Group #30772194 — subgroups

Subgroups — #30772194

Order 17152.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22561965	C₂	1	1	yes	maximal, characteristic
2	#22559895	C₂	1	1	yes	maximal, characteristic
3	#22558290	C₂	1	4	yes	maximal
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523954	C₄	1	1	yes	characteristic
7	C₂×#14523953	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	#16863242	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
16	#14523983	—	2	4	no	—
17	C₂×C₆₇⋊C₁₆	—	2	2	no	—
18	#14523954	C₈	1	4	yes	—
19	#14523953	—	2	4	no	—
20	C₈×Dic₆₇	—	2	2	no	—
21	C₈×Dic₆₇	—	2	2	no	—
22	C₆₇⋊#95	—	2	2	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	4	no	—
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	D₄	1	2	yes	—
29	C₄×C₈×C₆₇	Q₈	1	2	yes	—
30	C₂²×Dic₆₇	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
34	C₄×Dic₆₇	—	2	2	no	—
35	C₄×Dic₆₇	M₁₆	1	4	yes	—
36	C₆₇⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	#41	1	2	yes	—
44	C₂×C₈×C₆₇	#41	1	2	yes	—
45	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₆₇	C₂×C₈	1	2	yes	—
47	C₁₆×C₆₇	—	4	2	no	—
48	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#117	1	2	yes	—
53	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
55	C₂×Dic₆₇	—	2	4	no	—
56	C₂×Dic₆₇	—	4	4	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	#124	1	1	yes	characteristic
59	C₈×C₆₇	#124	1	1	yes	characteristic
60	C₈×C₆₇	#125	1	2	yes	—
61	C₆₇⋊C₈	—	2	4	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₆₇	#432	1	2	yes	—
65	C₄×C₆₇	—	2	4	no	—
66	C₄×C₆₇	#382	1	1	yes	characteristic
67	C₄×C₆₇	#423	1	1	yes	characteristic
68	C₄×C₆₇	#431	1	2	yes	—
69	Dic₆₇	—	4	4	no	—
70	#12884	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	#1831	1	1	yes	characteristic
73	C₂×C₆₇	#2153	1	2	yes	—
74	#2150	—	67	1	no	—
75	#2150	D₆₇	1	1	yes	characteristic
76	#1726	—	67	1	no	—
77	#1376	—	67	4	no	—
78	C₆₇	#12884	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₃₂	—	67	2	no	—
80	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
81	C₂×#97	—	67	1	no	—
82	C₂×#97	—	67	2	no	—
83	C₂×C₄×C₈	—	67	2	no	—
84	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
85	#376	—	67	1	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	C₄⋊C₁₆	—	134	2	no	—
88	C₂²×C₈	—	67	2	no	—
89	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
91	C₂×C₄²	—	67	2	no	—
92	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	134	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	134	2	no	—
97	#97	—	67	4	no	—
98	#97	—	134	4	no	—
99	C₄×C₈	—	134	2	no	—
100	C₄×C₈	—	134	2	no	—
101	C₄×C₈	Dic₁₃₄	1	2	yes	—
102	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
103	#95	—	134	2	no	—
104	C₂²×C₄	—	134	2	no	—
105	C₂²×C₄	#11761611	1	2	yes	—
106	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	67	4	no	—
109	C₂×C₈	—	134	2	no	—
110	C₂×C₈	—	134	2	no	—
111	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
113	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₂×C₈	#11761627	1	2	yes	—
115	C₂×C₈	#11761629	1	2	yes	—
116	C₄²	—	67	4	no	—
117	C₄²	—	134	2	no	—
118	C₄²	Dic₆₇⋊C₄	1	2	yes	—
119	C₄²	C₈×D₆₇	1	2	yes	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	268	2	no	—
122	C₂³	C₂×#11761611	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	268	4	no	—
125	C₂×C₄	—	134	4	no	—
126	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
127	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#14524001	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
130	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	134	4	no	—
133	C₈	#14524030	1	1	yes	characteristic
134	C₈	#14524032	1	1	yes	characteristic
135	C₈	#14524034	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
138	C₂²	#16863452	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	268	4	no	—
141	C₄	#16863261	1	1	yes	characteristic
142	C₄	#16863419	1	1	yes	characteristic
143	C₄	#16863448	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#22560325	1	1	yes	characteristic
146	C₂	#22561974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561965

C₂

yes

maximal, characteristic

#22559895

C₂

yes

maximal, characteristic

#22558290

C₂

yes

maximal

C₆₇×#2150

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523954

C₄

yes

characteristic

C₂×#14523953

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863242

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523954

C₈

yes

—

#14523953

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

M₁₆

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12884

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1831

yes

characteristic

C₂×C₆₇

#2153

yes

—

#2150

—

#2150

D₆₇

yes

characteristic

#1726

—

#1376

—

C₆₇

#12884

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

#97

—

#97

—

134

—

C₄×C₈

—

134

—

100

C₄×C₈

—

134

—

101

C₄×C₈

Dic₁₃₄

yes

—

102

C₄×C₈

C₂²⋊D₆₇

yes

—

103

#95

—

134

—

104

C₂²×C₄

—

134

—

105

C₂²×C₄

#11761611

yes

—

106

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

134

—

110

C₂×C₈

—

134

—

111

C₂×C₈

Dic₆₇⋊C₄

yes

—

112

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

113

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

114

C₂×C₈

#11761627

yes

—

115

C₂×C₈

#11761629

yes

—

116

C₄²

—

117

C₄²

—

134

—

118

C₄²

Dic₆₇⋊C₄

yes

—

119

C₄²

C₈×D₆₇

yes

—

120

C₁₆

—

121

C₁₆

—

268

—

122

C₂³

C₂×#11761611

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

134

—

126

C₂×C₄

Dic₆₇⋊C₈

yes

—

127

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

128

C₂×C₄

#14524001

yes

—

129

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

130

C₂×C₄

M₁₆×D₆₇

yes

—

131

C₈

—

132

C₈

—

134

—

133

C₈

#14524030

yes

characteristic

134

C₈

#14524032

yes

characteristic

135

C₈

#14524034

yes

—

136

C₂²

—

137

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

138

C₂²

#16863452

yes

—

139

C₄

—

140

C₄

—

268

—

141

C₄

#16863261

yes

characteristic

142

C₄

#16863419

yes

characteristic

143

C₄

#16863448

yes

—

144

C₂

—

145

C₂

#22560325

yes

characteristic

146

C₂

#22561974

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#12884
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
78. C₆₇	#12884	semidirect

H ≅ G/N

type

78. C₆₇

#12884

semidirect

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

subgroup	index
#22561965	2
#22559895	2
#22558290	2
C₆₇×#2150	2
#12884	67

subgroup

index

C₆₇×#2150

Subgroups — #30772194

Conjugacy classes of subgroups

Complements

Maximal subgroups