Group #30639505 — subgroups

Subgroups — #30639505

Order 16256.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861144	C₂	1	2	yes	maximal
2	#22549900	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#22549640	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	#16861144	—	2	4	no	—
9	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
10	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₁₂₇⋊C₈)	C₄	1	2	yes	—
12	C₄⋊(C₁₂₇⋊C₈)	C₄	1	2	yes	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	#97×C₁₂₇	—	2	1	no	—
17	C₄×C₈×C₁₂₇	—	2	1	no	—
18	#14523232	—	2	2	no	—
19	#14523232	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	1	no	—
28	M₁₆×C₁₂₇	—	2	1	no	—
29	M₁₆×C₁₂₇	D₄	1	2	yes	—
30	M₁₆×C₁₂₇	Q₈	1	2	yes	—
31	C₂×C₈×C₁₂₇	—	2	1	no	—
32	C₂×C₈×C₁₂₇	—	2	1	no	—
33	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₁₂₇⋊C₈	—	2	4	no	—
38	C₁₂₇⋊C₈	—	4	2	no	—
39	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	4	1	no	—
42	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
52	C₈×C₁₂₇	—	2	2	no	—
53	C₈×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	—	2	1	no	—
55	C₂²×C₁₂₇	—	2	1	no	—
56	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₂₇	#125	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₁₂₇	—	2	1	no	—
60	C₄×C₁₂₇	—	2	1	no	—
61	C₄×C₁₂₇	—	2	1	no	—
62	C₄×C₁₂₇	#108	1	2	yes	complemented
63	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₁₂₇	#121	1	1	yes	characteristic
66	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₂₇	—	2	1	no	—
68	C₂×C₁₂₇	—	2	1	no	—
69	C₂×C₁₂₇	#389	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×#108	1	1	yes	characteristic
71	C₂×C₁₂₇	#430	1	1	yes	characteristic
72	#1950	—	127	1	no	maximal, Sylow
73	C₁₂₇	#1950	1	1	yes	characteristic, complemented, Sylow
74	C₂×#108	—	127	2	no	—
75	#423	—	127	1	no	—
76	C₂×C₄⋊C₈	—	127	1	no	—
77	C₄×M₁₆	D₁₂₇	1	1	yes	characteristic
78	#341	—	127	2	no	—
79	C₂×M₁₆	—	127	2	no	—
80	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂²×C₈	—	127	2	no	—
83	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
84	#108	—	254	4	no	—
85	C₄⋊C₈	—	127	2	no	—
86	C₄⋊C₈	—	127	2	no	—
87	C₄⋊C₈	—	254	1	no	—
88	C₄⋊C₈	—	254	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
92	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	254	2	no	—
97	M₁₆	—	254	2	no	—
98	M₁₆	D₅₀₈	1	2	yes	—
99	M₁₆	Dic₂₅₄	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	127	4	no	—
103	C₂×C₈	—	254	2	no	—
104	C₂×C₈	—	254	2	no	—
105	C₂×C₈	—	254	2	no	—
106	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
108	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₂³	C₂×Dic₂₅₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
114	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
115	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
116	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
121	C₂×C₄	#14523248	1	1	yes	characteristic
122	C₂×C₄	#14523248	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	508	2	no	—
126	C₈	—	254	4	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
130	C₂²	#16861201	1	1	yes	characteristic
131	C₂²	#16861209	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#16861144	1	2	yes	complemented
136	C₄	C₄×D₅₀₈	1	1	yes	characteristic
137	C₄	C₄×Dic₂₅₄	1	1	yes	characteristic
138	C₄	#16861183	1	1	yes	characteristic
139	C₄	#16861188	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#22549774	1	1	yes	characteristic
143	C₂	C₂×#16861144	1	1	yes	characteristic
144	C₂	#22549924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861144

C₂

yes

maximal

#22549900

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic

#22549640

C₂

yes

maximal

C₄×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861144

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

D₄

yes

—

M₁₆×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

#125

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#108

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₄×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1950

yes

characteristic, complemented, Sylow

C₂×#108

—

127

—

#423

—

127

—

C₂×C₄⋊C₈

—

127

—

C₄×M₁₆

D₁₂₇

yes

characteristic

#341

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#108

—

254

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

—

254

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

M₁₆

D₅₀₈

yes

—

M₁₆

Dic₂₅₄

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

127

—

103

C₂×C₈

—

254

—

104

C₂×C₈

—

254

—

105

C₂×C₈

—

254

—

106

C₄²

C₄×D₁₂₇

yes

characteristic

107

C₄²

C₄×D₁₂₇

yes

characteristic

108

C₄²

C₂²⋊D₁₂₇

yes

characteristic

109

C₄²

C₂²⋊D₁₂₇

yes

characteristic

110

C₂³

C₂×Dic₂₅₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

114

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

115

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

116

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

C₂×C₄

C₂×D₅₀₈

yes

characteristic

120

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

121

C₂×C₄

#14523248

yes

characteristic

122

C₂×C₄

#14523248

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

508

—

126

C₈

—

254

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

130

C₂²

#16861201

yes

characteristic

131

C₂²

#16861209

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#16861144

yes

complemented

136

C₄

C₄×D₅₀₈

yes

characteristic

137

C₄

C₄×Dic₂₅₄

yes

characteristic

138

C₄

#16861183

yes

characteristic

139

C₄

#16861188

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#22549774

yes

characteristic

143

C₂

C₂×#16861144

yes

characteristic

144

C₂

#22549924

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#1950
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₄×M₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₂₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄×M₁₆

D₁₂₇

hypocenter

C₁₂₇

#1950

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
62. C₄×C₁₂₇	#108	semidirect
73. C₁₂₇	#1950	semidirect
135. C₄	#16861144	semidirect

H ≅ G/N

type

62. C₄×C₁₂₇

#108

semidirect

73. C₁₂₇

#1950

semidirect

135. C₄

#16861144

semidirect

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

subgroup	index
C₂×#16861144	2
#22549900	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549640	2
C₄×M₁₆×C₁₂₇	2
#1950	127

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

C₄×M₁₆×C₁₂₇

127

Subgroups — #30639505

Conjugacy classes of subgroups

Complements

Maximal subgroups