Group #30638855 — subgroups

Subgroups — #30638855

Order 16256.

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

normal subgroups only

This group has 172 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	#22549914	C₂	1	1	yes	maximal, characteristic
2	#22549831	C₂	1	1	yes	maximal, characteristic
3	#22549648	C₂	1	2	yes	maximal
4	#22549633	C₂	1	2	yes	maximal
5	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	1	yes	characteristic
7	#16861168	C₂²	1	2	yes	—
8	#16861168	C₂²	1	1	yes	characteristic
9	#16861143	—	2	1	no	—
10	#16861141	—	2	1	no	—
11	#16861119	C₄	1	2	yes	complemented
12	C₈×Dic₁₂₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#108×C₁₂₇	—	2	1	no	—
16	#108×C₁₂₇	—	2	1	no	—
17	#14523232	—	2	1	no	—
18	#14523232	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	1	no	—
22	C₂²⋊Dic₁₂₇	—	2	1	no	—
23	C₂²⋊Dic₁₂₇	—	2	2	no	—
24	C₄×Dic₁₂₇	—	2	2	no	—
25	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	2	no	—
28	M₁₆×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
36	C₂×Dic₁₂₇	—	2	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₁₂₇⋊C₈	—	2	2	no	—
39	C₂³×C₁₂₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
46	C₈×C₁₂₇	—	4	2	no	—
47	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₂₇	—	4	1	no	—
52	C₂²×C₁₂₇	#117	1	1	yes	characteristic
53	C₂²×C₁₂₇	#124	1	1	yes	characteristic
54	C₂²×C₁₂₇	#125	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₂₇	#117	1	1	yes	characteristic
57	C₄×C₁₂₇	#124	1	1	yes	characteristic
58	C₄×C₁₂₇	#125	1	1	yes	characteristic
59	Dic₁₂₇	—	4	2	no	—
60	Dic₁₂₇	—	8	2	no	—
61	C₂×C₁₂₇	—	2	1	no	—
62	C₂×C₁₂₇	—	2	1	no	—
63	C₂×C₁₂₇	—	2	1	no	—
64	C₂×C₁₂₇	#382	1	1	yes	characteristic
65	#1833	—	127	1	no	maximal, Sylow
66	C₁₂₇	#1833	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	D₁₂₇	1	1	yes	characteristic
68	#427	—	127	1	no	—
69	#405	—	127	1	no	—
70	#344	—	127	2	no	—
71	H(3,ℤ/4)	—	127	2	no	—
72	C₂×M₁₆	—	127	1	no	—
73	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
74	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
75	#117	—	127	1	no	—
76	#117	—	127	2	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	254	1	no	—
80	#106	—	254	1	no	—
81	#97	—	127	2	no	—
82	C₄×C₈	—	127	2	no	—
83	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	254	1	no	—
87	M₁₆	—	254	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	127	2	no	—
90	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
94	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
95	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
96	C₄⋊C₄	—	254	2	no	—
97	C₄⋊C₄	—	254	1	no	—
98	C₂²⋊C₄	—	254	2	no	—
99	C₂²⋊C₄	—	254	1	no	—
100	C₄²	—	127	2	no	—
101	C₄²	—	254	2	no	—
102	C₂³	#14523249	1	1	yes	characteristic
103	C₂×C₄	—	508	2	no	—
104	C₂×C₄	—	254	2	no	—
105	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
108	C₂×C₄	#14523247	1	1	yes	characteristic
109	C₂×C₄	#14523247	1	1	yes	characteristic
110	C₂×C₄	#14523249	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	254	2	no	—
113	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
114	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
115	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
116	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#16861167	1	1	yes	characteristic
119	C₂²	#16861198	1	1	yes	characteristic
120	C₂²	#16861200	1	1	yes	characteristic
121	C₄	—	1016	2	no	—
122	C₄	—	508	2	no	—
123	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
124	C₄	#16861167	1	1	yes	characteristic
125	C₄	#16861196	1	1	yes	characteristic
126	C₄	#16861200	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22549741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549914

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal, characteristic

#22549648

C₂

yes

maximal

#22549633

C₂

yes

maximal

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

—

#16861168

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₂₇

C₄

yes

complemented

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

—

#108×C₁₂₇

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#382

yes

characteristic

#1833

—

127

maximal, Sylow

C₁₂₇

#1833

yes

characteristic, complemented, Sylow

C₂×#108

D₁₂₇

yes

characteristic

#427

—

127

—

#405

—

127

—

#344

—

127

—

H(3,ℤ/4)

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

#117

—

127

—

#108

—

#108

—

#107

—

254

—

#106

—

254

—

#97

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

characteristic

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

100

C₄²

—

127

—

101

C₄²

—

254

—

102

C₂³

#14523249

yes

characteristic

103

C₂×C₄

—

508

—

104

C₂×C₄

—

254

—

105

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

107

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

108

C₂×C₄

#14523247

yes

characteristic

109

C₂×C₄

#14523247

yes

characteristic

110

C₂×C₄

#14523249

yes

characteristic

111

C₈

—

112

C₈

—

254

—

113

C₈

Dic₁₂₇⋊C₄

yes

characteristic

114

C₈

Dic₁₂₇⋊C₄

yes

characteristic

115

C₈

Dic₁₂₇⋊C₄

yes

characteristic

116

C₈

Dic₁₂₇⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#16861167

yes

characteristic

119

C₂²

#16861198

yes

characteristic

120

C₂²

#16861200

yes

characteristic

121

C₄

—

1016

—

122

C₄

—

508

—

123

C₄

C₂×Dic₁₂₇⋊C₄

yes

characteristic

124

C₄

#16861167

yes

characteristic

125

C₄

#16861196

yes

characteristic

126

C₄

#16861200

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#22549741

yes

characteristic

	subgroup	G/H
center	C₄	#16861167
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂×C₁₂₇	#382
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#1833
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#16861167

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#108×C₁₂₇

C₂

C₂×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #16861119	C₄	semidirect
12. C₈×Dic₁₂₇	C₄	semidirect
66. C₁₂₇	#1833	semidirect

H ≅ G/N

type

11. #16861119

C₄

semidirect

12. C₈×Dic₁₂₇

C₄

semidirect

66. C₁₂₇

#1833

semidirect

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

subgroup	index
#22549914	2
#22549831	2
#22549648	2
#22549633	2
C₂×#108×C₁₂₇	2
#1833	127

subgroup

index

C₂×#108×C₁₂₇

127

Subgroups — #30638855

Conjugacy classes of subgroups

Complements

Maximal subgroups