Group #30755334 — subgroups

Subgroups — #30755334

Order 17152.

558 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16863773	C₂	1	1	yes	maximal, characteristic
2	#22558878	C₂	1	4	yes	maximal, complemented
3	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₃₆	—	2	2	no	—
6	C₂×#14524040	C₂²	1	2	yes	—
7	#16863773	—	2	2	no	—
8	#16863773	C₄	1	4	yes	—
9	#16863449	C₂²	1	2	yes	—
10	#16863139	—	2	8	no	—
11	#16863118	—	2	2	no	—
12	#423×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#341×C₆₇	—	2	2	no	—
15	C₂²×D₂₆₈	—	2	2	no	—
16	C₂²×D₂₆₈	—	2	4	no	—
17	C₂×D₅₃₆	—	2	8	no	—
18	C₂×D₅₃₆	—	4	8	no	—
19	#14524040	—	2	2	no	—
20	#14524040	—	2	4	no	—
21	#14524040	C₂×C₄	1	4	yes	—
22	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	4	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₂³×D₆₇	—	4	2	no	—
32	C₂×D₂₆₈	—	2	4	no	—
33	C₂×D₂₆₈	—	2	8	no	—
34	C₂×D₂₆₈	—	4	8	no	—
35	C₂×D₂₆₈	—	4	4	no	—
36	C₂×D₂₆₈	—	4	2	no	—
37	C₂×D₂₆₈	—	8	8	no	—
38	D₅₃₆	—	4	16	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	#41	1	2	yes	—
42	M₁₆×C₆₇	—	4	4	no	—
43	C₂×C₈×C₆₇	—	2	4	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×D₄	1	2	yes	—
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂³×C₆₇	#124	1	1	yes	characteristic
50	C₂²×D₆₇	—	4	4	no	—
51	C₂²×D₆₇	—	8	8	no	—
52	C₂²×D₆₇	—	8	4	no	—
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₆₇	#121	1	2	yes	—
57	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
59	D₂₆₈	—	4	8	no	—
60	D₂₆₈	—	8	8	no	—
61	D₂₆₈	—	8	8	no	—
62	D₂₆₈	—	8	4	no	—
63	C₈×C₆₇	—	2	4	no	—
64	C₈×C₆₇	—	8	2	no	—
65	C₆₇⋊C₈	—	4	4	no	—
66	C₂²×C₆₇	—	2	2	no	—
67	C₂²×C₆₇	#390	1	1	yes	characteristic
68	C₂²×C₆₇	#490	1	2	yes	—
69	C₂×D₆₇	—	8	4	no	—
70	C₂×D₆₇	—	8	8	no	—
71	C₂×D₆₇	—	16	8	no	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	#418	1	2	yes	—
74	C₄×C₆₇	#472	1	4	yes	—
75	C₄×C₆₇	#496	1	2	yes	—
76	#9780	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1961	1	1	yes	characteristic
79	C₂×C₆₇	#1978	1	2	yes	—
80	D₆₇	—	16	4	no	—
81	C₂×#493	—	67	1	no	—
82	#1837	D₆₇	1	1	yes	characteristic, complemented
83	#1545	—	67	4	no	—
84	#1452	—	67	1	no	—
85	C₆₇	#9780	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₈	—	134	2	no	—
87	C₂×#127	—	67	2	no	—
88	#493	—	67	4	no	—
89	#493	—	134	2	no	—
90	#431	—	67	2	no	—
91	#423	C₂×D₆₇	1	2	yes	—
92	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	#341	—	2	2	no	—
94	#331	—	134	8	no	—
95	#322	—	134	2	no	—
96	C₂²×D₄	—	134	2	no	—
97	C₂²×D₄	—	134	4	no	—
98	C₂×D₈	—	268	8	no	—
99	C₂×D₈	—	134	8	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
102	#127	—	67	4	no	—
103	#127	—	134	4	no	—
104	#127	—	134	2	no	—
105	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	67	4	no	—
108	C₂²⋊C₈	—	134	2	no	—
109	C₄×C₈	—	134	2	no	—
110	C₄×C₈	C₄×D₆₇	1	2	yes	—
111	C₄×C₈	D₂₆₈	1	2	yes	—
112	C₂⁴	—	268	2	no	—
113	C₂×D₄	—	268	4	no	—
114	C₂×D₄	—	268	2	no	—
115	C₂×D₄	—	268	8	no	—
116	C₂×D₄	—	536	8	no	—
117	C₂×D₄	—	134	4	no	—
118	C₂×D₄	—	134	8	no	—
119	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
120	C₂²×C₄	#11761629	1	2	yes	—
121	D₈	—	268	16	no	—
122	M₁₆	—	4	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	134	4	no	—
126	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
127	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
128	C₄²	D₄×D₆₇	1	2	yes	—
129	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
130	C₂³	—	268	4	no	—
131	C₂³	—	536	8	no	—
132	C₂³	—	536	4	no	—
133	C₂³	#14524032	1	1	yes	characteristic
134	D₄	—	268	8	no	—
135	D₄	—	536	8	no	—
136	D₄	—	536	4	no	—
137	D₄	—	536	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	#14524005	1	2	yes	—
141	C₂×C₄	#14524018	1	2	yes	—
142	C₂×C₄	#14524039	1	1	yes	characteristic
143	C₂×C₄	#14524068	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	8	2	no	—
146	C₈	—	268	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	536	4	no	—
149	C₂²	—	536	8	no	—
150	C₂²	—	1072	8	no	—
151	C₂²	#16863297	1	1	yes	characteristic
152	C₂²	#16863762	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#16863400	1	2	yes	—
155	C₄	#16863667	1	4	yes	—
156	C₄	#16863786	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	1072	4	no	—
159	C₂	#22561032	1	1	yes	characteristic
160	C₂	#22561117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863773

