Group #30639477 — subgroups

Subgroups — #30639477

Order 16256.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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	1	yes	maximal, characteristic
2	#22549912	C₂	1	1	yes	maximal, characteristic
3	#22549904	C₂	1	1	yes	maximal, characteristic
4	#22549903	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#22549636	C₂	1	1	yes	maximal, characteristic
7	#339×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	1	yes	characteristic
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16861168	C₂²	1	1	yes	characteristic
12	#16861144	—	2	2	no	—
13	#16861142	C₄	1	2	yes	complemented
14	#16861140	C₄	1	2	yes	complemented
15	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
16	Dic₁₂₇⋊C₈	—	2	2	no	—
17	C₄×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	#117×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	#14523232	—	2	1	no	—
22	#14523232	—	2	1	no	—
23	C₂×C₁₂₇⋊C₈	—	2	1	no	—
24	C₂×C₁₂₇⋊C₈	—	2	1	no	—
25	C₂×C₁₂₇⋊C₈	—	2	1	no	—
26	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
29	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
30	C₂×C₁₂₇⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₁₂₇	—	2	1	no	—
33	C₄×Dic₁₂₇	—	2	1	no	—
34	C₂²×C₄×C₁₂₇	—	2	1	no	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₂₇	—	2	1	no	—
37	M₁₆×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	2	1	no	—
39	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
41	C₄²×C₁₂₇	—	2	1	no	—
42	C₄²×C₁₂₇	—	2	1	no	—
43	C₄²×C₁₂₇	Q₈	1	2	yes	—
44	C₂×Dic₁₂₇	—	2	2	no	—
45	C₁₂₇⋊C₈	—	2	2	no	—
46	C₁₂₇⋊C₈	—	4	1	no	—
47	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂³×C₁₂₇	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	—	2	4	no	—
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
59	C₈×C₁₂₇	—	4	1	no	—
60	C₂²×C₁₂₇	—	2	2	no	—
61	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₂₇	#125	1	1	yes	characteristic
64	C₄×C₁₂₇	—	2	4	no	—
65	C₄×C₁₂₇	—	4	2	no	—
66	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₁₂₇	#121	1	1	yes	characteristic
69	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₁₂₇	—	4	2	no	—
71	C₂×C₁₂₇	—	2	1	no	—
72	C₂×C₁₂₇	—	2	1	no	—
73	C₂×C₁₂₇	#389	1	1	yes	characteristic
74	C₂×C₁₂₇	#443	1	2	yes	—
75	#1945	—	127	1	no	maximal, Sylow
76	C₁₂₇	#1945	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	127	1	no	—
78	#427	—	127	1	no	—
79	#424	—	127	1	no	—
80	#424	—	127	1	no	—
81	C₂×C₄×C₈	—	127	1	no	—
82	#339	—	127	1	no	—
83	#339	D₁₂₇	1	1	yes	characteristic
84	C₂×M₁₆	—	127	1	no	—
85	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂²×C₈	—	127	1	no	—
87	C₂²×C₈	—	127	1	no	—
88	#117	—	127	1	no	—
89	#117	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
91	#108	—	254	2	no	—
92	#107	—	127	2	no	—
93	#106	—	127	2	no	—
94	C₄⋊C₈	—	254	2	no	—
95	C₄⋊C₈	—	254	2	no	—
96	C₄×C₈	—	254	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	254	1	no	—
102	M₁₆	—	254	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	127	1	no	—
105	C₂×C₈	—	127	1	no	—
106	C₂×C₈	—	127	2	no	—
107	C₂×C₈	—	127	2	no	—
108	C₂×C₈	—	127	2	no	—
109	C₂×C₈	—	254	1	no	—
110	C₂×C₈	—	254	1	no	—
111	C₂×C₈	—	254	1	no	—
112	C₄⋊C₄	—	127	2	no	—
113	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	254	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	254	1	no	—
119	C₄²	Dic₂₅₄	1	2	yes	—
120	C₂³	#14523247	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	254	2	no	—
125	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
130	C₂×C₄	#14523248	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	127	2	no	—
133	C₈	—	127	2	no	—
134	C₈	—	508	1	no	—
135	C₈	—	254	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₁₂₇)⋊C₄	1	1	yes	characteristic
138	C₂²	#16861186	1	1	yes	characteristic
139	C₂²	#16861199	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	508	2	no	—
143	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
144	C₄	C₄×Dic₂₅₄	1	1	yes	characteristic
145	C₄	#16861182	1	1	yes	characteristic
146	C₄	#16861208	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#22549773	1	1	yes	characteristic
150	C₂	#22549990	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861144

