Group #30639999 — subgroups

Subgroups — #30639999

Order 16256.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	#22549902	C₂	1	1	yes	maximal, characteristic
3	#22549886	C₂	1	2	yes	maximal
4	C₂×#16861127	C₂	1	2	yes	maximal
5	#341×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
7	C₂×Dic₅₀₈	—	2	4	no	—
8	C₂×Dic₅₀₈	—	2	2	no	—
9	C₂×#14523232	C₂²	1	1	yes	characteristic
10	#16861168	C₂²	1	1	yes	characteristic
11	Dic₁₂₇⋊C₈	—	2	2	no	—
12	#16861134	—	2	4	no	—
13	#16861127	—	2	4	no	—
14	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₅₄	—	2	2	no	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	2	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	D₄	1	4	yes	—
22	C₂²×Dic₁₂₇	—	2	2	no	—
23	Dic₅₀₈	—	2	8	no	—
24	#14523232	—	2	2	no	—
25	C₂×C₁₂₇⋊C₈	—	2	1	no	—
26	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
27	C₂²⋊Dic₁₂₇	—	2	1	no	—
28	C₄×Dic₁₂₇	—	2	1	no	—
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₂₇	—	2	2	no	—
31	M₁₆×C₁₂₇	—	2	2	no	—
32	C₂×C₈×C₁₂₇	—	2	2	no	—
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
35	Dic₂₅₄	—	2	4	no	—
36	Dic₂₅₄	—	2	4	no	—
37	Dic₂₅₄	—	4	2	no	—
38	Dic₂₅₄	—	4	2	no	—
39	C₂×Dic₁₂₇	—	2	2	no	—
40	C₂×Dic₁₂₇	—	2	4	no	—
41	C₂×Dic₁₂₇	—	4	2	no	—
42	C₂×Dic₁₂₇	—	4	2	no	—
43	C₁₂₇⋊C₈	—	4	1	no	—
44	C₂³×C₁₂₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#41	1	2	yes	—
49	C₈×C₁₂₇	—	2	2	no	—
50	C₈×C₁₂₇	—	4	2	no	—
51	C₂²×C₁₂₇	—	2	2	no	—
52	C₂²×C₁₂₇	#121	1	1	yes	characteristic
53	C₂²×C₁₂₇	#123	1	2	yes	—
54	C₄×C₁₂₇	#120	1	1	yes	characteristic
55	C₄×C₁₂₇	#121	1	2	yes	—
56	C₄×C₁₂₇	#124	1	1	yes	characteristic
57	Dic₁₂₇	—	4	4	no	—
58	Dic₁₂₇	—	4	2	no	—
59	C₂×C₁₂₇	—	2	2	no	—
60	C₂×C₁₂₇	#394	1	1	yes	characteristic
61	C₂×C₁₂₇	#473	1	2	yes	—
62	#2047	—	127	1	no	maximal, Sylow
63	C₁₂₇	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	127	1	no	—
65	#424	—	127	1	no	—
66	#419	—	127	2	no	—
67	C₂×#101	—	127	2	no	—
68	#341	D₁₂₇	1	1	yes	characteristic
69	C₂²×Q₈	—	127	2	no	—
70	C₂×Q₁₆	—	254	4	no	—
71	C₂×Q₁₆	—	254	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	C₄⋊C₈	—	254	2	no	—
77	#103	—	254	4	no	—
78	#101	—	254	4	no	—
79	C₂×Q₈	—	127	4	no	—
80	C₂×Q₈	—	254	2	no	—
81	C₂×Q₈	—	254	2	no	—
82	C₂×Q₈	—	254	2	no	—
83	C₂×Q₈	—	254	2	no	—
84	C₂×Q₈	—	254	2	no	—
85	C₂²×C₄	—	254	2	no	—
86	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	Q₁₆	—	254	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	254	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	254	1	no	—
94	C₂×C₈	D₅₀₈	1	2	yes	—
95	C₄⋊C₄	—	127	2	no	—
96	C₂²⋊C₄	—	254	1	no	—
97	C₄²	—	254	1	no	—
98	C₂³	#14523249	1	1	yes	characteristic
99	Q₈	—	508	2	no	—
100	Q₈	—	508	2	no	—
101	Q₈	—	254	4	no	—
102	Q₈	—	254	4	no	—
103	C₂×C₄	—	508	2	no	—
104	C₂×C₄	—	508	2	no	—
105	C₂×C₄	—	254	2	no	—
106	C₂×C₄	—	254	4	no	—
107	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
108	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
110	C₂×C₄	#14523248	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	508	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#16861184	1	1	yes	characteristic
116	C₂²	#16861194	1	2	yes	—
117	C₄	—	508	4	no	—
118	C₄	—	508	2	no	—
119	C₄	#16861179	1	1	yes	characteristic
120	C₄	#16861182	1	2	yes	—
121	C₄	#16861197	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#22549794	1	1	yes	characteristic
124	C₂	#22550152	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

#22549902

C₂

yes

maximal, characteristic

#22549886

C₂

yes

maximal

C₂×#16861127

C₂

yes

maximal

#341×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861134

—

#16861127

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

D₄

yes

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

#121

yes

—

C₄×C₁₂₇

#124

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#394

yes

characteristic

C₂×C₁₂₇

#473

yes

—

#2047

—

127

maximal, Sylow

C₁₂₇

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

#424

—

127

—

#419

—

127

—

C₂×#101

—

127

—

#341

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

C₄⋊C₈

—

254

—

#103

—

254

—

#101

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

Q₁₆

—

254

—

M₁₆

—

M₁₆

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

#14523249

yes

characteristic

Q₈

—

508

—

100

Q₈

—

508

—

101

Q₈

—

254

—

102

Q₈

—

254

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

254

—

106

C₂×C₄

—

254

—

107

C₂×C₄

D₄×D₁₂₇

yes

—

108

C₂×C₄

D₄×D₁₂₇

yes

characteristic

109

C₂×C₄

C₂×D₅₀₈

yes

characteristic

110

C₂×C₄

#14523248

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

508

—

114

C₂²

—

115

C₂²

#16861184

yes

characteristic

116

C₂²

#16861194

yes

—

117

C₄

—

508

—

118

C₄

—

508

—

119

C₄

#16861179

yes

characteristic

120

C₄

#16861182

yes

—

121

C₄

#16861197

yes

characteristic

122

C₂

—

123

C₂

#22549794

yes

characteristic

124

C₂

#22550152

yes

—

	subgroup	G/H
center	C₂²	#16861184
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#341×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂³×C₁₂₇	#41
hypercenter	#341	D₁₂₇
hypocenter	C₁₂₇	#2047
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#16861184

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#341×C₁₂₇

C₂

C₂²×C₁₂₇

C₂³×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
63. C₁₂₇	#2047	semidirect

H ≅ G/N

type

63. C₁₂₇

#2047

semidirect

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

subgroup	index
C₂²×Dic₅₀₈	2
#22549902	2
#22549886	2
C₂×#16861127	2
#341×C₁₂₇	2
#2047	127

subgroup

index

C₂²×Dic₅₀₈

#341×C₁₂₇

127

Subgroups — #30639999

Conjugacy classes of subgroups

Complements

Maximal subgroups