Group #30638726 — subgroups

Subgroups — #30638726

Order 16256.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#22549900	C₂	1	2	yes	maximal
2	C₂×#16861128	C₂	1	2	yes	maximal
3	#22549645	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	4	yes	—
6	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
8	#16861128	—	2	4	no	—
9	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₂₇	—	2	2	no	—
11	C₄×Q₈×C₁₂₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	#14523232	—	2	4	no	—
17	#14523232	—	2	4	no	—
18	C₂×C₁₂₇⋊C₈	—	2	4	no	—
19	C₂×Q₈×C₁₂₇	—	2	2	no	—
20	C₂×Q₈×C₁₂₇	—	2	2	no	—
21	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
22	C₂×Q₈×C₁₂₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	—	2	4	no	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄²×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	—	2	2	no	—
32	C₄²×C₁₂₇	D₄	1	2	yes	—
33	C₄²×C₁₂₇	D₄	1	2	yes	—
34	C₁₂₇⋊C₈	—	4	4	no	—
35	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₂₇	—	2	4	no	—
37	Q₈×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	—	2	2	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	—	4	1	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	—	2	1	no	—
54	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₂₇	#117	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	4	no	—
58	C₄×C₁₂₇	—	2	1	no	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	—	4	2	no	—
61	C₄×C₁₂₇	#101	1	2	yes	complemented
62	C₄×C₁₂₇	#121	1	2	yes	—
63	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₁₂₇	—	2	1	no	—
65	C₂×C₁₂₇	—	2	1	no	—
66	C₂×C₁₂₇	#380	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#101	1	1	yes	characteristic
68	C₂×C₁₂₇	#413	1	1	yes	characteristic
69	#1801	—	127	1	no	maximal, Sylow
70	C₁₂₇	#1801	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×Q₈	D₁₂₇	1	1	yes	characteristic
72	#423	—	127	2	no	—
73	C₂×#101	—	127	2	no	—
74	#343	—	127	2	no	—
75	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
76	C₂×M₁₆	—	127	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂×C₄²	Dic₁₂₇	1	2	yes	—
82	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
83	C₄⋊C₈	—	254	2	no	—
84	C₄⋊C₈	—	254	2	no	—
85	#101	—	254	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	D₅₀₈	1	2	yes	—
89	C₂×Q₈	Dic₂₅₄	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
92	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
95	M₁₆	—	254	4	no	—
96	M₁₆	—	254	4	no	—
97	C₂×C₈	—	254	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
103	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
104	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
118	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
120	C₂×C₄	#14523248	1	2	yes	—
121	C₈	—	508	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
126	C₂²	#16861168	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	#16861128	1	2	yes	complemented
132	C₄	#16861183	1	2	yes	—
133	C₄	#16861188	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#22549736	1	1	yes	characteristic
137	C₂	C₂×#16861128	1	1	yes	characteristic
138	C₂	#22549860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549900

C₂

yes

maximal

C₂×#16861128

C₂

yes

maximal

#22549645

C₂

yes

maximal

C₂×C₄×Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

—

#16861128

—

C₂²×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂×Q₈×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#101

yes

complemented

C₄×C₁₂₇

#121

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1801

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₂₇

yes

characteristic

#423

—

127

—

C₂×#101

—

127

—

#343

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

C₄⋊C₈

—

254

—

#101

—

254

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₅₀₈

yes

—

C₂×Q₈

Dic₂₅₄

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₁₂₇

yes

—

103

C₄²

C₂²⋊D₁₂₇

yes

—

104

C₂³

C₂²×Dic₁₂₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

115

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₁₂₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

118

C₂×C₄

C₂×D₅₀₈

yes

characteristic

119

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

120

C₂×C₄

#14523248

yes

—

121

C₈

—

508

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₁₂₇

yes

characteristic

126

C₂²

#16861168

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#16861128

yes

complemented

132

C₄

#16861183

yes

—

133

C₄

#16861188

yes

—

134

C₂

—

135

C₂

—

136

C₂

#22549736

yes

characteristic

137

C₂

C₂×#16861128

yes

characteristic

138

C₂

#22549860

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₂₇
commutator, prime radical	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄×Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	#1801
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₂₇

commutator, prime radical

C₂³×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄×Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄×Q₈

D₁₂₇

hypocenter

C₁₂₇

#1801

LCS

G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₁₂₇	#101	semidirect
70. C₁₂₇	#1801	semidirect
131. C₄	#16861128	semidirect

H ≅ G/N

type

61. C₄×C₁₂₇

#101

semidirect

70. C₁₂₇

#1801

semidirect

131. C₄

#16861128

semidirect

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

subgroup	index
#22549900	2
C₂×#16861128	2
#22549645	2
C₂×C₄×Q₈×C₁₂₇	2
#1801	127

subgroup

index

C₂×C₄×Q₈×C₁₂₇

127

Subgroups — #30638726

Conjugacy classes of subgroups

Complements

Maximal subgroups