Group #30770832 — subgroups

Subgroups — #30770832

Order 17152.

390 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal
4	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
10	#16863172	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#346×C₆₇	C₄	1	4	yes	complemented
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₈	1	4	yes	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₆₇⋊C₁₆	—	2	4	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄×C₆₇⋊C₈	C₈	1	4	yes	—
20	C₈×Dic₆₇	—	2	4	no	—
21	C₈×Dic₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	—	2	4	no	—
26	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
30	C₂×C₆₇⋊C₈	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
33	C₄×Dic₆₇	—	2	2	no	—
34	C₄×Dic₆₇	—	2	2	no	—
35	C₄×Dic₆₇	—	2	8	no	—
36	C₆₇⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₄²	1	2	yes	—
40	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	#41	1	4	yes	—
44	C₄²×C₆₇	C₄²	1	2	yes	—
45	C₄²×C₆₇	C₂×C₈	1	2	yes	—
46	C₁₆×C₆₇	—	2	4	no	—
47	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	4	no	—
49	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#117	1	2	yes	—
52	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
54	C₂×Dic₆₇	—	2	4	no	—
55	C₂×Dic₆₇	—	2	4	no	—
56	C₂×Dic₆₇	M₃₂	1	8	yes	—
57	C₈×C₆₇	#97	1	4	yes	—
58	C₈×C₆₇	#117	1	2	yes	—
59	C₈×C₆₇	#117	1	2	yes	—
60	C₆₇⋊C₈	—	2	4	no	—
61	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂²×C₆₇	#431	1	2	yes	—
63	C₂²×C₆₇	C₂×M₃₂	1	4	yes	—
64	C₄×C₆₇	—	2	4	no	—
65	C₄×C₆₇	#376	1	1	yes	characteristic
66	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
67	C₄×C₆₇	#431	1	2	yes	—
68	Dic₆₇	—	2	8	no	—
69	#12591	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	#1744	1	1	yes	characteristic
71	C₂×C₆₇	C₄×M₃₂	1	2	yes	—
72	C₂×C₆₇	#2163	1	4	yes	—
73	C₂×#346	—	67	1	no	—
74	C₂×#346	D₆₇	1	1	yes	characteristic
75	C₄²×C₈	—	67	1	no	—
76	#1374	—	67	4	no	—
77	C₆₇	#12591	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₁₆	—	67	2	no	—
79	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
80	C₂×C₄×C₈	—	67	1	no	—
81	C₂×C₄×C₈	—	67	2	no	—
82	C₂×C₄×C₈	—	67	2	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	C₄³	—	67	1	no	—
85	#346	—	67	4	no	—
86	#346	Dic₆₇	1	4	yes	complemented
87	C₂²×C₈	—	67	2	no	—
88	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
90	C₂×C₄²	—	67	4	no	—
91	C₂×C₄²	—	67	2	no	—
92	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	67	4	no	—
95	C₂×C₁₆	—	134	4	no	—
96	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
97	C₄×C₈	—	67	4	no	—
98	C₄×C₈	—	134	4	no	—
99	C₄×C₈	—	134	4	no	—
100	C₄×C₈	C₄×D₆₇	1	2	yes	—
101	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
102	C₂²×C₄	—	67	4	no	—
103	C₂²×C₄	C₈×D₆₇	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
105	C₂×C₈	—	67	4	no	—
106	C₂×C₈	—	134	2	no	—
107	C₂×C₈	—	134	2	no	—
108	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
109	C₂×C₈	#11761611	1	4	yes	—
110	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
111	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	#11761628	1	4	yes	—
113	C₄²	—	134	8	no	—
114	C₄²	—	134	2	no	—
115	C₄²	—	134	2	no	—
116	C₄²	C₄×Dic₆₇	1	2	yes	—
117	C₄²	C₈×D₆₇	1	2	yes	—
118	C₁₆	—	2	4	no	—
119	C₁₆	—	134	4	no	—
120	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	67	8	no	—
123	C₂×C₄	—	134	4	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
126	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
127	C₂×C₄	#14524000	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×#11761611	1	2	yes	—
130	C₈	—	134	4	no	—
131	C₈	#14523953	1	4	yes	—
132	C₈	#14524000	1	2	yes	—
133	C₈	#14524002	1	2	yes	—
134	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
135	C₂²	#16863447	1	2	yes	—
136	C₂²	M₃₂×D₆₇	1	4	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	134	8	no	—
139	C₄	#16863242	1	1	yes	characteristic
140	C₄	C₂×#14523953	1	1	yes	characteristic
141	C₄	#16863447	1	2	yes	—
142	C₂	#22559954	1	1	yes	characteristic
143	C₂	M₃₂×Dic₆₇	1	2	yes	—
144	C₂	#22562010	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863172

