Group #30638779 — subgroups

Subgroups — #30638779

Order 16256.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861217	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861144	C₂	1	2	yes	maximal, complemented
3	#22549858	C₂	1	2	yes	maximal
4	C₁₂₇⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	#16861217	—	2	4	no	—
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂×#14523231	C₄	1	2	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₂₇	C₄	1	2	yes	—
12	#16861144	—	2	4	no	—
13	#16861127	—	2	2	no	—
14	#16861125	—	2	2	no	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	#14523248	—	2	2	no	—
18	#14523248	—	2	2	no	—
19	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
20	C₂×D₅₀₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₂₇	Q₈	1	2	yes	—
22	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
23	#14523232	—	2	2	no	—
24	#14523232	—	2	2	no	—
25	#14523231	—	2	4	no	—
26	C₂×C₁₂₇⋊C₈	—	2	2	no	—
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
30	C₈×D₁₂₇	—	2	4	no	—
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₂₇	—	2	2	no	—
33	M₁₆×C₁₂₇	—	2	2	no	—
34	M₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₂²×D₁₂₇	C₄⋊C₄	1	2	yes	—
38	Dic₂₅₄	—	2	2	no	—
39	D₅₀₈	—	2	2	no	—
40	C₂²⋊D₁₂₇	—	2	4	no	—
41	C₂×Dic₁₂₇	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₂₇	—	2	2	no	—
43	C₄×D₁₂₇	—	2	2	no	—
44	C₁₂₇⋊C₈	—	4	2	no	—
45	C₁₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₁₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₈×C₁₂₇	—	2	2	no	—
53	C₈×C₁₂₇	—	4	2	no	—
54	C₂²×C₁₂₇	—	4	1	no	—
55	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₂₇	—	2	2	no	—
58	C₂×D₁₂₇	—	4	2	no	—
59	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₂₇	#117	1	1	yes	characteristic
61	C₄×C₁₂₇	#121	1	2	yes	—
62	Dic₁₂₇	—	2	2	no	—
63	C₂×C₁₂₇	—	2	1	no	—
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	#380	1	1	yes	characteristic
66	D₁₂₇	—	4	2	no	—
67	#1813	—	127	1	no	maximal, Sylow
68	C₁₂₇	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	127	1	no	—
70	C₂×#108	—	127	2	no	—
71	#413	—	127	2	no	—
72	#338	—	127	1	no	—
73	#338	D₁₂₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	127	1	no	—
75	#131	—	254	4	no	—
76	C₂×M₁₆	—	127	2	no	—
77	C₂×M₁₆	—	127	2	no	—
78	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²×C₈	—	127	1	no	—
81	C₂²×C₈	—	127	2	no	—
82	#108	—	254	4	no	—
83	#101	—	254	2	no	—
84	#100	—	254	2	no	—
85	#41	—	254	2	no	—
86	#41	—	254	2	no	—
87	C₂×Q₈	—	127	1	no	—
88	C₂×D₄	—	127	2	no	—
89	C₂×D₄	—	127	1	no	—
90	C₂²×C₄	—	127	2	no	—
91	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	254	4	no	—
96	M₁₆	—	254	2	no	—
97	M₁₆	—	254	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	127	1	no	—
100	C₂×C₈	—	127	1	no	—
101	C₂×C₈	—	127	4	no	—
102	C₂×C₈	—	254	4	no	—
103	C₂×C₈	—	254	2	no	—
104	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
105	C₂³	—	127	2	no	—
106	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
107	Q₈	—	254	2	no	—
108	D₄	—	254	4	no	—
109	D₄	—	254	2	no	—
110	C₂×C₄	—	127	2	no	—
111	C₂×C₄	—	254	2	no	—
112	C₂×C₄	—	254	2	no	—
113	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
115	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
116	C₂×C₄	#14523248	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	127	2	no	—
120	C₈	—	127	2	no	—
121	C₈	—	508	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	508	2	no	—
124	C₂²	—	254	2	no	—
125	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
126	C₂²	#16861187	1	2	yes	—
127	C₄	—	254	2	no	—
128	C₄	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
129	C₄	#16861166	1	1	yes	characteristic
130	C₄	#16861182	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	508	2	no	—
134	C₂	#22549735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861217

C₂

yes

maximal, characteristic

C₂×#16861144

C₂

yes

maximal, complemented

#22549858

C₂

yes

maximal

C₁₂₇⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

#16861217

—

C₂×#14523232

C₂²

yes

—

C₂×#14523231

C₄

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₂₇

C₄

yes

—

#16861144

—

#16861127

—

#16861125

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#14523248

—

#14523248

—

C₂×Dic₂₅₄

D₄

yes

characteristic

C₂×D₅₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

Q₈

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

#14523232

—

#14523232

—

#14523231

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₈×D₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂²×D₁₂₇

C₄⋊C₄

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₄⋊C₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂²⋊C₄

yes

—

C₁₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

C₂×Q₈

yes

characteristic

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

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#380

yes

characteristic

D₁₂₇

—

#1813

—

127

maximal, Sylow

C₁₂₇

#1813

yes

characteristic, complemented, Sylow

C₂×#131

—

127

—

C₂×#108

—

127

—

#413

—

127

—

#338

—

127

—

#338

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

#131

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

#108

—

254

—

#101

—

254

—

#100

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

100

C₂×C₈

—

127

—

101

C₂×C₈

—

127

—

102

C₂×C₈

—

254

—

103

C₂×C₈

—

254

—

104

C₂×C₈

C₄×D₁₂₇

yes

—

105

C₂³

—

127

—

106

C₂³

Q₈×D₁₂₇

yes

characteristic

107

Q₈

—

254

—

108

D₄

—

254

—

109

D₄

—

254

—

110

C₂×C₄

—

127

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

114

C₂×C₄

D₄×D₁₂₇

yes

—

115

C₂×C₄

D₄×D₁₂₇

yes

characteristic

116

C₂×C₄

#14523248

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

127

—

120

C₈

—

127

—

121

C₈

—

508

—

122

C₂²

—

123

C₂²

—

508

—

124

C₂²

—

254

—

125

C₂²

C₄⋊C₄×D₁₂₇

yes

characteristic

126

C₂²

#16861187

yes

—

127

C₄

—

254

—

128

C₄

C₂²⋊C₄×D₁₂₇

yes

characteristic

129

C₄

#16861166

yes

characteristic

130

C₄

#16861182

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

508

—

134

C₂

#22549735

yes

characteristic

	subgroup	G/H
center	C₄	#16861166
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#338×C₁₂₇	C₂
socle	C₂×C₁₂₇	#380
prime radical	C₂×#14523248	C₂²
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#338	D₁₂₇
hypocenter	C₁₂₇	#1813
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#16861166

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#338×C₁₂₇

C₂

C₂×C₁₂₇

C₂²

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#16861144	C₂	semidirect
4. C₁₂₇⋊#338	C₂	semidirect
5. #338×C₁₂₇	C₂	semidirect
68. C₁₂₇	#1813	semidirect
73. #338	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#16861144

C₂

semidirect

4. C₁₂₇⋊#338

C₂

semidirect

5. #338×C₁₂₇

C₂

semidirect

68. C₁₂₇

#1813

semidirect

73. #338

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861217	2
C₂×#16861144	2
#22549858	2
C₁₂₇⋊#338	2
#338×C₁₂₇	2
#1813	127

subgroup

index

C₂×#16861217

C₂×#16861144

#22549858

C₁₂₇⋊#338

#338×C₁₂₇

#1813

127

Subgroups — #30638779

Conjugacy classes of subgroups

Complements

Maximal subgroups