Group #30775836 — subgroups

Subgroups — #30775836

Order 17152.

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

normal subgroups only

This group has 250 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	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
2	#22561967	C₂	1	1	yes	maximal, characteristic
3	#22560995	C₂	1	2	yes	maximal
4	#22558319	C₂	1	2	yes	maximal
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	#16863438	—	2	2	no	—
8	C₂×#14523977	C₂²	1	2	yes	—
9	#16863351	C₂²	1	1	yes	characteristic
10	#16863222	—	2	2	no	—
11	Dic₆₇⋊C₁₆	—	2	2	no	—
12	#16863164	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₇	C₂²	1	2	yes	—
15	C₂×#11761612	D₄	1	1	yes	characteristic
16	#14524002	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	#14523977	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523977	—	4	2	no	—
23	#14523975	—	4	2	no	—
24	#14523953	—	2	1	no	—
25	C₈×Dic₆₇	—	2	1	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
30	#108×C₆₇	—	2	2	no	—
31	#108×C₆₇	Q₈	1	4	yes	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	#11761612	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
38	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
39	C₂²⋊Dic₆₇	—	2	1	no	—
40	C₄×Dic₆₇	—	2	1	no	—
41	C₆₇⋊C₁₆	—	4	1	no	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₆₇	—	2	4	no	—
44	M₁₆×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
50	C₁₆×C₆₇	—	2	2	no	—
51	C₂³×C₆₇	#135	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#137	1	2	yes	—
56	C₂×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	2	1	no	—
58	C₂×Dic₆₇	—	8	2	no	—
59	C₂×Dic₆₇	—	16	2	no	—
60	C₈×C₆₇	—	4	4	no	—
61	C₈×C₆₇	#123	1	1	yes	characteristic
62	C₈×C₆₇	#125	1	2	yes	—
63	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₆₇⋊C₈	—	4	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	#466	1	1	yes	characteristic
68	C₂²×C₆₇	#479	1	2	yes	—
69	C₄×C₆₇	#398	1	1	yes	characteristic
70	C₄×C₆₇	#470	1	1	yes	characteristic
71	C₄×C₆₇	#474	1	2	yes	—
72	Dic₆₇	—	4	1	no	—
73	Dic₆₇	—	4	1	no	—
74	Dic₆₇	—	16	2	no	—
75	#13517	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	—	2	2	no	—
77	C₂×C₆₇	#2086	1	1	yes	characteristic
78	C₂×C₆₇	#2221	1	1	yes	characteristic
79	C₂×C₆₇	#2222	1	1	yes	characteristic
80	C₂×#366	—	67	1	no	—
81	#2151	—	67	1	no	—
82	#1952	—	67	2	no	—
83	#1387	—	67	2	no	—
84	#1381	D₆₇	1	1	yes	characteristic
85	C₆₇	#13517	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	67	1	no	—
87	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
88	C₂×#108	C₂×D₆₇	1	2	yes	—
89	#428	—	134	2	no	—
90	C₂×#107	—	67	2	no	—
91	#405	—	67	1	no	—
92	#366	—	134	2	no	—
93	C₄⋊C₁₆	—	134	2	no	—
94	#343	—	134	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	67	1	no	—
97	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	#117	—	67	1	no	—
99	C₂×C₄⋊C₄	—	134	2	no	—
100	M₃₂	—	134	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	134	1	no	—
103	C₂×C₁₆	D₂₆₈	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	Dic₁₃₄	1	4	yes	—
106	#107	—	268	2	no	—
107	#107	—	134	2	no	—
108	#107	—	134	2	no	—
109	#106	—	268	2	no	—
110	#97	—	134	1	no	—
111	C₄×C₈	—	134	1	no	—
112	C₂²×C₄	—	268	2	no	—
113	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
114	M₁₆	—	2	4	no	—
115	M₁₆	—	4	2	no	—
116	M₁₆	—	134	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	134	1	no	—
119	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
121	C₂×C₈	#11761627	1	2	yes	—
122	C₂×C₈	#11761629	1	1	yes	characteristic
123	C₄⋊C₄	—	268	2	no	—
124	C₄⋊C₄	—	268	2	no	—
125	C₄⋊C₄	—	67	1	no	—
126	C₄⋊C₄	—	67	1	no	—
127	C₂²⋊C₄	—	134	1	no	—
128	C₄²	—	134	1	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	268	1	no	—
131	C₂³	#14524066	1	1	yes	characteristic
132	C₂×C₄	—	536	2	no	—
133	C₂×C₄	—	1072	2	no	—
134	C₂×C₄	—	134	1	no	—
135	C₂×C₄	—	134	1	no	—
136	C₂×C₄	#14524018	1	1	yes	characteristic
137	C₂×C₄	#14524020	1	2	yes	—
138	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
139	C₂×C₄	#14524072	1	2	yes	—
140	C₈	—	4	4	no	—
141	C₈	—	268	1	no	—
142	C₈	#14524027	1	1	yes	characteristic
143	C₈	#14524034	1	2	yes	—
144	C₈	#14524043	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16863633	1	1	yes	characteristic
148	C₂²	#16863705	1	2	yes	—
149	C₄	—	268	1	no	—
150	C₄	—	268	1	no	—
151	C₄	—	1072	2	no	—
152	C₄	#16863328	1	1	yes	characteristic
153	C₄	#16863655	1	1	yes	characteristic
154	C₄	#16863674	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#22561699	1	1	yes	characteristic
157	C₂	#22562330	1	1	yes	characteristic
158	C₂	#22562335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863222

