Group #30767638 — subgroups

Subgroups — #30767638

Order 17152.

206 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562117	C₂	1	2	yes	maximal
2	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22558279	C₂	1	1	yes	maximal, characteristic
4	#22558189	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₄	C₂²	1	2	yes	—
7	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₆₇⋊C₁₆	—	2	2	no	—
10	#16863171	—	2	2	no	—
11	#16863149	C₄	1	1	yes	characteristic
12	#16863128	C₄	1	2	yes	—
13	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₈²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₆₈	C₈	1	1	yes	characteristic
17	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	#14523977	C₈	1	1	yes	characteristic
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₆₇⋊C₈	—	2	2	no	—
22	#14523968	C₈	1	2	yes	—
23	C₈×Dic₆₇	—	2	2	no	—
24	C₂×C₁₆×C₆₇	—	2	1	no	—
25	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
31	Dic₂₆₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
34	Dic₆₇⋊C₄	—	2	2	no	—
35	C₄×Dic₆₇	—	2	2	no	—
36	C₆₇⋊C₁₆	—	4	2	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
42	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₆₇	—	2	2	no	—
44	C₁₆×C₆₇	—	2	1	no	—
45	C₁₆×C₆₇	Q₁₆	1	2	yes	complemented
46	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
49	C₂×Dic₆₇	—	2	2	no	—
50	C₂×Dic₆₇	—	2	2	no	—
51	Dic₁₃₄	—	2	2	no	—
52	Dic₁₃₄	—	2	2	no	—
53	C₈×C₆₇	—	2	2	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	M₃₂	1	2	yes	—
57	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₆₇	#135	1	1	yes	characteristic
59	C₆₇⋊C₈	—	4	2	no	—
60	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
63	C₄×C₆₇	#443	1	1	yes	characteristic
64	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
65	C₄×C₆₇	#504	1	1	yes	characteristic
66	Dic₆₇	—	4	2	no	—
67	Dic₆₇	—	4	2	no	—
68	#12054	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₆₇	#2169	1	1	yes	characteristic
71	C₂×C₆₇	#2170	1	1	yes	characteristic
72	#2183	—	67	2	no	—
73	C₈×Q₁₆	—	67	1	no	—
74	#1370	—	67	1	no	—
75	Q₈⋊C₁₆	—	67	2	no	—
76	C₈×C₁₆	D₆₇	1	1	yes	characteristic
77	C₆₇	#12054	1	1	yes	characteristic, complemented, Sylow
78	C₈×Q₈	—	67	2	no	—
79	C₄×Q₁₆	—	67	1	no	—
80	C₄⋊C₁₆	—	67	2	no	—
81	C₄⋊C₁₆	—	134	2	no	—
82	#346	—	134	2	no	—
83	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
84	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
85	#335	—	67	1	no	—
86	Q₈⋊C₈	—	67	2	no	—
87	C₈²	C₂×D₆₇	1	1	yes	characteristic
88	C₂×Q₁₆	—	67	1	no	—
89	C₄×Q₈	—	67	2	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	134	2	no	—
92	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
93	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
94	#107	—	67	1	no	—
95	C₄⋊C₈	—	67	2	no	—
96	C₄⋊C₈	—	134	2	no	—
97	#103	—	67	2	no	—
98	C₄×C₈	—	134	2	no	—
99	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×Q₈	—	67	2	no	—
103	Q₁₆	—	134	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	134	2	no	—
106	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
107	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
108	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
109	C₂×C₈	#11761628	1	1	yes	characteristic
110	C₄⋊C₄	—	67	2	no	—
111	C₄⋊C₄	—	134	2	no	—
112	C₄²	—	134	2	no	—
113	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	268	2	no	—
117	C₁₆	Dic₂₆₈	1	2	yes	complemented
118	Q₈	—	134	2	no	—
119	Q₈	—	134	2	no	—
120	C₂×C₄	—	134	2	no	—
121	C₂×C₄	—	134	2	no	—
122	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
124	C₂×C₄	#14524049	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	268	2	no	—
129	C₈	#14523982	1	2	yes	—
130	C₈	C₂×Dic₂₆₈	1	1	yes	characteristic
131	C₈	#14524065	1	1	yes	characteristic
132	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	268	2	no	—
135	C₄	—	268	2	no	—
136	C₄	C₄×Dic₂₆₈	1	1	yes	characteristic
137	C₄	#16863514	1	1	yes	characteristic
138	C₄	C₂×#14523982	1	1	yes	characteristic
139	C₄	#16863812	1	1	yes	characteristic
140	C₂	C₈×Dic₂₆₈	1	1	yes	characteristic
141	C₂	#22562042	1	1	yes	characteristic
142	C₂	#22562047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562117

