Group #30640220 — subgroups

Subgroups — #30640220

Order 16256.

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

normal subgroups only

This group has 224 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	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22549775	C₂	1	2	yes	maximal
4	#22549638	C₂	1	2	yes	maximal
5	#340×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
10	#16861143	—	2	2	no	—
11	Dic₁₂₇⋊C₈	—	2	2	no	—
12	C₁₂₇⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×Dic₁₂₇	—	2	2	no	—
16	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	4	4	no	—
23	C₄×Dic₁₂₇	—	2	1	no	—
24	C₄×Dic₁₂₇	—	2	1	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	Q₈	1	4	yes	—
31	C₂×Dic₁₂₇	—	2	1	no	—
32	C₂×Dic₁₂₇	—	2	1	no	—
33	C₂×Dic₁₂₇	—	2	1	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂×Dic₁₂₇	D₈	1	2	yes	—
39	C₁₂₇⋊C₈	—	4	1	no	—
40	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
48	C₈×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	#121	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₂₇	#135	1	1	yes	characteristic
53	C₂²×C₁₂₇	#137	1	2	yes	—
54	C₄×C₁₂₇	—	4	4	no	—
55	C₄×C₁₂₇	#123	1	1	yes	characteristic
56	C₄×C₁₂₇	#125	1	2	yes	—
57	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₂₇	—	2	2	no	—
59	Dic₁₂₇	—	4	1	no	—
60	Dic₁₂₇	—	8	2	no	—
61	C₂×C₁₂₇	#398	1	1	yes	characteristic
62	C₂×C₁₂₇	#466	1	1	yes	characteristic
63	C₂×C₁₂₇	#470	1	1	yes	characteristic
64	C₂×C₁₂₇	#474	1	2	yes	—
65	C₂×C₁₂₇	#479	1	2	yes	—
66	#2086	—	127	1	no	maximal, Sylow
67	C₁₂₇	#2086	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	127	1	no	—
69	C₂×C₄⋊C₈	—	127	1	no	—
70	#389	—	127	2	no	—
71	#340	—	127	2	no	—
72	#340	D₁₂₇	1	1	yes	characteristic
73	C₂²×C₈	—	127	1	no	—
74	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	127	2	no	—
76	C₂×C₄⋊C₄	—	254	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
78	C₂×C₄²	—	127	1	no	—
79	#107	—	254	2	no	—
80	C₄⋊C₈	—	254	2	no	—
81	#95	—	254	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	127	1	no	—
84	C₂²×C₄	—	127	1	no	—
85	C₂²×C₄	—	254	2	no	—
86	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	254	1	no	—
89	C₂×C₈	—	254	2	no	—
90	C₂×C₈	D₅₀₈	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	508	4	no	—
93	C₄⋊C₄	—	254	2	no	—
94	C₄⋊C₄	—	254	2	no	—
95	C₄⋊C₄	Dic₂₅₄	1	4	yes	—
96	C₄²	—	254	1	no	—
97	C₄²	—	254	1	no	—
98	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	127	2	no	—
103	C₂×C₄	—	508	2	no	—
104	C₂×C₄	—	508	2	no	—
105	C₂×C₄	—	508	2	no	—
106	C₂×C₄	—	254	1	no	—
107	C₂×C₄	—	254	2	no	—
108	C₂×C₄	—	254	1	no	—
109	C₂×C₄	—	254	1	no	—
110	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
112	C₂×C₄	#14523247	1	2	yes	—
113	C₂×C₄	#14523249	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	508	1	no	—
116	C₂²	#16861184	1	1	yes	characteristic
117	C₂²	#16861186	1	2	yes	—
118	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
119	C₂²	#16861232	1	1	yes	characteristic
120	C₂²	#16861238	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	1016	2	no	—
123	C₄	—	508	1	no	—
124	C₄	—	254	2	no	—
125	C₄	#16861193	1	1	yes	characteristic
126	C₄	#16861200	1	2	yes	—
127	C₄	#16861209	1	1	yes	characteristic
128	C₂	#22549808	1	1	yes	characteristic
129	C₂	#22550113	1	1	yes	characteristic
130	C₂	#22550135	1	1	yes	characteristic
131	C₂	#22550154	1	2	yes	—
132	C₂	#22550185	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861143

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549775

C₂

yes

maximal

#22549638

C₂

yes

maximal

#340×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861143

—

Dic₁₂₇⋊C₈

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₂²×C₁₂₇

C₂×D₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#123

yes

characteristic

C₄×C₁₂₇

#125

yes

—

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#479

yes

—

#2086

—

127

maximal, Sylow

C₁₂₇

#2086

yes

characteristic, complemented, Sylow

C₂×#107

—

127

—

C₂×C₄⋊C₈

—

127

—

#389

—

127

—

#340

—

127

—

#340

D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

#107

—

254

—

C₄⋊C₈

—

254

—

#95

—

254

—

C₂²×C₄

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

Dic₂₅₄

yes

—

C₄²

—

254

—

C₄²

—

254

—

C₂³

D₄×D₁₂₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

127

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

508

—

106

C₂×C₄

—

254

—

107

C₂×C₄

—

254

—

108

C₂×C₄

—

254

—

109

C₂×C₄

—

254

—

110

C₂×C₄

C₂×D₅₀₈

yes

characteristic

111

C₂×C₄

C₂×Dic₂₅₄

yes

—

112

C₂×C₄

#14523247

yes

—

113

C₂×C₄

#14523249

yes

characteristic

114

C₈

—

115

C₈

—

508

—

116

C₂²

#16861184

yes

characteristic

117

C₂²

#16861186

yes

—

118

C₂²

D₈×D₁₂₇

yes

characteristic

119

C₂²

#16861232

yes

characteristic

120

C₂²

#16861238

yes

—

121

C₄

—

122

C₄

—

1016

—

123

C₄

—

508

—

124

C₄

—

254

—

125

C₄

#16861193

yes

characteristic

126

C₄

#16861200

yes

—

127

C₄

#16861209

yes

characteristic

128

C₂

#22549808

yes

characteristic

129

C₂

#22550113

yes

characteristic

130

C₂

#22550135

yes

characteristic

131

C₂

#22550154

yes

—

132

C₂

#22550185

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#340×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#340	D₁₂₇
hypocenter	C₁₂₇	#2086
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#340×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
67. C₁₂₇	#2086	semidirect

H ≅ G/N

type

67. C₁₂₇

#2086

semidirect

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

subgroup	index
C₂×#16861143	2
C₂×Dic₁₂₇⋊C₈	2
#22549775	2
#22549638	2
#340×C₁₂₇	2
#2086	127

subgroup

index

C₂×Dic₁₂₇⋊C₈

#340×C₁₂₇

127

Subgroups — #30640220

Conjugacy classes of subgroups

Complements

Maximal subgroups