C₂

yes

maximal, characteristic

#22561967

C₂

yes

maximal, characteristic

#22560995

C₂

yes

maximal

#22558319

C₂

yes

maximal

C₆₇×#1381

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

#16863438

—

C₂×#14523977

C₂²

yes

—

#16863351

C₂²

yes

characteristic

#16863222

—

Dic₆₇⋊C₁₆

—

#16863164

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂×#11761612

D₄

yes

characteristic

#14524002

D₄

yes

characteristic

C₂×C₄⋊Dic₆₇

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

—

—

—

—

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₈

yes

characteristic

C₄⋊Dic₆₇

D₈

yes

characteristic

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#123

yes

characteristic

C₈×C₆₇

#125

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#466

yes

characteristic

C₂²×C₆₇

#479

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

#13517

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#13517

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

C₂×#108

C₂×D₆₇

yes

—

#428

—

134

—

C₂×#107

—

#405

—

#366

—

134

—

C₄⋊C₁₆

—

134

—

#343

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

134

—

100

M₃₂

—

134

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

134

—

103

C₂×C₁₆

D₂₆₈

yes

—

104

#108

—

105

#108

Dic₁₃₄

yes

—

106

#107

—

268

—

107

#107

—

134

—

108

#107

—

134

—

109

#106

—

268

—

110

#97

—

134

—

111

C₄×C₈

—

134

—

112

C₂²×C₄

—

268

—

113

C₂²×C₄

D₄×D₆₇

yes

characteristic

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

134

—

117

C₂×C₈

—

118

C₂×C₈

—

134

—

119

C₂×C₈

C₂×D₂₆₈

yes

characteristic

120

C₂×C₈

C₂×Dic₁₃₄

yes

—

121

C₂×C₈

#11761627

yes

—

122

C₂×C₈

#11761629

yes

characteristic

123

C₄⋊C₄

—

268

—

124

C₄⋊C₄

—

268

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

134

—

128

C₄²

—

134

—

129

C₁₆

—

130

C₁₆

—

268

—

131

C₂³

#14524066

yes

characteristic

132

C₂×C₄

—

536

—

133

C₂×C₄

—

1072

—

134

C₂×C₄

—

134

—

135

C₂×C₄

—

134

—

136

C₂×C₄

#14524018

yes

characteristic

137

C₂×C₄

#14524020

yes

—

138

C₂×C₄

D₈×D₆₇

yes

characteristic

139

C₂×C₄

#14524072

yes

—

140

C₈

—

141

C₈

—

268

—

142

C₈

#14524027

yes

characteristic

143

C₈

#14524034

yes

—

144

C₈

#14524043

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

#16863633

yes

characteristic

148

C₂²

#16863705

yes

—

149

C₄

—

268

—

150

C₄

—

268

—

151

C₄

—

1072

—

152

C₄

#16863328

yes

characteristic

153

C₄

#16863655

yes

characteristic

154

C₄

#16863674

yes

—

155

C₂

—

156

C₂

#22561699

yes

characteristic

157

C₂

#22562330

yes

characteristic

158

C₂

#22562335

yes

characteristic

	subgroup	G/H
center	C₂²	#16863633
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1381	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂³×C₆₇	#135
hypercenter	#1381	D₆₇
hypocenter	C₆₇	#13517
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16863633

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1381

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
85. C₆₇	#13517	semidirect

H ≅ G/N

type

85. C₆₇

#13517

semidirect

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

subgroup	index
C₂×#16863222	2
#22561967	2
#22560995	2
#22558319	2
C₆₇×#1381	2
#13517	67

subgroup

index

C₆₇×#1381

Subgroups — #30775836

Conjugacy classes of subgroups

Complements

Maximal subgroups