C₂

yes

maximal, characteristic

#22558878

C₂

yes

maximal, complemented

#22558501

C₂

yes

maximal, characteristic, complemented

C₆₇×#1837

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₃₆

—

C₂×#14524040

C₂²

yes

—

#16863773

—

#16863773

C₄

yes

—

#16863449

C₂²

yes

—

#16863139

—

#16863118

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#341×C₆₇

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

#14524040

—

#14524040

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#490

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#418

yes

—

C₄×C₆₇

#472

yes

—

C₄×C₆₇

#496

yes

—

#9780

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1961

yes

characteristic

C₂×C₆₇

#1978

yes

—

D₆₇

—

C₂×#493

—

#1837

D₆₇

yes

characteristic, complemented

#1545

—

#1452

—

C₆₇

#9780

yes

characteristic, complemented, Sylow

C₂²×D₈

—

134

—

C₂×#127

—

#493

—

#493

—

134

—

#431

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#341

—

#331

—

134

—

#322

—

134

—

C₂²×D₄

—

134

—

C₂²×D₄

—

134

—

C₂×D₈

—

268

—

C₂×D₈

—

134

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₆₇

yes

—

102

#127

—

103

#127

—

134

—

104

#127

—

134

—

105

C₂×C₄²

C₂²×D₆₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

134

—

109

C₄×C₈

—

134

—

110

C₄×C₈

C₄×D₆₇

yes

—

111

C₄×C₈

D₂₆₈

yes

—

112

C₂⁴

—

268

—

113

C₂×D₄

—

268

—

114

C₂×D₄

—

268

—

115

C₂×D₄

—

268

—

116

C₂×D₄

—

536

—

117

C₂×D₄

—

134

—

118

C₂×D₄

—

134

—

119

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

120

C₂²×C₄

#11761629

yes

—

121

D₈

—

268

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

134

—

126

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

127

C₄²

C₂×C₄×D₆₇

yes

characteristic

128

C₄²

D₄×D₆₇

yes

—

129

C₄²

C₂×D₂₆₈

yes

characteristic

130

C₂³

—

268

—

131

C₂³

—

536

—

132

C₂³

—

536

—

133

C₂³

#14524032

yes

characteristic

134

D₄

—

268

—

135

D₄

—

536

—

136

D₄

—

536

—

137

D₄

—

536

—

138

C₂×C₄

—

139

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

140

C₂×C₄

#14524005

yes

—

141

C₂×C₄

#14524018

yes

—

142

C₂×C₄

#14524039

yes

characteristic

143

C₂×C₄

#14524068

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

268

—

147

C₂²

—

148

C₂²

—

536

—

149

C₂²

—

536

—

150

C₂²

—

1072

—

151

C₂²

#16863297

yes

characteristic

152

C₂²

#16863762

yes

—

153

C₄

—

154

C₄

#16863400

yes

—

155

C₄

#16863667

yes

—

156

C₄

#16863786

yes

—

157

C₂

—

158

C₂

—

1072

—

159

C₂

#22561032

yes

characteristic

160

C₂

#22561117

yes

—

	subgroup	G/H
center	C₂²	#16863297
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1837	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂×#16863773	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1837	D₆₇
hypocenter	C₆₇	#9780
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#16863297

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1837

C₂

C₂²×C₆₇

C₂

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
2. #22558878	C₂	semidirect
3. #22558501	C₂	semidirect
4. C₆₇×#1837	C₂	semidirect
82. #1837	D₆₇	semidirect
85. C₆₇	#9780	semidirect

H ≅ G/N

type

2. #22558878

C₂

semidirect

3. #22558501

C₂

semidirect

4. C₆₇×#1837

C₂

semidirect

82. #1837

D₆₇

semidirect

85. C₆₇

#9780

semidirect

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

subgroup	index
C₂×#16863773	2
#22558878	2
#22558501	2
C₆₇×#1837	2
#9780	67

subgroup

index

C₆₇×#1837

Subgroups — #30755334

Conjugacy classes of subgroups

Complements

Maximal subgroups