Group #30775834 — subgroups

Subgroups — #30775834

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	#22560994	C₂	1	2	yes	maximal
3	#22558318	C₂	1	2	yes	maximal
4	#22558303	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863439	—	2	2	no	—
8	C₂×#14523977	C₂²	1	2	yes	—
9	#16863222	—	2	2	no	—
10	#16863164	—	2	2	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	#405×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	2	no	—
15	C₂×#11761612	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	#14523979	—	2	2	no	—
18	#14523979	Q₈	1	4	yes	—
19	#14523977	—	2	2	no	—
20	#14523977	—	2	2	no	—
21	#14523976	—	4	2	no	—
22	#14523974	—	4	2	no	—
23	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	#117×C₆₇	D₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	2	no	—
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	—	2	1	no	—
29	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
30	#97×C₆₇	—	2	1	no	—
31	C₄×C₈×C₆₇	—	2	1	no	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	#11761612	—	2	4	no	—
34	#11761612	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₆₇	#41	1	2	yes	—
44	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
47	C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₄²×C₆₇	—	2	1	no	—
49	C₁₆×C₆₇	—	2	2	no	—
50	C₁₆×C₆₇	—	4	1	no	—
51	C₂³×C₆₇	#135	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#137	1	2	yes	—
58	C₂×Dic₆₇	—	8	2	no	—
59	C₂×Dic₆₇	—	16	2	no	—
60	C₈×C₆₇	—	4	1	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	4	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	—	4	1	no	—
70	C₄×C₆₇	—	4	1	no	—
71	C₄×C₆₇	#398	1	1	yes	characteristic
72	C₄×C₆₇	#470	1	1	yes	characteristic
73	C₄×C₆₇	#474	1	2	yes	—
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	D₆₇	1	1	yes	characteristic
82	#1952	—	67	2	no	—
83	#1387	—	67	2	no	—
84	#1381	—	67	1	no	—
85	C₆₇	#13517	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
87	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
88	C₂×#108	—	67	2	no	—
89	#428	—	134	2	no	—
90	C₂×#107	—	67	2	no	—
91	#405	C₂×D₆₇	1	1	yes	characteristic
92	#366	—	134	2	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	#343	—	134	2	no	—
95	C₂×M₁₆	—	134	2	no	—
96	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	#117	D₂₆₈	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	134	2	no	—
100	M₃₂	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
104	#108	—	67	4	no	—
105	#108	—	134	2	no	—
106	#107	—	268	2	no	—
107	#107	—	134	2	no	—
108	#107	—	134	2	no	—
109	#106	—	268	2	no	—
110	#97	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₂²×C₄	—	268	2	no	—
113	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	268	2	no	—
116	M₁₆	—	134	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
120	C₂×C₈	Q₈×D₆₇	1	2	yes	—
121	C₂×C₈	#11761627	1	2	yes	—
122	C₂×C₈	#11761628	1	1	yes	characteristic
123	C₄⋊C₄	—	268	2	no	—
124	C₄⋊C₄	—	268	2	no	—
125	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
126	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	4	1	no	—
131	C₂³	#14524065	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	536	2	no	—
135	C₂×C₄	—	1072	2	no	—
136	C₂×C₄	#14524017	1	1	yes	characteristic
137	C₂×C₄	#14524023	1	2	yes	—
138	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
139	C₂×C₄	#14524076	1	2	yes	—
140	C₈	—	4	1	no	—
141	C₈	—	268	4	no	—
142	C₈	#14524024	1	1	yes	characteristic
143	C₈	#14524033	1	2	yes	—
144	C₈	#14524041	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16863632	1	1	yes	characteristic
148	C₂²	#16863704	1	2	yes	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	1072	2	no	—
152	C₄	#16863326	1	1	yes	characteristic
153	C₄	#16863654	1	1	yes	characteristic
154	C₄	#16863677	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#22561697	1	1	yes	characteristic
157	C₂	#22562329	1	1	yes	characteristic
158	C₂	#22562334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863222

C₂

yes

maximal, characteristic

#22560994

C₂

yes

maximal

#22558318

C₂

yes

maximal

#22558303

C₂

yes

maximal, characteristic

C₆₇×#2151

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

—

#16863439

—

C₂×#14523977

C₂²

yes

—

#16863222

—

#16863164

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂×#11761612

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

Q₈

yes

—

—

—

—

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

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

C₄⋊C₄×C₆₇

D₈

yes

characteristic

C₄⋊C₄×C₆₇

D₈

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

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

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13517

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#13517

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

C₂×#108

—

#428

—

134

—

C₂×#107

—

#405

C₂×D₆₇

yes

characteristic

#366

—

134

—

C₄⋊C₁₆

—

#343

—

134

—

C₂×M₁₆

—

134

—

C₂×M₁₆

D₂₆₈

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

D₂₆₈

yes

characteristic

C₂×C₄⋊C₄

—

134

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₆₇

yes

—

104

#108

—

105

#108

—

134

—

106

#107

—

268

—

107

#107

—

134

—

108

#107

—

134

—

109

#106

—

268

—

110

#97

—

111

C₄×C₈

—

112

C₂²×C₄

—

268

—

113

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

114

M₁₆

—

115

M₁₆

—

268

—

116

M₁₆

—

134

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

120

C₂×C₈

Q₈×D₆₇

yes

—

121

C₂×C₈

#11761627

yes

—

122

C₂×C₈

#11761628

yes

characteristic

123

C₄⋊C₄

—

268

—

124

C₄⋊C₄

—

268

—

125

C₄⋊C₄

D₅₃₆

yes

characteristic

126

C₄⋊C₄

D₅₃₆

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

—

129

C₁₆

—

130

C₁₆

—

131

C₂³

#14524065

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

536

—

135

C₂×C₄

—

1072

—

136

C₂×C₄

#14524017

yes

characteristic

137

C₂×C₄

#14524023

yes

—

138

C₂×C₄

C₂×D₅₃₆

yes

characteristic

139

C₂×C₄

#14524076

yes

—

140

C₈

—

141

C₈

—

268

—

142

C₈

#14524024

yes

characteristic

143

C₈

#14524033

yes

—

144

C₈

#14524041

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

#16863632

yes

characteristic

148

C₂²

#16863704

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

1072

—

152

C₄

#16863326

yes

characteristic

153

C₄

#16863654

yes

characteristic

154

C₄

#16863677

yes

—

155

C₂

—

156

C₂

#22561697

yes

characteristic

157

C₂

#22562329

yes

characteristic

158

C₂

#22562334

yes

characteristic

	subgroup	G/H
center	C₂²	#16863632
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2151	C₂
socle	C₂²×C₆₇	#466
prime radical	C₂³×C₆₇	#135
hypercenter	#2151	D₆₇
hypocenter	C₆₇	#13517
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#16863632

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2151

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

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
#22560994	2
#22558318	2
#22558303	2
C₆₇×#2151	2
#13517	67

subgroup

index

C₆₇×#2151

Subgroups — #30775834

Conjugacy classes of subgroups

Complements

Maximal subgroups