C₂

yes

maximal, characteristic

C₄×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#22558286

C₂

yes

maximal

C₂×#346×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₆₇

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

C₄²×Dic₆₇

C₄

yes

characteristic

#16863172

C₄

yes

complemented

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₈

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₈

yes

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄²

yes

—

C₄²×C₆₇

C₂×C₈

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

M₃₂

yes

—

C₈×C₆₇

#97

yes

—

C₈×C₆₇

#117

yes

—

C₈×C₆₇

#117

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₇

#431

yes

—

C₂²×C₆₇

C₂×M₃₂

yes

—

C₄×C₆₇

—

C₄×C₆₇

#376

yes

characteristic

C₄×C₆₇

C₂×#97

yes

characteristic

C₄×C₆₇

#431

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

C₄×M₃₂

yes

—

C₂×C₆₇

#2163

yes

—

C₂×#346

—

C₂×#346

D₆₇

yes

characteristic

C₄²×C₈

—

#1374

—

C₆₇

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄³

—

#346

—

#346

Dic₆₇

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₂×Dic₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

134

—

C₄×C₈

—

134

—

100

C₄×C₈

C₄×D₆₇

yes

—

101

C₄×C₈

C₂×Dic₆₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

C₈×D₆₇

yes

—

104

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

C₄×Dic₆₇

yes

—

109

C₂×C₈

#11761611

yes

—

110

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

111

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

112

C₂×C₈

#11761628

yes

—

113

C₄²

—

134

—

114

C₄²

—

134

—

115

C₄²

—

134

—

116

C₄²

C₄×Dic₆₇

yes

—

117

C₄²

C₈×D₆₇

yes

—

118

C₁₆

—

119

C₁₆

—

134

—

120

C₂³

C₂×C₈×D₆₇

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

C₈×Dic₆₇

yes

—

126

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

127

C₂×C₄

#14524000

yes

—

128

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

129

C₂×C₄

C₂×#11761611

yes

—

130

C₈

—

134

—

131

C₈

#14523953

yes

—

132

C₈

#14524000

yes

—

133

C₈

#14524002

yes

—

134

C₂²

C₂×C₈×Dic₆₇

yes

characteristic

135

C₂²

#16863447

yes

—

136

C₂²

M₃₂×D₆₇

yes

—

137

C₄

—

138

C₄

—

134

—

139

C₄

#16863242

yes

characteristic

140

C₄

C₂×#14523953

yes

characteristic

141

C₄

#16863447

yes

—

142

C₂

#22559954

yes

characteristic

143

C₂

M₃₂×Dic₆₇

yes

—

144

C₂

#22562010

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#12591
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₇

commutator

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₂×#346×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂²×C₈

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
10. #16863172	C₄	semidirect
13. #346×C₆₇	C₄	semidirect
77. C₆₇	#12591	semidirect
86. #346	Dic₆₇	semidirect

H ≅ G/N

type

10. #16863172

C₄

semidirect

13. #346×C₆₇

C₄

semidirect

77. C₆₇

#12591

semidirect

86. #346

Dic₆₇

semidirect

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

subgroup	index
C₂×#16863172	2
C₄×C₈×Dic₆₇	2
#22558286	2
C₂×#346×C₆₇	2
#12591	67

subgroup

index

C₂×#16863172

C₄×C₈×Dic₆₇

#22558286

C₂×#346×C₆₇

#12591

Subgroups — #30770832

Conjugacy classes of subgroups

Complements

Maximal subgroups