C₂

yes

maximal, characteristic

#22549912

C₂

yes

maximal, characteristic

#22549904

C₂

yes

maximal, characteristic

#22549903

C₂

yes

maximal, characteristic

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#22549636

C₂

yes

maximal, characteristic

#339×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

#16861144

—

#16861142

C₄

yes

complemented

#16861140

C₄

yes

complemented

C₄⋊(C₁₂₇⋊C₈)

—

Dic₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×C₁₂₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

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

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#125

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₄×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#389

yes

characteristic

C₂×C₁₂₇

#443

yes

—

#1945

—

127

maximal, Sylow

C₁₂₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

127

—

#427

—

127

—

#424

—

127

—

#424

—

127

—

C₂×C₄×C₈

—

127

—

#339

—

127

—

#339

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#108

—

254

—

#107

—

127

—

#106

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

—

254

—

C₄×C₈

—

254

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

254

—

102

M₁₆

—

254

—

103

C₂×C₈

—

104

C₂×C₈

—

127

—

105

C₂×C₈

—

127

—

106

C₂×C₈

—

127

—

107

C₂×C₈

—

127

—

108

C₂×C₈

—

127

—

109

C₂×C₈

—

254

—

110

C₂×C₈

—

254

—

111

C₂×C₈

—

254

—

112

C₄⋊C₄

—

127

—

113

C₄⋊C₄

C₄×D₁₂₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

254

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

254

—

119

C₄²

Dic₂₅₄

yes

—

120

C₂³

#14523247

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

254

—

125

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

126

C₂×C₄

D₄×D₁₂₇

yes

characteristic

127

C₂×C₄

D₄×D₁₂₇

yes

characteristic

128

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

129

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

130

C₂×C₄

#14523248

yes

characteristic

131

C₈

—

132

C₈

—

127

—

133

C₈

—

127

—

134

C₈

—

508

—

135

C₈

—

254

—

136

C₂²

—

137

C₂²

(C₂²⋊D₁₂₇)⋊C₄

yes

characteristic

138

C₂²

#16861186

yes

characteristic

139

C₂²

#16861199

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

508

—

143

C₄

C₄⋊C₄×D₁₂₇

yes

characteristic

144

C₄

C₄×Dic₂₅₄

yes

characteristic

145

C₄

#16861182

yes

characteristic

146

C₄

#16861208

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#22549773

yes

characteristic

150

C₂

#22549990

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#339×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄×D₄
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#339	D₁₂₇
hypocenter	C₁₂₇	#1945
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#339×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₄×D₄

prime radical

C₂³×C₁₂₇

#41

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #16861142	C₄	semidirect
14. #16861140	C₄	semidirect
76. C₁₂₇	#1945	semidirect

H ≅ G/N

type

13. #16861142

C₄

semidirect

14. #16861140

C₄

semidirect

76. C₁₂₇

#1945

semidirect

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

subgroup	index
C₂×#16861144	2
#22549912	2
#22549904	2
#22549903	2
C₂×C₄×C₁₂₇⋊C₈	2
#22549636	2
#339×C₁₂₇	2
#1945	127

subgroup

index

C₂×C₄×C₁₂₇⋊C₈

#339×C₁₂₇

127

Subgroups — #30639477

Conjugacy classes of subgroups

Complements

Maximal subgroups