Group #30640284 — subgroups

Subgroups — #30640284

Order 16256.

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

normal subgroups only

This group has 210 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	#22549925	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549903	C₂	1	2	yes	maximal, complemented
3	#22549894	C₂	1	2	yes	maximal
4	#22549844	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	C₂×#14523232	C₂²	1	2	yes	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16861145	—	2	2	no	—
10	C₄⋊(C₁₂₇⋊C₈)	—	2	4	no	—
11	#16861137	—	2	4	no	—
12	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	#117×C₁₂₇	C₂²	1	2	yes	—
15	#14523248	—	2	2	no	—
16	#14523248	—	2	2	no	—
17	C₂×Dic₂₅₄	Q₈	1	1	yes	characteristic
18	C₂×D₅₀₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
20	C₂×C₄×D₁₂₇	—	2	1	no	—
21	#14523232	—	2	2	no	—
22	#14523232	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	1	no	—
24	C₂×C₁₂₇⋊C₈	—	2	2	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄²×C₁₂₇	D₄	1	4	yes	—
32	C₂²×D₁₂₇	—	2	1	no	—
33	Dic₂₅₄	—	2	2	no	—
34	D₅₀₈	—	2	2	no	—
35	C₂²⋊D₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	2	1	no	—
37	C₄×D₁₂₇	—	4	1	no	—
38	C₄×D₁₂₇	—	4	1	no	—
39	C₁₂₇⋊C₈	—	4	1	no	—
40	C₁₂₇⋊C₈	—	4	2	no	—
41	C₂³×C₁₂₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₈×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	—	4	1	no	—
49	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₂₇	#123	1	2	yes	—
51	C₂×D₁₂₇	—	4	2	no	—
52	C₂×D₁₂₇	—	4	1	no	—
53	C₄×C₁₂₇	—	4	4	no	—
54	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₂₇	#123	1	1	yes	characteristic
56	C₄×C₁₂₇	#127	1	1	yes	characteristic
57	Dic₁₂₇	—	4	1	no	—
58	C₂×C₁₂₇	—	2	2	no	—
59	C₂×C₁₂₇	—	2	1	no	—
60	C₂×C₁₂₇	#399	1	1	yes	characteristic
61	D₁₂₇	—	8	1	no	—
62	#2098	—	127	1	no	maximal, Sylow
63	C₁₂₇	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	127	1	no	—
65	#424	—	127	2	no	—
66	#421	—	127	2	no	—
67	#408	—	127	1	no	—
68	H(3,ℤ/4)	D₁₂₇	1	1	yes	characteristic, complemented
69	C₂×#41	—	127	1	no	—
70	C₂×M₁₆	—	127	2	no	—
71	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂²×C₈	—	127	1	no	—
73	#117	C₂×D₁₂₇	1	2	yes	—
74	#108	—	254	2	no	—
75	C₄⋊C₈	—	254	4	no	—
76	#104	—	254	4	no	—
77	C₂²⋊C₈	—	254	2	no	—
78	#41	—	254	2	no	—
79	#41	—	254	2	no	—
80	C₂×Q₈	—	127	1	no	—
81	C₂×D₄	—	127	1	no	—
82	C₂×D₄	—	254	1	no	—
83	C₂²×C₄	—	254	1	no	—
84	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	254	2	no	—
87	M₁₆	—	254	2	no	—
88	C₂×C₈	—	254	1	no	—
89	C₂×C₈	—	254	2	no	—
90	C₂×C₈	—	254	2	no	—
91	C₂×C₈	D₅₀₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₅₀₈	1	4	yes	—
95	C₂³	—	254	1	no	—
96	C₂³	#14523247	1	1	yes	characteristic
97	Q₈	—	254	2	no	—
98	D₄	—	508	2	no	—
99	D₄	—	254	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	508	1	no	—
102	C₂×C₄	—	508	1	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
105	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	#14523247	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	508	1	no	—
110	C₈	—	508	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	508	2	no	—
113	C₂²	—	508	1	no	—
114	C₂²	#16861189	1	1	yes	characteristic
115	C₂²	#16861193	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	508	1	no	—
118	C₄	#16861189	1	2	yes	—
119	C₄	#16861193	1	1	yes	characteristic
120	C₄	#16861206	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1016	1	no	—
124	C₂	#22549813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549925

C₂

yes

maximal, characteristic, complemented

#22549903

C₂

yes

maximal, complemented

#22549894

C₂

yes

maximal

#22549844

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861145

—

C₄⋊(C₁₂₇⋊C₈)

—

#16861137

—

(C₂×D₁₂₇)⋊C₈

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

#14523248

—

#14523248

—

C₂×Dic₂₅₄

Q₈

yes

characteristic

C₂×D₅₀₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#123

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#399

yes

characteristic

D₁₂₇

—

#2098

—

127

maximal, Sylow

C₁₂₇

#2098

yes

characteristic, complemented, Sylow

#430

—

127

—

#424

—

127

—

#421

—

127

—

#408

—

127

—

H(3,ℤ/4)

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

#117

C₂×D₁₂₇

yes

—

#108

—

254

—

C₄⋊C₈

—

254

—

#104

—

254

—

C₂²⋊C₈

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

254

—

M₁₆

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₅₀₈

yes

—

C₂³

—

254

—

C₂³

#14523247

yes

characteristic

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

100

C₂×C₄

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

508

—

103

C₂×C₄

—

254

—

104

C₂×C₄

C₂×D₅₀₈

yes

—

105

C₂×C₄

C₂×D₅₀₈

yes

characteristic

106

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

107

C₂×C₄

#14523247

yes

—

108

C₈

—

109

C₈

—

508

—

110

C₈

—

508

—

111

C₂²

—

112

C₂²

—

508

—

113

C₂²

—

508

—

114

C₂²

#16861189

yes

characteristic

115

C₂²

#16861193

yes

—

116

C₄

—

117

C₄

—

508

—

118

C₄

#16861189

yes

—

119

C₄

#16861193

yes

characteristic

120

C₄

#16861206

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1016

—

124

C₂

#22549813

yes

characteristic

	subgroup	G/H
center	C₄	#16861206
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	H(3,ℤ/4)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#399
prime radical	C₂×#14523248	C₂²
hypercenter	H(3,ℤ/4)	D₁₂₇
hypocenter	C₁₂₇	#2098
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16861206

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

H(3,ℤ/4)×C₁₂₇

C₂

C₂×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #22549925	C₂	semidirect
2. #22549903	C₂	semidirect
5. H(3,ℤ/4)×C₁₂₇	C₂	semidirect
63. C₁₂₇	#2098	semidirect
68. H(3,ℤ/4)	D₁₂₇	semidirect

H ≅ G/N

type

1. #22549925

C₂

semidirect

2. #22549903

C₂

semidirect

5. H(3,ℤ/4)×C₁₂₇

C₂

semidirect

63. C₁₂₇

#2098

semidirect

68. H(3,ℤ/4)

D₁₂₇

semidirect

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

subgroup	index
#22549925	2
#22549903	2
#22549894	2
#22549844	2
H(3,ℤ/4)×C₁₂₇	2
#2098	127

subgroup

index

H(3,ℤ/4)×C₁₂₇

127

Subgroups — #30640284

Conjugacy classes of subgroups

Complements

Maximal subgroups