Group #30640150 — subgroups

Subgroups — #30640150

Order 16256.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861134	C₂	1	2	yes	maximal
2	#22549782	C₂	1	1	yes	maximal, characteristic
3	#22549742	C₂	1	1	yes	maximal, characteristic
4	#22549639	C₂	1	2	yes	maximal
5	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	#16861134	—	2	4	no	—
10	C₁₂₇⋊#95	—	2	2	no	—
11	C₁₂₇⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₄	—	2	1	no	—
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂×Dic₂₅₄	—	2	2	no	—
17	C₂×Dic₂₅₄	D₄	1	2	yes	—
18	C₂²×Dic₁₂₇	—	2	1	no	—
19	C₂²×Dic₁₂₇	—	2	2	no	—
20	C₂²×Dic₁₂₇	—	2	1	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₄²×C₁₂₇	—	2	2	no	—
30	Dic₂₅₄	—	2	4	no	—
31	Dic₂₅₄	—	4	2	no	—
32	C₂×Dic₁₂₇	—	2	2	no	—
33	C₂×Dic₁₂₇	—	4	1	no	—
34	C₂×Dic₁₂₇	—	4	1	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	1	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂×Dic₁₂₇	—	4	2	no	—
39	C₂×Dic₁₂₇	—	4	2	no	—
40	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	2	no	—
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	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
48	C₈×C₁₂₇	—	4	2	no	—
49	C₂²×C₁₂₇	#121	1	1	yes	characteristic
50	C₂²×C₁₂₇	#123	1	1	yes	characteristic
51	C₂²×C₁₂₇	#124	1	1	yes	characteristic
52	C₂²×C₁₂₇	#135	1	2	yes	—
53	C₂²×C₁₂₇	#137	1	2	yes	—
54	C₄×C₁₂₇	—	4	2	no	—
55	C₄×C₁₂₇	#123	1	2	yes	—
56	C₄×C₁₂₇	#126	1	2	yes	—
57	Dic₁₂₇	—	4	2	no	—
58	Dic₁₂₇	—	8	2	no	—
59	Dic₁₂₇	—	8	1	no	—
60	C₂×C₁₂₇	#397	1	1	yes	characteristic
61	C₂×C₁₂₇	#464	1	2	yes	—
62	C₂×C₁₂₇	#485	1	2	yes	—
63	C₂×C₁₂₇	#488	1	1	yes	characteristic
64	C₂×C₁₂₇	#491	1	1	yes	characteristic
65	#2074	—	127	1	no	maximal, Sylow
66	C₁₂₇	#2074	1	1	yes	characteristic, complemented, Sylow
67	C₂×#103	—	127	2	no	—
68	#391	—	127	1	no	—
69	#382	—	127	1	no	—
70	#340	—	127	2	no	—
71	#336	D₁₂₇	1	1	yes	characteristic
72	C₂²×Q₈	—	127	1	no	—
73	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
74	C₂×C₄⋊C₄	—	127	1	no	—
75	C₂×C₄⋊C₄	—	127	2	no	—
76	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
77	#103	—	254	4	no	—
78	#95	—	254	2	no	—
79	#95	—	254	2	no	—
80	C₂×Q₈	—	127	2	no	—
81	C₂×Q₈	—	254	2	no	—
82	C₂×Q₈	—	254	2	no	—
83	C₂×Q₈	—	254	1	no	—
84	C₂²×C₄	—	254	2	no	—
85	C₂²×C₄	—	254	1	no	—
86	C₂²×C₄	—	254	1	no	—
87	C₂²×C₄	D₅₀₈	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	254	2	no	—
93	C₄⋊C₄	—	254	2	no	—
94	C₄⋊C₄	—	254	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
97	Q₈	—	508	2	no	—
98	Q₈	—	254	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	508	2	no	—
103	C₂×C₄	—	508	1	no	—
104	C₂×C₄	—	508	2	no	—
105	C₂×C₄	—	508	2	no	—
106	C₂×C₄	—	508	2	no	—
107	C₂×C₄	—	508	1	no	—
108	C₂×C₄	—	508	1	no	—
109	C₂×C₄	—	254	2	no	—
110	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
111	C₂×C₄	#14523247	1	2	yes	—
112	C₂×C₄	#14523248	1	2	yes	—
113	C₂×C₄	#14523249	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#16861184	1	1	yes	characteristic
116	C₂²	#16861193	1	1	yes	characteristic
117	C₂²	#16861197	1	1	yes	characteristic
118	C₂²	#16861231	1	2	yes	—
119	C₂²	#16861242	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	1016	2	no	—
122	C₄	—	1016	1	no	—
123	C₄	—	508	2	no	—
124	C₄	#16861191	1	2	yes	—
125	C₄	#16861204	1	2	yes	—
126	C₂	#22549803	1	1	yes	characteristic
127	C₂	#22550099	1	2	yes	—
128	C₂	#22550218	1	2	yes	—
129	C₂	#22550231	1	1	yes	characteristic
130	C₂	#22550245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861134

C₂

yes

maximal

#22549782

C₂

yes

maximal, characteristic

#22549742

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

#336×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861134

—

C₁₂₇⋊#95

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

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

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#126

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#397

yes

characteristic

C₂×C₁₂₇

#464

yes

—

C₂×C₁₂₇

#485

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2074

yes

characteristic, complemented, Sylow

C₂×#103

—

127

—

#391

—

127

—

#382

—

127

—

#340

—

127

—

#336

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#95

—

254

—

#95

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

508

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

508

—

106

C₂×C₄

—

508

—

107

C₂×C₄

—

508

—

108

C₂×C₄

—

508

—

109

C₂×C₄

—

254

—

110

C₂×C₄

D₄×D₁₂₇

yes

characteristic

111

C₂×C₄

#14523247

yes

—

112

C₂×C₄

#14523248

yes

—

113

C₂×C₄

#14523249

yes

characteristic

114

C₈

—

115

C₂²

#16861184

yes

characteristic

116

C₂²

#16861193

yes

characteristic

117

C₂²

#16861197

yes

characteristic

118

C₂²

#16861231

yes

—

119

C₂²

#16861242

yes

—

120

C₄

—

121

C₄

—

1016

—

122

C₄

—

1016

—

123

C₄

—

508

—

124

C₄

#16861191

yes

—

125

C₄

#16861204

yes

—

126

C₂

#22549803

yes

characteristic

127

C₂

#22550099

yes

—

128

C₂

#22550218

yes

—

129

C₂

#22550231

yes

characteristic

130

C₂

#22550245

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#336×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#336	D₁₂₇
hypocenter	C₁₂₇	#2074
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#336×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
66. C₁₂₇	#2074	semidirect

H ≅ G/N

type

66. C₁₂₇

#2074

semidirect

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

subgroup	index
C₂×#16861134	2
#22549782	2
#22549742	2
#22549639	2
#336×C₁₂₇	2
#2074	127

subgroup

index

#336×C₁₂₇

127

Subgroups — #30640150

Conjugacy classes of subgroups

Complements

Maximal subgroups