Group #30767671 — subgroups

Subgroups — #30767671

Order 17152.

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

normal subgroups only

This group has 212 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	C₁₆×Dic₁₃₄	C₂	1	2	yes	maximal
2	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22558273	C₂	1	1	yes	maximal, characteristic
4	#22558189	C₂	1	2	yes	maximal
5	C₆₇×#1312	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	C₁₆×Dic₆₇	—	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	2	yes	—
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
30	Dic₂₆₈	C₁₆	1	4	yes	complemented
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
33	Dic₆₇⋊C₄	—	2	2	no	—
34	C₄×Dic₆₇	—	2	2	no	—
35	C₆₇⋊C₁₆	—	4	2	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
41	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₆₇	—	2	2	no	—
43	C₁₆×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	2	no	—
48	C₂×Dic₆₇	—	2	2	no	—
49	Dic₁₃₄	—	2	2	no	—
50	Dic₁₃₄	C₂×C₁₆	1	4	yes	—
51	C₈×C₆₇	—	2	1	no	—
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₈×C₆₇	C₂×C₁₆	1	2	yes	—
55	C₈×C₆₇	#137	1	2	yes	—
56	C₆₇⋊C₈	—	4	2	no	—
57	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
58	C₄×C₆₇	—	2	1	no	—
59	C₄×C₆₇	#441	1	1	yes	characteristic
60	C₄×C₆₇	#444	1	1	yes	characteristic
61	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
62	C₄×C₆₇	#504	1	1	yes	characteristic
63	Dic₆₇	—	2	4	no	—
64	Dic₆₇	—	4	2	no	—
65	#12059	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	#1579	1	1	yes	characteristic
67	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
68	C₂×C₆₇	#2171	1	1	yes	characteristic
69	Q₈×C₁₆	—	67	2	no	—
70	C₈×Q₁₆	—	67	1	no	—
71	#1367	—	67	1	no	—
72	Q₈⋊C₁₆	—	67	2	no	—
73	#1312	D₆₇	1	1	yes	characteristic
74	C₆₇	#12059	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	67	2	no	—
76	C₄×Q₁₆	—	67	1	no	—
77	C₄⋊C₁₆	—	67	2	no	—
78	C₄⋊C₁₆	—	134	2	no	—
79	C₄×C₁₆	—	134	2	no	—
80	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
82	#335	—	67	1	no	—
83	Q₈⋊C₈	—	67	2	no	—
84	C₈²	C₂×D₆₇	1	1	yes	characteristic
85	C₂×Q₁₆	—	67	1	no	—
86	C₄×Q₈	—	67	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	134	2	no	—
89	C₂×C₁₆	D₂₆₈	1	2	yes	—
90	#107	—	67	1	no	—
91	C₄⋊C₈	—	67	2	no	—
92	C₄⋊C₈	—	134	2	no	—
93	#103	—	67	2	no	—
94	C₄×C₈	—	134	2	no	—
95	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
96	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×Q₈	—	67	2	no	—
99	Q₁₆	—	67	4	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	134	2	no	—
102	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
103	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
105	C₂×C₈	#11761628	1	1	yes	characteristic
106	C₄⋊C₄	—	67	2	no	—
107	C₄⋊C₄	—	134	2	no	—
108	C₄²	—	134	2	no	—
109	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	268	2	no	—
113	Q₈	—	67	4	no	—
114	Q₈	—	134	2	no	—
115	C₂×C₄	—	134	2	no	—
116	C₂×C₄	—	134	2	no	—
117	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
119	C₂×C₄	#14524049	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	268	2	no	—
124	C₈	C₁₆×D₆₇	1	2	yes	—
125	C₈	#14524076	1	2	yes	—
126	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	268	2	no	—
129	C₄	—	134	4	no	—
130	C₄	#16863504	1	1	yes	characteristic
131	C₄	#16863519	1	1	yes	characteristic
132	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
133	C₄	#16863812	1	1	yes	characteristic
134	C₂	#22559029	1	1	yes	characteristic
135	C₂	C₁₆×D₂₆₈	1	1	yes	characteristic
136	C₂	#22562052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₃₄

C₂

yes

maximal

C₈×Dic₂₆₈

C₂

yes

maximal, characteristic

#22558273

C₂

yes

maximal, characteristic

#22558189

C₂

yes

maximal

C₆₇×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₂²

yes

—

C₄×Dic₂₆₈

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

—

Dic₆₇⋊C₁₆

—

C₁₆×Dic₆₇

—

#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

—

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

—

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

—

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

#137

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂²×C₁₆

yes

characteristic

C₄×C₆₇

#504

yes

characteristic

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

D₄×C₁₆

yes

characteristic

C₂×C₆₇

#2171

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₆₇

yes

characteristic

C₆₇

#12059

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₄×C₁₆

—

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

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

#103

—

C₄×C₈

—

134

—

C₄×C₈

C₄×D₆₇

yes

characteristic

C₄×C₈

C₄×D₆₇

yes

characteristic

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

134

—

102

C₂×C₈

C₈×D₆₇

yes

characteristic

103

C₂×C₈

C₈×D₆₇

yes

characteristic

104

C₂×C₈

C₂×D₂₆₈

yes

characteristic

105

C₂×C₈

#11761628

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

134

—

108

C₄²

—

134

—

109

C₄²

C₂×C₄×D₆₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

268

—

113

Q₈

—

114

Q₈

—

134

—

115

C₂×C₄

—

134

—

116

C₂×C₄

—

134

—

117

C₂×C₄

C₄×D₂₆₈

yes

characteristic

118

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

119

C₂×C₄

#14524049

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

268

—

124

C₈

C₁₆×D₆₇

yes

—

125

C₈

#14524076

yes

—

126

C₂²

C₈×D₂₆₈

yes

characteristic

127

C₄

—

128

C₄

—

268

—

129

C₄

—

134

—

130

C₄

#16863504

yes

characteristic

131

C₄

#16863519

yes

characteristic

132

C₄

C₂×C₁₆×D₆₇

yes

characteristic

133

C₄

#16863812

yes

characteristic

134

C₂

#22559029

yes

characteristic

135

C₂

C₁₆×D₂₆₈

yes

characteristic

136

C₂

#22562052

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1312	C₂
socle, prime radical	C₂²×C₆₇	C₈×D₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#12059
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₂₆₈

commutator

C₄×C₆₇

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1312

C₂

socle, prime radical

C₂²×C₆₇

C₈×D₄

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
30. Dic₂₆₈	C₁₆	semidirect
74. C₆₇	#12059	semidirect

H ≅ G/N

type

30. Dic₂₆₈

C₁₆

semidirect

74. C₆₇

#12059

semidirect

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

subgroup	index
C₁₆×Dic₁₃₄	2
C₈×Dic₂₆₈	2
#22558273	2
#22558189	2
C₆₇×#1312	2
#12059	67

subgroup

index

C₁₆×Dic₁₃₄

C₈×Dic₂₆₈

#22558273

#22558189

C₆₇×#1312

#12059

Subgroups — #30767671

Conjugacy classes of subgroups

Complements

Maximal subgroups