Group #30775966 — subgroups

Subgroups — #30775966

Order 17152.

234 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22560384	C₂	1	1	yes	maximal, characteristic
2	#22558325	C₂	1	2	yes	maximal
3	#22558322	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	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
11	#427×C₆₇	C₂²	1	1	yes	characteristic
12	#424×C₆₇	—	2	1	no	—
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	#339×C₆₇	—	2	1	no	—
15	C₂×#11761612	—	2	2	no	—
16	#14523983	—	2	2	no	—
17	#14523983	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	#14523979	—	2	4	no	—
20	#14523979	—	4	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₆₇	—	2	1	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	#117×C₆₇	—	2	1	no	—
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	#108×C₆₇	—	2	2	no	—
28	#107×C₆₇	—	2	1	no	—
29	#106×C₆₇	Q₈	1	2	yes	—
30	C₄⋊C₈×C₆₇	—	4	2	no	—
31	#97×C₆₇	—	2	2	no	—
32	#11761612	—	4	2	no	—
33	#11761612	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₆₇⋊C₁₆	—	4	2	no	—
36	C₂²×C₄×C₆₇	—	2	1	no	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₆₇	—	4	1	no	—
39	M₁₆×C₆₇	—	4	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	—	4	1	no	—
43	C₂×C₈×C₆₇	D₈	1	2	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	#41	1	2	yes	—
46	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂²⋊C₄×C₆₇	—	4	1	no	—
50	C₄²×C₆₇	—	2	1	no	—
51	C₄²×C₆₇	—	4	1	no	—
52	C₄²×C₆₇	SD₁₆	1	2	yes	—
53	C₂³×C₆₇	#137	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	4	2	no	—
56	C₂×C₄×C₆₇	—	8	1	no	—
57	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
63	C₈×C₆₇	—	2	2	no	—
64	C₈×C₆₇	—	8	1	no	—
65	C₈×C₆₇	#125	1	2	yes	—
66	C₈×C₆₇	#126	1	2	yes	—
67	C₆₇⋊C₈	—	8	2	no	—
68	C₂²×C₆₇	—	4	1	no	—
69	C₂²×C₆₇	#474	1	1	yes	characteristic
70	C₂²×C₆₇	#483	1	1	yes	characteristic
71	C₂²×C₆₇	#491	1	1	yes	characteristic
72	C₄×C₆₇	—	4	2	no	—
73	C₄×C₆₇	—	8	2	no	—
74	C₄×C₆₇	#400	1	1	yes	characteristic
75	C₄×C₆₇	#475	1	1	yes	characteristic
76	C₄×C₆₇	#480	1	1	yes	characteristic
77	C₄×C₆₇	#486	1	1	yes	characteristic
78	#13540	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	—	2	1	no	—
80	C₂×C₆₇	—	2	1	no	—
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	#2099	1	1	yes	characteristic
83	#1946	D₆₇	1	1	yes	characteristic
84	#1849	—	67	1	no	—
85	#1389	—	67	2	no	—
86	#1388	—	67	2	no	—
87	#1375	—	67	1	no	—
88	C₆₇	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	67	2	no	—
90	C₂×#108	—	67	2	no	—
91	C₂×#108	—	134	1	no	—
92	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
93	#427	C₂×D₆₇	1	1	yes	characteristic
94	#424	—	2	1	no	—
95	#423	—	134	1	no	—
96	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
97	#339	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	134	2	no	—
100	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
104	M₃₂	—	134	2	no	—
105	M₃₂	—	134	2	no	—
106	C₂×C₁₆	—	134	2	no	—
107	#108	—	2	2	no	—
108	#108	—	268	2	no	—
109	#108	—	134	4	no	—
110	#107	—	2	1	no	—
111	#106	Dic₁₃₄	1	2	yes	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	268	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	—	268	2	no	—
120	M₁₆	—	268	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	#11761613	1	2	yes	—
126	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
127	C₂×C₈	#11761627	1	2	yes	—
128	C₂×C₈	#11761628	1	1	yes	characteristic
129	C₂×C₈	#11761629	1	2	yes	—
130	C₄⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	4	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	#11761615	1	2	yes	—
135	C₁₆	—	268	2	no	—
136	C₂³	#14524073	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#14524022	1	1	yes	characteristic
141	C₂×C₄	#14524024	1	1	yes	characteristic
142	C₂×C₄	#14524032	1	1	yes	characteristic
143	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
144	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
145	C₂×C₄	#14524073	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	536	2	no	—
149	C₈	#14524034	1	2	yes	—
150	C₈	#14524038	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#16863676	1	1	yes	characteristic
153	C₂²	#16863724	1	1	yes	characteristic
154	C₂²	#16863766	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#16863335	1	1	yes	characteristic
158	C₄	#16863681	1	1	yes	characteristic
159	C₄	#16863707	1	1	yes	characteristic
160	C₄	#16863744	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#22561767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560384

C₂

yes

maximal, characteristic

#22558325

C₂

yes

maximal

#22558322

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

—

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

—

#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

Q₈

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

D₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

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

C₂²⋊Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#137

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#125

yes

—

C₈×C₆₇

#126

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

#2099

yes

characteristic

#1946

D₆₇

yes

—

—

—

—

C₆₇

#13540

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

#339

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

100

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

101

C₂²×C₈

C₂²×D₆₇

yes

characteristic

102

#117

—

103

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

104

M₃₂

—

134

—

105

M₃₂

—

134

—

106

C₂×C₁₆

—

134

—

107

#108

—

108

#108

—

268

—

109

#108

—

134

—

110

#107

—

111

#106

Dic₁₃₄

yes

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

268

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

268

—

120

M₁₆

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

268

—

125

C₂×C₈

#11761613

yes

—

126

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

127

C₂×C₈

#11761627

yes

—

128

C₂×C₈

#11761628

yes

characteristic

129

C₂×C₈

#11761629

yes

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

#11761615

yes

—

135

C₁₆

—

268

—

136

C₂³

#14524073

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#14524022

yes

characteristic

141

C₂×C₄

#14524024

yes

characteristic

142

C₂×C₄

#14524032

yes

characteristic

143

C₂×C₄

C₂×#11761613

yes

characteristic

144

C₂×C₄

C₂×#11761615

yes

characteristic

145

C₂×C₄

#14524073

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

536

—

149

C₈

#14524034

yes

—

150

C₈

#14524038

yes

—

151

C₂²

—

152

C₂²

#16863676

yes

characteristic

153

C₂²

#16863724

yes

characteristic

154

C₂²

#16863766

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

#16863335

yes

characteristic

158

C₄

#16863681

yes

characteristic

159

C₄

#16863707

yes

characteristic

160

C₄

#16863744

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#22561767

yes

characteristic

	subgroup	G/H
center	C₄	#16863744
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1946	C₂
socle	C₂×C₆₇	#2099
prime radical	C₂³×C₆₇	#137
hypercenter	#1946	D₆₇
hypocenter	C₆₇	#13540
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#16863744

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
88. C₆₇	#13540	semidirect

H ≅ G/N

type

88. C₆₇

#13540

semidirect

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

subgroup	index
#22560384	2
#22558325	2
#22558322	2
#22558289	2
C₆₇×#1946	2
#13540	67

subgroup

index

C₆₇×#1946

Subgroups — #30775966

Conjugacy classes of subgroups

Complements

Maximal subgroups