Group #30639653 — subgroups

Subgroups — #30639653

Order 16256.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861209	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861132	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	#16861209	—	2	2	no	—
8	#16861209	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
12	#16861132	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	—	2	1	no	—
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	#95×C₁₂₇	—	2	1	no	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	—	2	4	no	—
19	C₂×Dic₂₅₄	—	2	4	no	—
20	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
21	C₂²×Dic₁₂₇	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	4	no	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₂₇	—	2	4	no	—
25	C₄⋊Dic₁₂₇	—	2	4	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	D₄	1	4	yes	—
32	C₄²×C₁₂₇	—	4	2	no	—
33	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₄²×C₁₂₇	D₄	1	2	yes	—
35	Dic₂₅₄	—	2	8	no	—
36	Dic₂₅₄	—	4	2	no	—
37	Dic₂₅₄	—	4	2	no	—
38	C₂×Dic₁₂₇	—	2	4	no	—
39	C₂×Dic₁₂₇	—	4	4	no	—
40	C₁₂₇⋊C₈	—	4	2	no	—
41	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄×C₁₂₇	—	4	2	no	—
45	C₂×C₄×C₁₂₇	—	4	2	no	—
46	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
49	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	#41	1	2	yes	—
54	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₂₇	#121	1	2	yes	—
56	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₂₇	#137	1	2	yes	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	—	4	4	no	—
61	C₄×C₁₂₇	#103	1	4	yes	complemented
62	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
63	C₄×C₁₂₇	#124	1	1	yes	characteristic
64	C₄×C₁₂₇	#127	1	1	yes	characteristic
65	Dic₁₂₇	—	4	4	no	—
66	C₂×C₁₂₇	#390	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#103	1	1	yes	characteristic
68	C₂×C₁₂₇	#419	1	1	yes	characteristic
69	C₂×C₁₂₇	#461	1	2	yes	—
70	C₂×C₁₂₇	#462	1	2	yes	—
71	#1981	—	127	1	no	maximal, Sylow
72	C₁₂₇	#1981	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄⋊Q₈	—	127	1	no	—
74	C₂×C₄⋊C₈	—	127	1	no	—
75	C₂×#103	—	127	4	no	—
76	C₄×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
77	C₂²×Q₈	—	127	2	no	—
78	C₂²×C₈	—	127	2	no	—
79	C₄⋊Q₈	—	127	4	no	—
80	C₄⋊Q₈	—	254	2	no	—
81	C₂×C₄⋊C₄	—	254	1	no	—
82	C₂×C₄⋊C₄	—	254	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
86	C₄⋊C₈	—	254	2	no	—
87	#103	—	254	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	127	4	no	—
90	C₂×Q₈	—	254	4	no	—
91	C₂×Q₈	—	254	4	no	—
92	C₂×Q₈	—	254	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	254	2	no	—
95	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂×C₈	—	254	4	no	—
99	C₂×C₈	—	254	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	254	4	no	—
102	C₄⋊C₄	—	254	4	no	—
103	C₄⋊C₄	D₅₀₈	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₁₂₇	1	2	yes	—
106	C₄²	D₅₀₈	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
108	Q₈	—	508	2	no	—
109	Q₈	—	508	2	no	—
110	Q₈	—	254	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	508	4	no	—
116	C₂×C₄	—	254	4	no	—
117	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
119	C₂×C₄	#14523236	1	2	yes	—
120	C₂×C₄	#14523238	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
124	C₂×C₄	#14523248	1	2	yes	—
125	C₈	—	508	2	no	—
126	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
127	C₂²	#16861183	1	2	yes	—
128	C₂²	C₂×#14523236	1	1	yes	characteristic
129	C₂²	C₂×#14523238	1	1	yes	characteristic
130	C₂²	#16861239	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	508	4	no	—
134	C₄	#16861132	1	4	yes	complemented
135	C₄	C₄×D₅₀₈	1	2	yes	—
136	C₄	#16861197	1	1	yes	characteristic
137	C₄	#16861206	1	1	yes	characteristic
138	C₂	#22549778	1	1	yes	characteristic
139	C₂	C₂×#16861132	1	1	yes	characteristic
140	C₂	#22549884	1	1	yes	characteristic
141	C₂	#22550081	1	2	yes	—
142	C₂	#22550086	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#16861132

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861209

—

#16861209

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

—

C₄⋊(C₁₂₇⋊C₈)

—

#16861132

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#121

yes

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#103

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#462

yes

—

#1981

—

127

maximal, Sylow

C₁₂₇

#1981

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×C₄⋊C₈

—

127

—

C₂×#103

—

127

—

C₄×C₄⋊C₄

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

#103

—

254

—

#95

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

254

—

C₂×C₈

—

254

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

254

—

102

C₄⋊C₄

—

254

—

103

C₄⋊C₄

D₅₀₈

yes

—

104

C₄²

—

105

C₄²

C₄×D₁₂₇

yes

—

106

C₄²

D₅₀₈

yes

—

107

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

108

Q₈

—

508

—

109

Q₈

—

508

—

110

Q₈

—

254

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

508

—

116

C₂×C₄

—

254

—

117

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

119

C₂×C₄

#14523236

yes

—

120

C₂×C₄

#14523238

yes

—

121

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

122

C₂×C₄

C₂×D₅₀₈

yes

characteristic

123

C₂×C₄

C₂×D₅₀₈

yes

—

124

C₂×C₄

#14523248

yes

—

125

C₈

—

508

—

126

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

127

C₂²

#16861183

yes

—

128

C₂²

C₂×#14523236

yes

characteristic

129

C₂²

C₂×#14523238

yes

characteristic

130

C₂²

#16861239

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

508

—

134

C₄

#16861132

yes

complemented

135

C₄

C₄×D₅₀₈

yes

—

136

C₄

#16861197

yes

characteristic

137

C₄

#16861206

yes

characteristic

138

C₂

#22549778

yes

characteristic

139

C₂

C₂×#16861132

yes

characteristic

140

C₂

#22549884

yes

characteristic

141

C₂

#22550081

yes

—

142

C₂

#22550086

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄×C₄⋊C₄×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#1981
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₄×C₄⋊C₄×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₂₇

hypocenter

C₁₂₇

#1981

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #16861209	C₄	semidirect
61. C₄×C₁₂₇	#103	semidirect
72. C₁₂₇	#1981	semidirect
134. C₄	#16861132	semidirect

H ≅ G/N

type

8. #16861209

C₄

semidirect

61. C₄×C₁₂₇

#103

semidirect

72. C₁₂₇

#1981

semidirect

134. C₄

#16861132

semidirect

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

subgroup	index
C₂×#16861209	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×#16861132	2
C₄×C₄⋊C₄×C₁₂₇	2
#1981	127

subgroup

index

C₂×#16861209

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×#16861132

C₄×C₄⋊C₄×C₁₂₇

#1981

127

Subgroups — #30639653

Conjugacy classes of subgroups

Complements

Maximal subgroups