Group #30640172 — subgroups

Subgroups — #30640172

Order 16256.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#16861143	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549878	C₂	1	2	yes	maximal
3	#22549842	C₂	1	1	yes	maximal, characteristic
4	#22549644	C₂	1	2	yes	maximal, complemented
5	#341×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	C₂×#14523232	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	#16861143	—	2	2	no	—
10	#16861134	—	2	2	no	—
11	#16861131	—	2	2	no	—
12	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#14523248	—	2	2	no	—
16	#14523248	—	2	2	no	—
17	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
18	C₂×D₅₀₈	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₂×C₄×D₁₂₇	—	2	1	no	—
22	#14523232	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₄⋊Dic₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	2	no	—
28	M₁₆×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂²×D₁₂₇	—	2	1	no	—
33	Dic₂₅₄	—	2	2	no	—
34	D₅₀₈	—	2	2	no	—
35	C₂²⋊D₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂×Dic₁₂₇	—	8	2	no	—
39	C₄×D₁₂₇	—	4	1	no	—
40	C₄×D₁₂₇	—	4	1	no	—
41	C₁₂₇⋊C₈	—	4	1	no	—
42	C₂³×C₁₂₇	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#41	1	2	yes	—
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₈×C₁₂₇	—	2	2	no	—
48	C₈×C₁₂₇	—	4	2	no	—
49	C₂²×C₁₂₇	—	2	1	no	—
50	C₂²×C₁₂₇	—	2	1	no	—
51	C₂²×C₁₂₇	#121	1	1	yes	characteristic
52	C₂²×C₁₂₇	#126	1	2	yes	—
53	C₂×D₁₂₇	—	4	1	no	—
54	C₂×D₁₂₇	—	4	1	no	—
55	C₂×D₁₂₇	—	4	1	no	—
56	C₄×C₁₂₇	#123	1	2	yes	—
57	C₄×C₁₂₇	#123	1	1	yes	characteristic
58	C₄×C₁₂₇	#124	1	1	yes	characteristic
59	Dic₁₂₇	—	4	1	no	—
60	Dic₁₂₇	—	8	2	no	—
61	C₂×C₁₂₇	—	2	2	no	—
62	C₂×C₁₂₇	#397	1	1	yes	characteristic
63	C₂×C₁₂₇	#472	1	1	yes	characteristic
64	C₂×C₁₂₇	#473	1	1	yes	characteristic
65	D₁₂₇	—	8	1	no	—
66	#2078	—	127	1	no	maximal, Sylow
67	C₁₂₇	#2078	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	127	1	no	—
69	#417	—	127	2	no	—
70	#408	—	127	1	no	—
71	#343	—	127	2	no	—
72	#341	D₁₂₇	1	1	yes	characteristic, complemented
73	C₂×#41	—	127	1	no	—
74	C₂×M₁₆	—	127	1	no	—
75	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
76	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	127	2	no	—
78	#107	—	254	2	no	—
79	#103	—	254	2	no	—
80	#102	—	254	2	no	—
81	C₂²⋊C₈	—	254	2	no	—
82	#41	—	254	2	no	—
83	#41	—	254	2	no	—
84	C₂×Q₈	—	127	1	no	—
85	C₂×D₄	—	127	1	no	—
86	C₂×D₄	—	254	1	no	—
87	C₂²×C₄	—	254	2	no	—
88	C₂²×C₄	—	254	1	no	—
89	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	254	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	254	1	no	—
96	C₂×C₈	D₅₀₈	1	2	yes	—
97	C₄⋊C₄	—	254	2	no	—
98	C₄⋊C₄	—	254	2	no	—
99	C₂³	—	254	1	no	—
100	C₂³	#14523249	1	1	yes	characteristic
101	Q₈	—	254	2	no	—
102	D₄	—	508	2	no	—
103	D₄	—	254	2	no	—
104	C₂×C₄	—	1016	2	no	—
105	C₂×C₄	—	508	2	no	—
106	C₂×C₄	—	508	1	no	—
107	C₂×C₄	—	508	1	no	—
108	C₂×C₄	—	254	1	no	—
109	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
111	C₂×C₄	#14523247	1	2	yes	—
112	C₂×C₄	#14523248	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	508	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	508	1	no	—
119	C₂²	—	508	1	no	—
120	C₂²	—	508	1	no	—
121	C₂²	#16861184	1	1	yes	characteristic
122	C₂²	#16861204	1	2	yes	—
123	C₄	—	1016	2	no	—
124	C₄	—	508	1	no	—
125	C₄	#16861191	1	2	yes	—
126	C₄	#16861193	1	1	yes	characteristic
127	C₄	#16861197	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	1016	1	no	—
130	C₂	#22549803	1	1	yes	characteristic
131	C₂	#22550147	1	1	yes	characteristic
132	C₂	#22550152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861143

C₂

yes

maximal, characteristic, complemented

#22549878

C₂

yes

maximal

#22549842

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal, complemented

#341×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861143

—

#16861134

—

#16861131

—

(C₂×D₁₂₇)⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#14523248

—

#14523248

—

C₂×Dic₂₅₄

D₄

yes

characteristic

C₂×D₅₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#126

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

#2078

—

127

maximal, Sylow

C₁₂₇

#2078

yes

characteristic, complemented, Sylow

C₂×#107

—

127

—

#417

—

127

—

#408

—

127

—

#343

—

127

—

#341

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

#107

—

254

—

#103

—

254

—

#102

—

254

—

C₂²⋊C₈

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂³

—

254

—

100

C₂³

#14523249

yes

characteristic

101

Q₈

—

254

—

102

D₄

—

508

—

103

D₄

—

254

—

104

C₂×C₄

—

1016

—

105

C₂×C₄

—

508

—

106

C₂×C₄

—

508

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

254

—

109

C₂×C₄

D₄×D₁₂₇

yes

characteristic

110

C₂×C₄

C₂×D₅₀₈

yes

characteristic

111

C₂×C₄

#14523247

yes

—

112

C₂×C₄

#14523248

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

508

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

508

—

119

C₂²

—

508

—

120

C₂²

—

508

—

121

C₂²

#16861184

yes

characteristic

122

C₂²

#16861204

yes

—

123

C₄

—

1016

—

124

C₄

—

508

—

125

C₄

#16861191

yes

—

126

C₄

#16861193

yes

characteristic

127

C₄

#16861197

yes

characteristic

128

C₂

—

129

C₂

—

1016

—

130

C₂

#22549803

yes

characteristic

131

C₂

#22550147

yes

characteristic

132

C₂

#22550152

yes

characteristic

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#341×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂×#14523248	C₂²
hypercenter	#341	D₁₂₇
hypocenter	C₁₂₇	#2078
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#16861184

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#341×C₁₂₇

C₂

C₂²×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#16861143	C₂	semidirect
4. #22549644	C₂	semidirect
5. #341×C₁₂₇	C₂	semidirect
67. C₁₂₇	#2078	semidirect
72. #341	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#16861143

C₂

semidirect

4. #22549644

C₂

semidirect

5. #341×C₁₂₇

C₂

semidirect

67. C₁₂₇

#2078

semidirect

72. #341

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861143	2
#22549878	2
#22549842	2
#22549644	2
#341×C₁₂₇	2
#2078	127

subgroup

index

#341×C₁₂₇

127

Subgroups — #30640172

Conjugacy classes of subgroups

Complements

Maximal subgroups