Group #30775967 — subgroups

Subgroups — #30775967

Order 17152.

234 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22560962	C₂	1	1	yes	maximal, characteristic
2	#22560384	C₂	1	1	yes	maximal, characteristic
3	#22558324	C₂	1	2	yes	maximal
4	#22558321	C₂	1	2	yes	maximal
5	C₆₇×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	1	yes	characteristic
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863434	C₂²	1	1	yes	characteristic
10	#16863422	—	2	1	no	—
11	#16863420	—	2	1	no	—
12	#16863156	—	2	1	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	1	no	—
16	C₂×#11761612	—	2	2	no	—
17	#14524002	—	2	1	no	—
18	#14523979	—	2	2	no	—
19	#14523979	—	2	4	no	—
20	#14523979	—	4	2	no	—
21	#14523977	—	2	1	no	—
22	#14523975	Q₈	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
24	Dic₆₇⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	2	no	—
29	M₃₂×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	#97×C₆₇	—	2	2	no	—
32	#11761612	—	4	1	no	—
33	#11761612	—	4	2	no	—
34	#11761612	—	4	2	no	—
35	#11761612	—	4	1	no	—
36	C₂×C₆₇⋊C₈	—	4	1	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₄⋊Dic₆₇	—	2	2	no	—
39	C₂²⋊Dic₆₇	—	4	1	no	—
40	C₄×Dic₆₇	—	4	1	no	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	D₈	1	2	yes	—
46	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₆₇	#41	1	2	yes	—
50	C₄²×C₆₇	—	2	1	no	—
51	C₄²×C₆₇	SD₁₆	1	2	yes	—
52	C₁₆×C₆₇	—	4	2	no	—
53	C₂³×C₆₇	#137	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	8	1	no	—
56	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
62	C₂×Dic₆₇	—	4	2	no	—
63	C₈×C₆₇	—	2	2	no	—
64	C₈×C₆₇	#125	1	2	yes	—
65	C₈×C₆₇	#126	1	2	yes	—
66	C₆₇⋊C₈	—	8	1	no	—
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	#400	1	1	yes	characteristic
74	C₄×C₆₇	#475	1	1	yes	characteristic
75	C₄×C₆₇	#480	1	1	yes	characteristic
76	C₄×C₆₇	#486	1	1	yes	characteristic
77	Dic₆₇	—	8	2	no	—
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	—	67	1	no	—
84	#1849	—	67	1	no	—
85	#1389	—	67	2	no	—
86	#1388	—	67	2	no	—
87	#1375	D₆₇	1	1	yes	characteristic
88	C₆₇	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
90	C₂×#108	—	67	2	no	—
91	C₂×#108	—	67	1	no	—
92	C₂×#108	—	134	1	no	—
93	#427	—	67	1	no	—
94	#424	—	134	1	no	—
95	#423	—	134	1	no	—
96	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
97	#339	—	134	1	no	—
98	C₂×M₁₆	—	134	1	no	—
99	C₂×M₁₆	—	134	2	no	—
100	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	#117	—	134	1	no	—
103	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	268	2	no	—
108	#108	—	134	2	no	—
109	#108	—	134	4	no	—
110	#107	—	134	1	no	—
111	#106	—	67	2	no	—
112	C₄⋊C₈	—	268	2	no	—
113	C₄⋊C₈	—	268	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
117	M₁₆	—	268	1	no	—
118	M₁₆	—	268	2	no	—
119	M₁₆	—	268	2	no	—
120	M₁₆	—	268	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	268	1	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	D₅₃₆	1	2	yes	—
126	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
127	C₂×C₈	#11761627	1	2	yes	—
128	C₂×C₈	#11761627	1	1	yes	characteristic
129	C₂×C₈	#11761628	1	2	yes	—
130	C₄⋊C₄	—	134	2	no	—
131	C₂²⋊C₄	—	268	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	268	1	no	—
134	C₄²	#11761617	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#14524076	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	268	2	no	—
140	C₂×C₄	#14524023	1	1	yes	characteristic
141	C₂×C₄	#14524027	1	1	yes	characteristic
142	C₂×C₄	#14524031	1	1	yes	characteristic
143	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
144	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
145	C₂×C₄	#14524076	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	536	1	no	—
148	C₈	—	536	2	no	—
149	C₈	#14524033	1	2	yes	—
150	C₈	#14524036	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#16863677	1	1	yes	characteristic
153	C₂²	#16863727	1	1	yes	characteristic
154	C₂²	#16863765	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	536	2	no	—
157	C₄	#16863336	1	1	yes	characteristic
158	C₄	#16863682	1	1	yes	characteristic
159	C₄	#16863710	1	1	yes	characteristic
160	C₄	#16863743	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#22561768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560962