C₂

yes

maximal

C₈×Dic₂₆₈

C₂

yes

maximal, characteristic

#22558279

C₂

yes

maximal, characteristic

#22558189

C₂

yes

maximal

C₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₂²

yes

—

C₄×Dic₂₆₈

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

—

Dic₆₇⋊C₁₆

—

#16863171

—

#16863149

C₄

yes

characteristic

#16863128

C₄

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₈

yes

characteristic

C₄×Dic₁₃₄

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

#14523977

C₈

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

#14523968

C₈

yes

—

C₈×Dic₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

C₂×C₈

yes

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

C₂×C₈

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

Q₁₆

yes

complemented

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₃₂

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

characteristic

C₈×C₆₇

#135

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₁₆

yes

characteristic

C₄×C₆₇

#443

yes

characteristic

C₄×C₆₇

C₂×M₃₂

yes

characteristic

C₄×C₆₇

#504

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#12054

—

maximal, Sylow

C₂×C₆₇

C₈×Q₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

D₆₇

yes

characteristic

C₆₇

#12054

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

#346

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

D₂₆₈

yes

characteristic

C₂×C₁₆

D₂₆₈

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

#103

—

C₄×C₈

—

134

—

C₄×C₈

C₄×D₆₇

yes

characteristic

100

C₄×C₈

C₄×D₆₇

yes

characteristic

101

C₄×C₈

C₂²×D₆₇

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

134

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

C₈×D₆₇

yes

characteristic

107

C₂×C₈

C₈×D₆₇

yes

characteristic

108

C₂×C₈

C₂×D₂₆₈

yes

characteristic

109

C₂×C₈

#11761628

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₄²

—

134

—

113

C₄²

C₂×C₄×D₆₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

268

—

117

C₁₆

Dic₂₆₈

yes

complemented

118

Q₈

—

134

—

119

Q₈

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

C₄×D₂₆₈

yes

characteristic

123

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

124

C₂×C₄

#14524049

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

268

—

129

C₈

#14523982

yes

—

130

C₈

C₂×Dic₂₆₈

yes

characteristic

131

C₈

#14524065

yes

characteristic

132

C₂²

C₈×D₂₆₈

yes

characteristic

133

C₄

—

134

C₄

—

268

—

135

C₄

—

268

—

136

C₄

C₄×Dic₂₆₈

yes

characteristic

137

C₄

#16863514

yes

characteristic

138

C₄

C₂×#14523982

yes

characteristic

139

C₄

#16863812

yes

characteristic

140

C₂

C₈×Dic₂₆₈

yes

characteristic

141

C₂

#22562042

yes

characteristic

142

C₂

#22562047

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×C₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	C₈×D₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₈×C₁₆	D₆₇
hypocenter	C₆₇	#12054
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₈×C₁₆×C₆₇

C₂

socle, prime radical

C₂²×C₆₇

C₈×D₄

cosocle

C₄×C₈×C₆₇

C₂³

hypercenter

C₈×C₁₆

D₆₇

hypocenter

C₆₇

#12054

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

N	H ≅ G/N	type
45. C₁₆×C₆₇	Q₁₆	semidirect
77. C₆₇	#12054	semidirect
117. C₁₆	Dic₂₆₈	semidirect

H ≅ G/N

type

45. C₁₆×C₆₇

Q₁₆

semidirect

77. C₆₇

#12054

semidirect

117. C₁₆

Dic₂₆₈

semidirect

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

subgroup	index
#22562117	2
C₈×Dic₂₆₈	2
#22558279	2
#22558189	2
C₈×C₁₆×C₆₇	2
#12054	67

subgroup

index

C₈×Dic₂₆₈

C₈×C₁₆×C₆₇

Subgroups — #30767638

Conjugacy classes of subgroups

Complements

Maximal subgroups