Group #30639511 — subgroups

Subgroups — #30639511

Order 16256.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22549918	C₂	1	2	yes	maximal
2	C₂×#16861141	C₂	1	1	yes	maximal, characteristic
3	#22549831	C₂	1	1	yes	maximal, characteristic
4	#22549644	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	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	#16861143	—	2	2	no	—
10	#16861141	—	2	2	no	—
11	#16861141	—	2	1	no	—
12	#16861141	—	2	1	no	—
13	#16861119	—	2	1	no	—
14	C₈×Dic₁₂₇	—	2	1	no	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	#108×C₁₂₇	C₄	1	4	yes	complemented
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	#14523232	—	2	1	no	—
20	#14523232	Q₈	1	2	yes	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
25	C₂²⋊Dic₁₂₇	—	2	1	no	—
26	C₄×Dic₁₂₇	—	2	1	no	—
27	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₂₇	—	2	2	no	—
29	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
35	C₂×Dic₁₂₇	—	2	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	8	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×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
46	C₈×C₁₂₇	—	2	4	no	—
47	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
48	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₁₂₇	—	2	2	no	—
50	C₂²×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	#121	1	1	yes	characteristic
52	C₂²×C₁₂₇	#125	1	1	yes	characteristic
53	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₁₂₇	—	4	2	no	—
58	Dic₁₂₇	—	8	2	no	—
59	C₂×C₁₂₇	—	2	1	no	—
60	C₂×C₁₂₇	—	2	1	no	—
61	C₂×C₁₂₇	#389	1	1	yes	characteristic
62	C₂×C₁₂₇	#471	1	2	yes	—
63	#1951	—	127	1	no	maximal, Sylow
64	C₁₂₇	#1951	1	1	yes	characteristic, complemented, Sylow
65	C₂×#108	D₁₂₇	1	1	yes	characteristic
66	#428	—	127	2	no	—
67	C₂×#106	—	127	1	no	—
68	#405	—	127	1	no	—
69	#343	—	127	2	no	—
70	C₂×M₁₆	—	127	1	no	—
71	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
72	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
73	#117	—	127	1	no	—
74	C₂×C₄⋊C₄	—	127	2	no	—
75	#108	Dic₁₂₇	1	4	yes	complemented
76	#107	—	254	2	no	—
77	#106	—	254	2	no	—
78	#106	—	254	1	no	—
79	#106	—	254	1	no	—
80	#97	—	254	1	no	—
81	C₄×C₈	—	254	1	no	—
82	C₂²×C₄	—	254	2	no	—
83	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	127	2	no	—
86	M₁₆	—	254	1	no	—
87	M₁₆	C₂×Dic₁₂₇	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	254	1	no	—
90	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
91	C₂×C₈	D₅₀₈	1	1	yes	characteristic
92	C₂×C₈	D₅₀₈	1	1	yes	characteristic
93	C₂×C₈	Dic₂₅₄	1	2	yes	—
94	C₄⋊C₄	—	127	2	no	—
95	C₄⋊C₄	—	254	2	no	—
96	C₄⋊C₄	—	254	2	no	—
97	C₂²⋊C₄	—	254	1	no	—
98	C₄²	—	254	1	no	—
99	C₂³	#14523249	1	1	yes	characteristic
100	C₂×C₄	—	1016	2	no	—
101	C₂×C₄	—	508	2	no	—
102	C₂×C₄	—	254	2	no	—
103	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
108	C₂×C₄	#14523247	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	254	2	no	—
111	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
112	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₁₂₇	1	1	yes	characteristic
115	C₂²	#16861184	1	1	yes	characteristic
116	C₂²	#16861200	1	1	yes	characteristic
117	C₄	—	1016	2	no	—
118	C₄	—	508	2	no	—
119	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
120	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
121	C₄	#16861189	1	1	yes	characteristic
122	C₄	#16861208	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#22549775	1	1	yes	characteristic
126	C₂	#22550140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549918

C₂

yes

maximal

C₂×#16861141

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal, characteristic

#22549644

C₂

yes

maximal

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₁₂₇

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

C₄

yes

complemented

C₂²×Dic₁₂₇

—

#14523232

—

#14523232

Q₈

yes

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

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×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

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#389

yes

characteristic

C₂×C₁₂₇

#471

yes

—

#1951

—

127

maximal, Sylow

C₁₂₇

#1951

yes

characteristic, complemented, Sylow

C₂×#108

D₁₂₇

yes

characteristic

#428

—

127

—

C₂×#106

—

127

—

#405

—

127

—

#343

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

C₂×C₄⋊C₄

—

127

—

#108

Dic₁₂₇

yes

complemented

#107

—

254

—

#106

—

254

—

#106

—

254

—

#106

—

254

—

#97

—

254

—

C₄×C₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

C₂×Dic₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

#14523249

yes

characteristic

100

C₂×C₄

—

1016

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

254

—

103

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

104

C₂×C₄

D₄×D₁₂₇

yes

characteristic

105

C₂×C₄

C₂×D₅₀₈

yes

characteristic

106

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

107

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

108

C₂×C₄

#14523247

yes

characteristic

109

C₈

—

110

C₈

—

254

—

111

C₈

C₄⋊Dic₁₂₇

yes

—

112

C₈

C₄⋊Dic₁₂₇

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₁₂₇

yes

characteristic

115

C₂²

#16861184

yes

characteristic

116

C₂²

#16861200

yes

characteristic

117

C₄

—

1016

—

118

C₄

—

508

—

119

C₄

C₂×C₄⋊Dic₁₂₇

yes

characteristic

120

C₄

D₄×Dic₁₂₇

yes

characteristic

121

C₄

#16861189

yes

characteristic

122

C₄

#16861208

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#22549775

yes

characteristic

126

C₂

#22550140

yes

—

	subgroup	G/H
center	C₂²	#16861184
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#1951
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#16861184

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#108×C₁₂₇

C₂

C₂²×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₁₂₇	C₄	semidirect
64. C₁₂₇	#1951	semidirect
75. #108	Dic₁₂₇	semidirect

H ≅ G/N

type

17. #108×C₁₂₇

C₄

semidirect

64. C₁₂₇

#1951

semidirect

75. #108

Dic₁₂₇

semidirect

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

subgroup	index
#22549918	2
C₂×#16861141	2
#22549831	2
#22549644	2
C₂×#108×C₁₂₇	2
#1951	127

subgroup

index

C₂×#108×C₁₂₇

127

Subgroups — #30639511

Conjugacy classes of subgroups

Complements

Maximal subgroups