Group #30767624 — subgroups

Subgroups — #30767624

Order 17152.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22562111	C₂	1	2	yes	maximal
2	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22558187	C₂	1	2	yes	maximal
4	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1370	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₄	C₂²	1	2	yes	—
7	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
8	Dic₆₇⋊C₁₆	—	2	2	no	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₆₇	—	2	2	no	—
12	#16863149	C₄	1	1	yes	characteristic
13	#16863128	C₄	1	2	yes	—
14	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
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	1	no	—
19	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
21	#14523977	C₈	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
23	Dic₆₇⋊C₈	—	2	2	no	—
24	#14523968	C₈	1	2	yes	—
25	C₈×Dic₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	—	2	2	no	—
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	C₁₆	1	4	yes	complemented
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	—	2	2	no	—
37	C₆₇⋊C₁₆	—	2	1	no	—
38	C₆₇⋊C₁₆	Q₁₆	1	2	yes	complemented
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
44	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₆₇	—	4	2	no	—
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	C₂×C₁₆	1	4	yes	—
53	C₈×C₆₇	—	2	2	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	C₂×C₁₆	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	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₆₇	#504	1	1	yes	characteristic
66	Dic₆₇	—	2	4	no	—
67	Dic₆₇	—	4	2	no	—
68	C₁₆×Q₁₆	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₆₇	#2170	1	1	yes	characteristic
72	Q₈×C₁₆	—	67	2	no	—
73	C₈×Q₁₆	—	67	1	no	—
74	#1370	D₆₇	1	1	yes	characteristic
75	Q₈⋊C₁₆	—	67	2	no	—
76	C₈×C₁₆	—	67	1	no	—
77	C₆₇	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₈×Q₈	—	67	2	no	—
79	C₄×Q₁₆	—	67	1	no	—
80	C₄⋊C₁₆	—	134	2	no	—
81	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
82	C₄×C₁₆	—	67	1	no	—
83	C₄×C₁₆	—	67	1	no	—
84	C₄×C₁₆	—	134	2	no	—
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	2	no	—
91	C₂×C₁₆	—	67	1	no	—
92	C₂×C₁₆	—	67	1	no	—
93	C₂×C₁₆	—	134	1	no	—
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	—	67	4	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	D₄×D₆₇	1	1	yes	characteristic
109	C₂×C₈	#11761629	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	—	4	2	no	—
115	C₁₆	—	67	2	no	—
116	C₁₆	—	134	2	no	—
117	C₁₆	—	134	1	no	—
118	Q₈	—	67	4	no	—
119	Q₈	—	134	2	no	—
120	C₂×C₄	—	134	2	no	—
121	C₂×C₄	—	134	2	no	—
122	C₂×C₄	#14524005	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
124	C₂×C₄	#14524051	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	C₁₆×D₆₇	1	2	yes	—
130	C₈	Q₁₆×D₆₇	1	1	yes	characteristic
131	C₈	#14524062	1	1	yes	characteristic
132	C₂²	#16863461	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	268	2	no	—
135	C₄	—	134	4	no	—
136	C₄	#16863490	1	1	yes	characteristic
137	C₄	#16863512	1	1	yes	characteristic
138	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
139	C₄	#16863814	1	1	yes	characteristic
140	C₂	#22559015	1	1	yes	characteristic
141	C₂	#22562028	1	1	yes	characteristic
142	C₂	#22562045	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562111

C₂

yes

maximal

C₈×Dic₂₆₈

C₂

yes

maximal, characteristic

#22558187

C₂

yes

maximal

C₈×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic

C₆₇×#1370

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₂²

yes

—

C₄×Dic₂₆₈

C₄

yes

characteristic

Dic₆₇⋊C₁₆

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₁₆×Dic₆₇

—

#16863149

C₄

yes

characteristic

#16863128

C₄

yes

—

C₄⋊C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₈

yes

characteristic

C₄×Dic₁₃₄

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

#14523977

C₈

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

#14523968

C₈

yes

—

C₈×Dic₆₇

—

C₂×C₁₆×C₆₇

—

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

C₁₆

yes

complemented

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

C₂×C₈

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

Q₁₆

yes

complemented

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

—

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

C₂×C₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₁₆

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

C₂²×C₁₆

yes

characteristic

C₄×C₆₇

#504

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₁₆×Q₁₆

—

maximal, Sylow

C₂×C₆₇

C₈×Q₁₆

yes

characteristic

C₂×C₆₇

D₄×C₁₆

yes

characteristic

C₂×C₆₇

#2170

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

D₆₇

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₆₇

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

134

—

C₄⋊C₁₆

C₂×D₆₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

134

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

#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

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

C₈×D₆₇

yes

characteristic

107

C₂×C₈

C₈×D₆₇

yes

characteristic

108

C₂×C₈

D₄×D₆₇

yes

characteristic

109

C₂×C₈

#11761629

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

—

134

—

117

C₁₆

—

134

—

118

Q₈

—

119

Q₈

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

#14524005

yes

characteristic

123

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

124

C₂×C₄

#14524051

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

268

—

129

C₈

C₁₆×D₆₇

yes

—

130

C₈

Q₁₆×D₆₇

yes

characteristic

131

C₈

#14524062

yes

characteristic

132

C₂²

#16863461

yes

characteristic

133

C₄

—

134

C₄

—

268

—

135

C₄

—

134

—

136

C₄

#16863490

yes

characteristic

137

C₄

#16863512

yes

characteristic

138

C₄

C₂×C₁₆×D₆₇

yes

characteristic

139

C₄

#16863814

yes

characteristic

140

C₂

#22559015

yes

characteristic

141

C₂

#22562028

yes

characteristic

142

C₂

#22562045

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1370	C₂
socle, prime radical	C₂²×C₆₇	C₈×D₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1370	D₆₇
hypocenter	C₆₇	C₁₆×Q₁₆
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₆₇

commutator

C₄×C₆₇

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1370

C₂

socle, prime radical

C₂²×C₆₇

C₈×D₄

cosocle

C₄×C₈×C₆₇

C₂³

hypercenter

#1370

D₆₇

hypocenter

C₆₇

C₁₆×Q₁₆

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
31. Dic₂₆₈	C₁₆	semidirect
38. C₆₇⋊C₁₆	Q₁₆	semidirect
77. C₆₇	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

31. Dic₂₆₈

C₁₆

semidirect

38. C₆₇⋊C₁₆

Q₁₆

semidirect

77. C₆₇

C₁₆×Q₁₆

semidirect

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

subgroup	index
#22562111	2
C₈×Dic₂₆₈	2
#22558187	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1370	2
C₁₆×Q₁₆	67

subgroup

index

#22562111

C₈×Dic₂₆₈

#22558187

C₈×C₆₇⋊C₁₆

C₆₇×#1370

C₁₆×Q₁₆

Subgroups — #30767624

Conjugacy classes of subgroups

Complements

Maximal subgroups