Group #30775798 — subgroups

Subgroups — #30775798

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	#22561981	C₂	1	2	yes	maximal
2	#22560384	C₂	1	1	yes	maximal, characteristic
3	#22558312	C₂	1	2	yes	maximal
4	#22558289	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1946	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863420	—	2	1	no	—
10	#16863218	—	2	4	no	—
11	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
12	#427×C₆₇	C₂²	1	1	yes	characteristic
13	#424×C₆₇	—	2	1	no	—
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	#339×C₆₇	—	2	1	no	—
16	C₂×#11761612	—	2	2	no	—
17	#14523983	—	2	2	no	—
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
20	#14523979	—	2	2	no	—
21	#14523979	—	2	2	no	—
22	#14523979	—	4	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
24	C₂×M₁₆×C₆₇	—	2	1	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	#117×C₆₇	—	2	1	no	—
28	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
29	#108×C₆₇	—	2	2	no	—
30	#107×C₆₇	—	2	1	no	—
31	#106×C₆₇	D₄	1	2	yes	—
32	C₄⋊C₈×C₆₇	—	4	2	no	—
33	#97×C₆₇	—	2	2	no	—
34	#11761612	—	4	2	no	—
35	#11761612	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₆₇⋊C₁₆	—	2	4	no	—
38	C₂²×C₄×C₆₇	—	2	1	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₆₇	—	4	1	no	—
41	M₁₆×C₆₇	—	4	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	4	1	no	—
45	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
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	C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂²⋊C₄×C₆₇	—	4	1	no	—
52	C₄²×C₆₇	—	2	1	no	—
53	C₄²×C₆₇	—	4	1	no	—
54	C₄²×C₆₇	SD₁₆	1	2	yes	—
55	C₂³×C₆₇	#137	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	—	8	1	no	—
59	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₈×C₆₇	—	2	2	no	—
66	C₈×C₆₇	—	8	1	no	—
67	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
68	C₈×C₆₇	#125	1	2	yes	—
69	C₆₇⋊C₈	—	8	2	no	—
70	C₂²×C₆₇	—	4	1	no	—
71	C₂²×C₆₇	#461	1	1	yes	characteristic
72	C₂²×C₆₇	#483	1	1	yes	characteristic
73	C₂²×C₆₇	#501	1	1	yes	characteristic
74	C₄×C₆₇	—	4	2	no	—
75	C₄×C₆₇	—	8	2	no	—
76	C₄×C₆₇	#398	1	1	yes	characteristic
77	C₄×C₆₇	#460	1	1	yes	characteristic
78	C₄×C₆₇	#481	1	1	yes	characteristic
79	C₄×C₆₇	#498	1	1	yes	characteristic
80	#13511	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	#2084	1	1	yes	characteristic
85	#2155	—	67	2	no	—
86	#1946	D₆₇	1	1	yes	characteristic
87	#1849	—	67	1	no	—
88	#1385	—	67	2	no	—
89	#1375	—	67	1	no	—
90	C₆₇	#13511	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	67	2	no	—
92	C₂×#108	—	67	2	no	—
93	C₂×#108	—	134	1	no	—
94	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
95	#427	C₂×D₆₇	1	1	yes	characteristic
96	#424	—	2	1	no	—
97	#423	—	134	1	no	—
98	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
99	#365	—	134	4	no	—
100	#339	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	134	2	no	—
103	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
105	#117	—	2	1	no	—
106	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
107	M₃₂	—	134	2	no	—
108	M₃₂	—	134	2	no	—
109	C₂×C₁₆	—	67	4	no	—
110	#108	—	2	2	no	—
111	#108	—	268	2	no	—
112	#108	—	134	2	no	—
113	#108	—	134	2	no	—
114	#107	—	2	1	no	—
115	#106	D₂₆₈	1	2	yes	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	—	268	2	no	—
118	#97	—	2	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	268	2	no	—
124	M₁₆	—	268	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	268	2	no	—
129	C₂×C₈	#11761616	1	2	yes	—
130	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
131	C₂×C₈	Q₈×D₆₇	1	2	yes	—
132	C₂×C₈	#11761627	1	2	yes	—
133	C₂×C₈	#11761628	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	#11761615	1	2	yes	—
139	C₁₆	—	134	4	no	—
140	C₂³	#14524073	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	#14524017	1	1	yes	characteristic
145	C₂×C₄	#14524024	1	1	yes	characteristic
146	C₂×C₄	#14524041	1	1	yes	characteristic
147	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
149	C₂×C₄	#14524068	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	8	1	no	—
152	C₈	—	536	2	no	—
153	C₈	#14524023	1	2	yes	—
154	C₈	#14524033	1	2	yes	—
155	C₂²	—	4	1	no	—
156	C₂²	#16863601	1	1	yes	characteristic
157	C₂²	#16863724	1	1	yes	characteristic
158	C₂²	#16863806	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	8	2	no	—
161	C₄	#16863326	1	1	yes	characteristic
162	C₄	#16863596	1	1	yes	characteristic
163	C₄	#16863712	1	1	yes	characteristic
164	C₄	#16863792	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#22561685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561981

