Group #30640289 — subgroups

Subgroups — #30640289

Order 16256.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22549774	C₂	1	1	yes	maximal, characteristic
2	#22549746	C₂	1	1	yes	maximal, characteristic
3	#22549639	C₂	1	2	yes	maximal
4	#22549639	C₂	1	2	yes	maximal
5	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
7	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₂₇⋊C₄	—	2	1	no	—
12	C₁₂₇⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	—	2	1	no	—
16	C₂²×Dic₁₂₇	—	2	1	no	—
17	C₂²×Dic₁₂₇	—	2	2	no	—
18	C₄⋊Dic₁₂₇	—	2	1	no	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	4	no	—
21	C₄⋊Dic₁₂₇	—	4	2	no	—
22	C₄⋊Dic₁₂₇	—	4	2	no	—
23	C₄⋊Dic₁₂₇	Q₈	1	2	yes	—
24	Dic₁₂₇⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	C₂×Dic₁₂₇	—	2	2	no	—
33	C₂×Dic₁₂₇	—	4	1	no	—
34	C₂×Dic₁₂₇	—	4	1	no	—
35	C₂×Dic₁₂₇	—	4	1	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	1	no	—
38	C₂×Dic₁₂₇	—	4	2	no	—
39	C₂×Dic₁₂₇	—	4	2	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	1	no	—
43	C₂×C₄×C₁₂₇	—	2	1	no	—
44	C₂×C₄×C₁₂₇	—	2	1	no	—
45	C₂×C₄×C₁₂₇	—	2	1	no	—
46	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	#41	1	2	yes	—
50	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₈×C₁₂₇	—	4	2	no	—
53	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₂₇	#123	1	1	yes	characteristic
55	C₂²×C₁₂₇	#124	1	1	yes	characteristic
56	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₁₂₇	#137	1	1	yes	characteristic
59	C₂²×C₁₂₇	#137	1	1	yes	characteristic
60	C₄×C₁₂₇	—	2	2	no	—
61	C₄×C₁₂₇	—	2	2	no	—
62	C₄×C₁₂₇	#125	1	2	yes	—
63	C₄×C₁₂₇	#126	1	2	yes	—
64	Dic₁₂₇	—	4	2	no	—
65	Dic₁₂₇	—	8	1	no	—
66	Dic₁₂₇	—	8	2	no	—
67	C₂×C₁₂₇	#400	1	1	yes	characteristic
68	C₂×C₁₂₇	#474	1	1	yes	characteristic
69	C₂×C₁₂₇	#475	1	1	yes	characteristic
70	C₂×C₁₂₇	#480	1	1	yes	characteristic
71	C₂×C₁₂₇	#483	1	1	yes	characteristic
72	C₂×C₁₂₇	#486	1	1	yes	characteristic
73	C₂×C₁₂₇	#491	1	1	yes	characteristic
74	#2099	—	127	1	no	maximal, Sylow
75	C₁₂₇	#2099	1	1	yes	characteristic, complemented, Sylow
76	#389	—	127	1	no	—
77	#384	—	127	1	no	—
78	#340	—	127	2	no	—
79	#340	—	127	2	no	—
80	#336	D₁₂₇	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
82	C₂×C₄⋊C₄	—	127	1	no	—
83	C₂×C₄⋊C₄	—	127	2	no	—
84	C₂×C₄⋊C₄	—	127	1	no	—
85	C₂×C₄⋊C₄	—	254	1	no	—
86	C₂×C₄⋊C₄	—	254	1	no	—
87	C₂×C₄⋊C₄	—	254	1	no	—
88	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
89	#95	—	254	1	no	—
90	C₂²×C₄	—	254	1	no	—
91	C₂²×C₄	—	254	2	no	—
92	C₂²×C₄	—	254	1	no	—
93	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
94	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	127	2	no	—
100	C₄⋊C₄	—	508	2	no	—
101	C₄⋊C₄	—	508	2	no	—
102	C₄⋊C₄	—	508	2	no	—
103	C₄⋊C₄	—	254	2	no	—
104	C₄⋊C₄	—	254	4	no	—
105	C₄⋊C₄	—	254	1	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	508	1	no	—
113	C₂×C₄	—	508	2	no	—
114	C₂×C₄	—	508	1	no	—
115	C₂×C₄	—	508	2	no	—
116	C₂×C₄	—	508	2	no	—
117	C₂×C₄	—	508	1	no	—
118	C₂×C₄	—	508	1	no	—
119	C₂×C₄	—	508	1	no	—
120	C₂×C₄	—	254	2	no	—
121	C₂×C₄	D₁₀₁₆	1	2	yes	—
122	C₂×C₄	#14523237	1	2	yes	—
123	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
124	C₂×C₄	#14523247	1	2	yes	—
125	C₂×C₄	#14523247	1	1	yes	characteristic
126	C₂×C₄	#14523248	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#16861189	1	1	yes	characteristic
129	C₂²	#16861193	1	1	yes	characteristic
130	C₂²	#16861197	1	1	yes	characteristic
131	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
132	C₂²	C₂×#14523237	1	1	yes	characteristic
133	C₂²	#16861242	1	1	yes	characteristic
134	C₂²	#16861242	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	1016	1	no	—
138	C₄	—	1016	2	no	—
139	C₄	—	508	2	no	—
140	C₄	#16861199	1	2	yes	—
141	C₄	#16861202	1	2	yes	—
142	C₂	#22549816	1	1	yes	characteristic
143	C₂	#22550157	1	1	yes	characteristic
144	C₂	#22550162	1	1	yes	characteristic
145	C₂	#22550190	1	1	yes	characteristic
146	C₂	#22550207	1	1	yes	characteristic
147	C₂	#22550223	1	1	yes	characteristic
148	C₂	#22550245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549774