C₂

yes

maximal, characteristic

#22560384

C₂

yes

maximal, characteristic

#22558324

C₂

yes

maximal

#22558321

C₂

yes

maximal

C₆₇×#1375

C₂

yes

maximal, characteristic

C₂×#14523979

—

C₂×#14523979

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863434

C₂²

yes

—

—

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

#97×C₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

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

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

#137

yes

characteristic

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

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#125

yes

—

C₈×C₆₇

#126

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#13540

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#13540

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

—

134

—

#427

—

#424

—

134

—

#423

—

134

—

C₂×#97

C₂×D₆₇

yes

characteristic

#339

—

134

—

C₂×M₁₆

—

134

—

C₂×M₁₆

—

134

—

100

C₂²×C₈

D₂₆₈

yes

characteristic

101

C₂²×C₈

C₂²×D₆₇

yes

characteristic

102

#117

—

134

—

103

C₂×C₄²

D₂₆₈

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

268

—

108

#108

—

134

—

109

#108

—

134

—

110

#107

—

134

—

111

#106

—

112

C₄⋊C₈

—

268

—

113

C₄⋊C₈

—

268

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

117

M₁₆

—

268

—

118

M₁₆

—

268

—

119

M₁₆

—

268

—

120

M₁₆

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

268

—

124

C₂×C₈

—

268

—

125

C₂×C₈

D₅₃₆

yes

—

126

C₂×C₈

Q₈×D₆₇

yes

characteristic

127

C₂×C₈

#11761627

yes

—

128

C₂×C₈

#11761627

yes

characteristic

129

C₂×C₈

#11761628

yes

—

130

C₄⋊C₄

—

134

—

131

C₂²⋊C₄

—

268

—

132

C₄²

—

133

C₄²

—

268

—

134

C₄²

#11761617

yes

—

135

C₁₆

—

136

C₂³

#14524076

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

268

—

140

C₂×C₄

#14524023

yes

characteristic

141

C₂×C₄

#14524027

yes

characteristic

142

C₂×C₄

#14524031

yes

characteristic

143

C₂×C₄

C₂×D₅₃₆

yes

characteristic

144

C₂×C₄

C₂×#11761617

yes

characteristic

145

C₂×C₄

#14524076

yes

characteristic

146

C₈

—

147

C₈

—

536

—

148

C₈

—

536

—

149

C₈

#14524033

yes

—

150

C₈

#14524036

yes

—

151

C₂²

—

152

C₂²

#16863677

yes

characteristic

153

C₂²

#16863727

yes

characteristic

154

C₂²

#16863765

yes

characteristic

155

C₄

—

156

C₄

—

536

—

157

C₄

#16863336

yes

characteristic

158

C₄

#16863682

yes

characteristic

159

C₄

#16863710

yes

characteristic

160

C₄

#16863743

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#22561768

yes

characteristic

	subgroup	G/H
center	C₄	#16863743
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1375	C₂
socle	C₂×C₆₇	#2099
prime radical	C₂³×C₆₇	#137
hypercenter	#1375	D₆₇
hypocenter	C₆₇	#13540
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#16863743

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1375

C₂

C₂×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

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
#22560962	2
#22560384	2
#22558324	2
#22558321	2
C₆₇×#1375	2
#13540	67

subgroup

index

C₆₇×#1375

Subgroups — #30775967

Conjugacy classes of subgroups

Complements

Maximal subgroups