C₂

yes

maximal

#22560384

C₂

yes

maximal, characteristic

#22558312

C₂

yes

maximal

#22558289

C₂

yes

maximal, characteristic

C₆₇×#1946

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523979

—

C₂×#14523979

C₂²

yes

—

#16863420

—

#16863218

—

C₂×#108×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

characteristic

#424×C₆₇

—

C₂×#97×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂×#11761612

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

Q₈

yes

—

#14523979

—

#14523979

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#107×C₆₇

—

#106×C₆₇

D₄

yes

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

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

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

#137

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#125

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#13511

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

134

—

C₂×#108

C₂×D₆₇

yes

characteristic

#427

C₂×D₆₇

yes

characteristic

#424

—

#423

—

134

—

C₂×#97

C₂×D₆₇

yes

characteristic

#365

—

134

—

100

#339

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

134

—

103

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

104

C₂²×C₈

C₂²×D₆₇

yes

characteristic

105

#117

—

106

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

107

M₃₂

—

134

—

108

M₃₂

—

134

—

109

C₂×C₁₆

—

110

#108

—

111

#108

—

268

—

112

#108

—

134

—

113

#108

—

134

—

114

#107

—

115

#106

D₂₆₈

yes

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

268

—

118

#97

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

268

—

124

M₁₆

—

268

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

268

—

129

C₂×C₈

#11761616

yes

—

130

C₂×C₈

C₂×D₂₆₈

yes

characteristic

131

C₂×C₈

Q₈×D₆₇

yes

—

132

C₂×C₈

#11761627

yes

—

133

C₂×C₈

#11761628

yes

characteristic

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

#11761615

yes

—

139

C₁₆

—

134

—

140

C₂³

#14524073

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14524017

yes

characteristic

145

C₂×C₄

#14524024

yes

characteristic

146

C₂×C₄

#14524041

yes

characteristic

147

C₂×C₄

C₂×#11761615

yes

characteristic

148

C₂×C₄

C₂×#11761616

yes

characteristic

149

C₂×C₄

#14524068

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

536

—

153

C₈

#14524023

yes

—

154

C₈

#14524033

yes

—

155

C₂²

—

156

C₂²

#16863601

yes

characteristic

157

C₂²

#16863724

yes

characteristic

158

C₂²

#16863806

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

#16863326

yes

characteristic

162

C₄

#16863596

yes

characteristic

163

C₄

#16863712

yes

characteristic

164

C₄

#16863792

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#22561685

yes

characteristic

	subgroup	G/H
center	C₄	#16863792
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1946	C₂
socle	C₂×C₆₇	#2084
prime radical	C₂³×C₆₇	#137
hypercenter	#1946	D₆₇
hypocenter	C₆₇	#13511
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#16863792

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1946

C₂

C₂×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
90. C₆₇	#13511	semidirect

H ≅ G/N

type

90. C₆₇

#13511

semidirect

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

subgroup	index
#22561981	2
#22560384	2
#22558312	2
#22558289	2
C₆₇×#1946	2
#13511	67

subgroup

index

C₆₇×#1946

Subgroups — #30775798

Conjugacy classes of subgroups

Complements

Maximal subgroups