C₂

yes

maximal, characteristic

#22549746

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

#22549639

C₂

yes

maximal

#336×C₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇⋊C₄

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

Q₈

yes

—

Dic₁₂₇⋊C₄

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

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×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#125

yes

—

C₄×C₁₂₇

#126

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2099

yes

characteristic, complemented, Sylow

#389

—

127

—

#384

—

127

—

#340

—

127

—

#340

—

127

—

#336

D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#95

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

127

—

100

C₄⋊C₄

—

508

—

101

C₄⋊C₄

—

508

—

102

C₄⋊C₄

—

508

—

103

C₄⋊C₄

—

254

—

104

C₄⋊C₄

—

254

—

105

C₄⋊C₄

—

254

—

106

C₄²

—

107

C₂³

C₂×D₅₀₈

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

508

—

113

C₂×C₄

—

508

—

114

C₂×C₄

—

508

—

115

C₂×C₄

—

508

—

116

C₂×C₄

—

508

—

117

C₂×C₄

—

508

—

118

C₂×C₄

—

508

—

119

C₂×C₄

—

508

—

120

C₂×C₄

—

254

—

121

C₂×C₄

D₁₀₁₆

yes

—

122

C₂×C₄

#14523237

yes

—

123

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

124

C₂×C₄

#14523247

yes

—

125

C₂×C₄

#14523247

yes

characteristic

126

C₂×C₄

#14523248

yes

—

127

C₈

—

128

C₂²

#16861189

yes

characteristic

129

C₂²

#16861193

yes

characteristic

130

C₂²

#16861197

yes

characteristic

131

C₂²

C₂×D₁₀₁₆

yes

characteristic

132

C₂²

C₂×#14523237

yes

characteristic

133

C₂²

#16861242

yes

characteristic

134

C₂²

#16861242

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

1016

—

138

C₄

—

1016

—

139

C₄

—

508

—

140

C₄

#16861199

yes

—

141

C₄

#16861202

yes

—

142

C₂

#22549816

yes

characteristic

143

C₂

#22550157

yes

characteristic

144

C₂

#22550162

yes

characteristic

145

C₂

#22550190

yes

characteristic

146

C₂

#22550207

yes

characteristic

147

C₂

#22550223

yes

characteristic

148

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	#2099
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
75. C₁₂₇	#2099	semidirect

H ≅ G/N

type

75. C₁₂₇

#2099

semidirect

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

subgroup	index
#22549774	2
#22549746	2
#22549639	2
#336×C₁₂₇	2
#2099	127

subgroup

index

#336×C₁₂₇

127

Subgroups — #30640289

Conjugacy classes of subgroups

Complements

Maximal subgroups