Group #30770600 — subgroups

Subgroups — #30770600

Order 17152.

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

normal subgroups only

This group has 274 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	#22562462	C₂	1	2	yes	maximal
2	#22562437	C₂	1	2	yes	maximal
3	#22559827	C₂	1	1	yes	maximal, characteristic
4	#22559806	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	—	2	4	no	—
7	#16863801	—	2	1	no	—
8	#16863801	—	2	1	no	—
9	#16863801	C₂²	1	1	yes	characteristic
10	#16863794	—	2	1	no	—
11	#16863776	C₂²	1	1	yes	characteristic
12	#16863776	C₂²	1	2	yes	—
13	#16863747	—	2	1	no	—
14	#16863201	—	2	4	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	2	no	—
20	#14524043	—	2	1	no	—
21	#14524043	—	2	1	no	—
22	#14524043	—	2	2	no	—
23	Dic₅₃₆	—	4	8	no	—
24	#14523977	—	2	2	no	—
25	#14523977	—	4	2	no	—
26	#14523977	—	4	2	no	—
27	#14523975	—	4	2	no	—
28	#14523968	—	4	2	no	—
29	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₂×Dic₁₃₄	—	4	2	no	—
35	C₂×Dic₁₃₄	—	4	1	no	—
36	Dic₂₆₈	—	4	4	no	—
37	Dic₂₆₈	—	4	4	no	—
38	Dic₂₆₈	—	4	4	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	C₄⋊Dic₆₇	—	4	1	no	—
42	C₄⋊Dic₆₇	—	4	1	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₄⋊Dic₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	D₈	1	2	yes	—
46	C₂×C₈×C₆₇	D₈	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₆₇	—	8	1	no	—
54	C₂×Dic₆₇	—	8	2	no	—
55	C₂×Dic₆₇	—	8	1	no	—
56	Dic₁₃₄	—	8	2	no	—
57	Dic₁₃₄	—	8	2	no	—
58	Dic₁₃₄	—	8	2	no	—
59	Dic₁₃₄	—	8	2	no	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	Q₃₂	1	4	yes	complemented
62	C₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₆₇	#493	1	1	yes	characteristic
64	C₄×C₆₇	#496	1	2	yes	—
65	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
66	C₄×C₆₇	#510	1	2	yes	—
67	Dic₆₇	—	16	1	no	—
68	Dic₆₇	—	16	2	no	—
69	Dic₆₇	—	16	1	no	—
70	#12528	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	#1721	1	1	yes	characteristic
72	C₂×C₆₇	#2247	1	1	yes	characteristic
73	C₂×C₆₇	#2251	1	1	yes	characteristic
74	#2247	—	67	2	no	—
75	#2241	—	67	2	no	—
76	#1713	—	67	1	no	—
77	#1707	—	67	1	no	—
78	#1312	D₆₇	1	1	yes	characteristic
79	C₆₇	#12528	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₃₂	—	134	4	no	—
81	#500	—	67	1	no	—
82	#500	—	134	1	no	—
83	#500	—	134	1	no	—
84	#498	—	134	1	no	—
85	#494	—	67	1	no	—
86	#494	—	67	2	no	—
87	#487	—	134	1	no	—
88	#358	—	134	4	no	—
89	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
90	C₈²	C₂×D₆₇	1	1	yes	characteristic
91	C₂×Q₁₆	—	134	2	no	—
92	C₂×Q₁₆	—	134	2	no	—
93	C₂×Q₁₆	—	134	2	no	—
94	C₄⋊Q₈	—	134	1	no	—
95	C₄⋊Q₈	—	134	1	no	—
96	C₄⋊Q₈	—	134	2	no	—
97	Q₃₂	—	268	8	no	—
98	C₂×C₁₆	D₂₆₈	1	4	yes	—
99	#107	—	268	2	no	—
100	#107	—	268	2	no	—
101	#107	—	134	2	no	—
102	#106	—	268	2	no	—
103	#103	—	268	2	no	—
104	C₄×C₈	D₂₆₈	1	1	yes	characteristic
105	C₄×C₈	D₂₆₈	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×Q₈	—	268	1	no	—
108	C₂×Q₈	—	268	2	no	—
109	C₂×Q₈	—	268	1	no	—
110	Q₁₆	—	268	4	no	—
111	Q₁₆	—	268	4	no	—
112	Q₁₆	—	268	4	no	—
113	C₂×C₈	D₅₃₆	1	2	yes	—
114	C₂×C₈	D₅₃₆	1	2	yes	—
115	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
116	C₄⋊C₄	—	268	1	no	—
117	C₄⋊C₄	—	268	1	no	—
118	C₄⋊C₄	—	268	1	no	—
119	C₄⋊C₄	—	268	1	no	—
120	C₄⋊C₄	—	268	2	no	—
121	C₄⋊C₄	—	268	2	no	—
122	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	536	2	no	—
125	Q₈	—	536	2	no	—
126	Q₈	—	536	2	no	—
127	Q₈	—	536	2	no	—
128	C₂×C₄	—	536	1	no	—
129	C₂×C₄	—	536	2	no	—
130	C₂×C₄	—	536	1	no	—
131	C₂×C₄	#14524040	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₅₃₆	1	4	yes	complemented
136	C₈	#14524071	1	4	yes	—
137	C₂²	#16863773	1	1	yes	characteristic
138	C₄	—	1072	1	no	—
139	C₄	—	1072	2	no	—
140	C₄	—	1072	1	no	—
141	C₄	#16863785	1	2	yes	—
142	C₄	C₂×Dic₅₃₆	1	2	yes	—
143	C₄	#16863839	1	2	yes	—
144	C₂	#22559875	1	1	yes	characteristic
145	C₂	#22562462	1	1	yes	characteristic
146	C₂	#22562480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562462

