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13520

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2087

yes

characteristic

#2161

D₆₇

yes

—

—

—

—

C₆₇

#13520

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

—

#430

—

134

—

C₂×#108

—

C₂×#108

C₂×D₆₇

yes

—

#405

—

#368

—

C₄⋊C₁₆

—

134

—

#344

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

134

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#108

—

#108

—

#108

—

268

—

#108

—

100

#108

—

134

—

101

#97

—

134

—

102

C₄×C₈

—

134

—

103

C₂²×C₄

D₄×D₆₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

268

—

107

M₁₆

—

134

—

108

M₁₆

—

134

—

109

C₂×C₈

—

110

C₂×C₈

—

268

—

111

C₂×C₈

—

134

—

112

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

113

C₂×C₈

Q₈×D₆₇

yes

—

114

C₂×C₈

#11761627

yes

characteristic

115

C₂×C₈

#11761628

yes

—

116

C₄⋊C₄

—

134

—

117

C₂²⋊C₄

—

118

C₄²

—

134

—

119

C₁₆

—

120

C₁₆

—

268

—

121

C₂³

Q₁₆×D₆₇

yes

characteristic

122

C₂×C₄

—

134

—

123

C₂×C₄

#14524014

yes

characteristic

124

C₂×C₄

#14524021

yes

—

125

C₂×C₄

#14524066

yes

characteristic

126

C₂×C₄

#14524067

yes

—

127

C₈

—

128

C₈

—

268

—

129

C₈

—

268

—

130

C₈

#14524026

yes

characteristic

131

C₈

#14524033

yes

—

132

C₈

#14524041

yes

characteristic

133

C₂²

—

134

C₂²

#16863634

yes

characteristic

135

C₂²

#16863690

yes

—

136

C₄

—

268

—

137

C₄

#16863327

yes

characteristic

138

C₄

#16863646

yes

characteristic

139

C₄

#16863699

yes

—

140

C₂

—

141

C₂

—

142

C₂

#22561703

yes

characteristic

	subgroup	G/H
center	C₄	#16863646
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2161	C₂
socle	C₂×C₆₇	#2087
prime radical	C₂³×C₆₇	C₂×Q₁₆
hypercenter	#2161	D₆₇
hypocenter	C₆₇	#13520
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#16863646

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2161

C₂

socle

C₂×C₆₇

#2087

prime radical

C₂³×C₆₇

C₂×Q₁₆

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
77. C₆₇	#13520	semidirect

H ≅ G/N

type

77. C₆₇

#13520

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22561969	2
#22560997	2
#22558313	2
#22558302	2
C₆₇×#2161	2
#13520	67

subgroup

index

C₆₇×#2161

Subgroups — #30775850

Conjugacy classes of subgroups

Complements

Maximal subgroups