Group #30639521 — subgroups

Subgroups — #30639521

Order 16256.

184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22549925	C₂	1	2	yes	maximal
2	C₂×#16861145	C₂	1	1	yes	maximal, characteristic
3	#22549831	C₂	1	1	yes	maximal, characteristic
4	#22549648	C₂	1	2	yes	maximal
5	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	2	yes	—
7	C₂×#14523232	C₂²	1	1	yes	characteristic
8	#16861168	C₂²	1	1	yes	characteristic
9	#16861145	—	2	4	no	—
10	#16861145	—	2	1	no	—
11	#16861145	—	2	1	no	—
12	#16861119	—	2	1	no	—
13	C₈×Dic₁₂₇	—	2	1	no	—
14	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	#108×C₁₂₇	C₄	1	4	yes	complemented
17	#14523232	—	2	2	no	—
18	#14523232	—	2	2	no	—
19	#14523232	—	2	1	no	—
20	#14523232	D₄	1	2	yes	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	1	no	—
24	C₂²⋊Dic₁₂₇	Q₈	1	2	yes	—
25	C₄×Dic₁₂₇	—	2	1	no	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	2	no	—
28	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₁₂₇⋊C₈	—	2	2	no	—
36	C₁₂₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
44	C₈×C₁₂₇	—	2	4	no	—
45	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₂₇	—	4	1	no	—
48	C₂²×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₂₇	#121	1	1	yes	characteristic
54	C₄×C₁₂₇	#125	1	1	yes	characteristic
55	Dic₁₂₇	—	4	2	no	—
56	C₂×C₁₂₇	—	2	1	no	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	—	2	1	no	—
59	C₂×C₁₂₇	#389	1	1	yes	characteristic
60	#1953	—	127	1	no	maximal, Sylow
61	C₁₂₇	#1953	1	1	yes	characteristic, complemented, Sylow
62	#430	—	127	2	no	—
63	C₂×#108	—	127	1	no	—
64	C₂×#108	D₁₂₇	1	1	yes	characteristic
65	#405	—	127	1	no	—
66	#344	—	127	2	no	—
67	C₂×M₁₆	—	127	1	no	—
68	C₂×M₁₆	—	127	2	no	—
69	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
70	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
71	#117	—	127	1	no	—
72	#108	—	254	4	no	—
73	#108	—	254	1	no	—
74	#108	—	254	1	no	—
75	#108	Dic₁₂₇	1	4	yes	complemented
76	#97	—	254	1	no	—
77	C₄×C₈	—	254	1	no	—
78	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	127	2	no	—
81	M₁₆	—	254	2	no	—
82	M₁₆	—	254	2	no	—
83	M₁₆	—	254	1	no	—
84	M₁₆	C₂×Dic₁₂₇	1	4	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	254	2	no	—
87	C₂×C₈	—	254	1	no	—
88	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
89	C₂×C₈	D₅₀₈	1	1	yes	characteristic
90	C₂×C₈	D₅₀₈	1	1	yes	characteristic
91	C₂×C₈	Dic₂₅₄	1	2	yes	—
92	C₄⋊C₄	—	254	1	no	—
93	C₂²⋊C₄	—	127	2	no	—
94	C₄²	—	254	1	no	—
95	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
96	C₂×C₄	—	254	2	no	—
97	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
99	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
101	C₂×C₄	#14523247	1	1	yes	characteristic
102	C₂×C₄	#14523249	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	508	2	no	—
105	C₈	—	254	2	no	—
106	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
107	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	Q₈×Dic₁₂₇	1	1	yes	characteristic
110	C₂²	#16861189	1	1	yes	characteristic
111	C₂²	#16861208	1	1	yes	characteristic
112	C₄	—	508	2	no	—
113	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
114	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
115	C₄	#16861184	1	1	yes	characteristic
116	C₄	#16861200	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#22549775	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549925

C₂

yes

maximal

C₂×#16861145

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal, characteristic

#22549648

C₂

yes

maximal

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

—

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

—

—

—

—

C₈×Dic₁₂₇

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

C₄

yes

—

—

—

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

Q₈

yes

—

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

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×Q₈

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

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#389

yes

characteristic

#1953

—

127

maximal, Sylow

C₁₂₇

#1953

yes

characteristic, complemented, Sylow

#430

—

127

—

C₂×#108

—

127

—

C₂×#108

D₁₂₇

yes

characteristic

#405

—

127

—

#344

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

#108

—

254

—

#108

—

254

—

#108

—

254

—

#108

Dic₁₂₇

yes

complemented

#97

—

254

—

C₄×C₈

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

—

254

—

M₁₆

—

254

—

M₁₆

C₂×Dic₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

127

—

C₄²

—

254

—

C₂³

Q₈×D₁₂₇

yes

characteristic

C₂×C₄

—

254

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

D₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

100

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

101

C₂×C₄

#14523247

yes

characteristic

102

C₂×C₄

#14523249

yes

characteristic

103

C₈

—

104

C₈

—

508

—

105

C₈

—

254

—

106

C₈

C₄⋊Dic₁₂₇

yes

—

107

C₈

C₄⋊Dic₁₂₇

yes

—

108

C₂²

—

109

C₂²

Q₈×Dic₁₂₇

yes

characteristic

110

C₂²

#16861189

yes

characteristic

111

C₂²

#16861208

yes

characteristic

112

C₄

—

508

—

113

C₄

C₂×C₄⋊Dic₁₂₇

yes

characteristic

114

C₄

D₄×Dic₁₂₇

yes

characteristic

115

C₄

#16861184

yes

characteristic

116

C₄

#16861200

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#22549775

yes

characteristic

	subgroup	G/H
center	C₄	#16861184
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂×C₁₂₇	#389
prime radical	C₂³×C₁₂₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#1953
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#16861184

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#108×C₁₂₇

C₂

socle

C₂×C₁₂₇

#389

prime radical

C₂³×C₁₂₇

C₂×Q₈

cosocle

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
16. #108×C₁₂₇	C₄	semidirect
61. C₁₂₇	#1953	semidirect
75. #108	Dic₁₂₇	semidirect

H ≅ G/N

type

16. #108×C₁₂₇

C₄

semidirect

61. C₁₂₇

#1953

semidirect

75. #108

Dic₁₂₇

semidirect

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

subgroup	index
#22549925	2
C₂×#16861145	2
#22549831	2
#22549648	2
C₂×#108×C₁₂₇	2
#1953	127

subgroup

index

C₂×#108×C₁₂₇

127

Subgroups — #30639521

Conjugacy classes of subgroups

Complements

Maximal subgroups