C₂

yes

maximal

#22562437

C₂

yes

maximal

#22559827

C₂

yes

maximal, characteristic

#22559806

C₂

yes

maximal, characteristic

C₆₇×#1312

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

#16863801

—

#16863801

—

#16863801

C₂²

yes

characteristic

#16863794

—

#16863776

C₂²

yes

characteristic

#16863776

C₂²

yes

—

#16863747

—

#16863201

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

#14524043

—

#14524043

—

Dic₅₃₆

—

—

—

—

—

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

Q₃₂

yes

complemented

C₈×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

#496

yes

—

C₄×C₆₇

C₂×Q₃₂

yes

—

C₄×C₆₇

#510

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#12528

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

#500

—

#500

—

134

—

#500

—

134

—

#498

—

134

—

#494

—

#494

—

#487

—

134

—

#358

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

Q₃₂

—

268

—

C₂×C₁₆

D₂₆₈

yes

—

#107

—

268

—

100

#107

—

268

—

101

#107

—

134

—

102

#106

—

268

—

103

#103

—

268

—

104

C₄×C₈

D₂₆₈

yes

characteristic

105

C₄×C₈

D₂₆₈

yes

characteristic

106

C₄×C₈

C₂²×D₆₇

yes

characteristic

107

C₂×Q₈

—

268

—

108

C₂×Q₈

—

268

—

109

C₂×Q₈

—

268

—

110

Q₁₆

—

268

—

111

Q₁₆

—

268

—

112

Q₁₆

—

268

—

113

C₂×C₈

D₅₃₆

yes

—

114

C₂×C₈

D₅₃₆

yes

—

115

C₂×C₈

C₂×D₂₆₈

yes

—

116

C₄⋊C₄

—

268

—

117

C₄⋊C₄

—

268

—

118

C₄⋊C₄

—

268

—

119

C₄⋊C₄

—

268

—

120

C₄⋊C₄

—

268

—

121

C₄⋊C₄

—

268

—

122

C₄²

C₂×D₂₆₈

yes

characteristic

123

C₁₆

—

124

Q₈

—

536

—

125

Q₈

—

536

—

126

Q₈

—

536

—

127

Q₈

—

536

—

128

C₂×C₄

—

536

—

129

C₂×C₄

—

536

—

130

C₂×C₄

—

536

—

131

C₂×C₄

#14524040

yes

characteristic

132

C₂×C₄

C₂×D₅₃₆

yes

characteristic

133

C₂×C₄

C₂×D₅₃₆

yes

characteristic

134

C₈

—

135

C₈

Dic₅₃₆

yes

complemented

136

C₈

#14524071

yes

—

137

C₂²

#16863773

yes

characteristic

138

C₄

—

1072

—

139

C₄

—

1072

—

140

C₄

—

1072

—

141

C₄

#16863785

yes

—

142

C₄

C₂×Dic₅₃₆

yes

—

143

C₄

#16863839

yes

—

144

C₂

#22559875

yes

characteristic

145

C₂

#22562462

yes

characteristic

146

C₂

#22562480

yes

characteristic

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1312	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#12528
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1312

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
61. C₈×C₆₇	Q₃₂	semidirect
79. C₆₇	#12528	semidirect
135. C₈	Dic₅₃₆	semidirect

H ≅ G/N

type

61. C₈×C₆₇

Q₃₂

semidirect

79. C₆₇

#12528

semidirect

135. C₈

Dic₅₃₆

semidirect

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

subgroup	index
#22562462	2
#22562437	2
#22559827	2
#22559806	2
C₆₇×#1312	2
#12528	67

subgroup

index

C₆₇×#1312

Subgroups — #30770600

Conjugacy classes of subgroups

Complements

Maximal subgroups