Group #30767682 — subgroups

Subgroups — #30767682

Order 17152.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22562118	C₂	1	2	yes	maximal
2	#22558273	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
4	#22558127	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
7	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
8	#16863172	—	2	2	no	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
13	#335×C₆₇	C₄	1	1	yes	characteristic
14	Q₈⋊C₈×C₆₇	C₄	1	2	yes	—
15	C₈²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₆₇⋊C₁₆	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	—	2	1	no	—
18	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
19	C₂×Q₁₆×C₆₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
21	#107×C₆₇	C₈	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	—	2	2	no	—
23	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
24	#103×C₆₇	C₈	1	2	yes	—
25	C₄×C₈×C₆₇	—	2	2	no	—
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	C₆₇⋊C₁₆	—	2	2	no	—
30	C₆₇⋊C₁₆	—	4	1	no	—
31	C₆₇⋊C₁₆	—	4	2	no	—
32	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
33	Q₁₆×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
42	C₄²×C₆₇	—	2	2	no	—
43	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₆₇	—	2	2	no	—
45	Q₈×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
51	C₈×C₆₇	—	2	1	no	—
52	C₈×C₆₇	—	2	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₈×C₆₇	—	4	2	no	—
55	C₈×C₆₇	M₃₂	1	2	yes	—
56	C₈×C₆₇	#137	1	2	yes	—
57	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
58	C₄×C₆₇	—	2	1	no	—
59	C₄×C₆₇	—	4	2	no	—
60	C₄×C₆₇	—	4	2	no	—
61	C₄×C₆₇	#441	1	1	yes	characteristic
62	C₄×C₆₇	#444	1	1	yes	characteristic
63	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
64	C₄×C₆₇	#504	1	1	yes	characteristic
65	#12061	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	#1579	1	1	yes	characteristic
67	C₂×C₆₇	#2169	1	1	yes	characteristic
68	C₂×C₆₇	#2171	1	1	yes	characteristic
69	#2183	—	67	2	no	—
70	C₈×Q₁₆	D₆₇	1	1	yes	characteristic
71	#1367	—	67	1	no	—
72	Q₈⋊C₁₆	—	67	2	no	—
73	#1312	—	67	1	no	—
74	C₆₇	#12061	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	C₂×D₆₇	1	2	yes	—
76	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
77	C₄⋊C₁₆	—	67	2	no	—
78	C₄⋊C₁₆	—	134	2	no	—
79	#346	—	134	2	no	—
80	C₄×C₁₆	—	67	1	no	—
81	C₄×C₁₆	—	67	1	no	—
82	#335	Dic₆₇	1	1	yes	characteristic
83	Q₈⋊C₈	Dic₆₇	1	2	yes	—
84	C₈²	C₂×D₆₇	1	1	yes	characteristic
85	C₂×Q₁₆	C₆₇⋊C₈	1	1	yes	characteristic
86	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
87	C₂×C₁₆	—	67	2	no	—
88	C₂×C₁₆	—	134	2	no	—
89	C₂×C₁₆	—	134	1	no	—
90	#107	C₆₇⋊C₈	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
93	#103	C₆₇⋊C₈	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
104	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
105	C₂×C₈	#11761627	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
110	C₁₆	—	268	1	no	—
111	C₁₆	—	268	2	no	—
112	C₁₆	—	134	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#14524050	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	#14523983	1	2	yes	—
125	C₈	#14524075	1	2	yes	—
126	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	#16863503	1	1	yes	characteristic
131	C₄	#16863518	1	1	yes	characteristic
132	C₄	C₂×#14523983	1	1	yes	characteristic
133	C₄	#16863813	1	1	yes	characteristic
134	C₂	#22559028	1	1	yes	characteristic
135	C₂	#22562041	1	1	yes	characteristic
136	C₂	#22562051	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562118

C₂

yes

maximal

#22558273

C₂

yes

maximal, characteristic

Q₈⋊(C₆₇⋊C₁₆)

C₂

yes

maximal

#22558127

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₄⋊(C₆₇⋊C₁₆)

C₂²

yes

—

#16863172

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×Q₈×C₆₇

C₂²

yes

—

C₄×Q₁₆×C₆₇

C₄

yes

characteristic

#335×C₆₇

C₄

yes

characteristic

Q₈⋊C₈×C₆₇

C₄

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₆₇

C₈

yes

characteristic

C₄×Q₈×C₆₇

C₂×C₄

yes

—

#107×C₆₇

C₈

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂×C₄

yes

—

#103×C₆₇

C₈

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

C₂×C₈

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

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

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₃₂

yes

—

C₈×C₆₇

#137

yes

—

C₂²×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×M₃₂

yes

C₄×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₈×Q₁₆

D₆₇

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₆₇

#12061

yes

characteristic, complemented, Sylow

C₈×Q₈

C₂×D₆₇

yes

—

C₄×Q₁₆

Dic₆₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

#346

—

134

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₆₇

yes

characteristic

Q₈⋊C₈

Dic₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

C₆₇⋊C₈

yes

characteristic

C₄×Q₈

C₂×Dic₆₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

—

134

—

#107

C₆₇⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₆₇

yes

—

#103

C₆₇⋊C₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

C₂×C₆₇⋊C₈

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₆₇⋊C₈

yes

characteristic

103

C₂×C₈

C₂×C₆₇⋊C₈

yes

characteristic

104

C₂×C₈

D₄×D₆₇

yes

characteristic

105

C₂×C₈

#11761627

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₆₇⋊C₈

yes

—

108

C₄²

—

109

C₄²

C₂²×Dic₆₇

yes

characteristic

110

C₁₆

—

268

—

111

C₁₆

—

268

—

112

C₁₆

—

134

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×Dic₆₇

yes

characteristic

118

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

119

C₂×C₄

#14524050

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

#14523983

yes

—

125

C₈

#14524075

yes

—

126

C₂²

D₄×C₆₇⋊C₈

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16863503

yes

characteristic

131

C₄

#16863518

yes

characteristic

132

C₄

C₂×#14523983

yes

characteristic

133

C₄

#16863813

yes

characteristic

134

C₂

#22559028

yes

characteristic

135

C₂

#22562041

yes

characteristic

136

C₂

#22562051

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×Q₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	C₈×D₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₈×Q₁₆	D₆₇
hypocenter	C₆₇	#12061
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₈×Q₁₆×C₆₇

C₂

socle, prime radical

C₂²×C₆₇

C₈×D₄

cosocle

C₄×C₈×C₆₇

C₂³

hypercenter

C₈×Q₁₆

D₆₇

hypocenter

C₆₇

#12061

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
74. C₆₇	#12061	semidirect

H ≅ G/N

type

74. C₆₇

#12061

semidirect

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

subgroup	index
#22562118	2
#22558273	2
Q₈⋊(C₆₇⋊C₁₆)	2
#22558127	2
C₈×Q₁₆×C₆₇	2
#12061	67

subgroup

index

Q₈⋊(C₆₇⋊C₁₆)

C₈×Q₁₆×C₆₇

Subgroups — #30767682

Conjugacy classes of subgroups

Complements

Maximal subgroups