Group #30639532 — subgroups

Subgroups — #30639532

Order 16256.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#16861209	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861134	C₂	1	4	yes	maximal
3	#22549746	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
6	#16861209	—	2	2	no	—
7	#16861209	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	#16861134	—	2	8	no	—
12	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	—	2	2	no	—
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂×Dic₂₅₄	—	2	4	no	—
17	C₂×Dic₂₅₄	—	2	4	no	—
18	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₂₇	—	2	2	no	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	8	no	—
22	C₄⋊Dic₁₂₇	—	2	4	no	—
23	C₄⋊Dic₁₂₇	—	4	4	no	—
24	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	4	yes	—
29	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₂₇	D₄	1	2	yes	—
31	Dic₂₅₄	—	2	8	no	—
32	Dic₂₅₄	—	4	4	no	—
33	C₂×Dic₁₂₇	—	2	4	no	—
34	C₂×Dic₁₂₇	—	4	2	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	4	no	—
38	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₂₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₁₂₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	#41	1	2	yes	—
46	C₈×C₁₂₇	—	2	4	no	—
47	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₂₇	#124	1	1	yes	characteristic
49	C₂²×C₁₂₇	#127	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₂₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₂₇	#103	1	8	yes	complemented
53	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
54	C₄×C₁₂₇	#121	1	2	yes	—
55	Dic₁₂₇	—	4	4	no	—
56	Dic₁₂₇	—	8	2	no	—
57	C₂×C₁₂₇	#390	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂×#103	1	2	yes	—
59	C₂×C₁₂₇	#487	1	2	yes	—
60	C₂×C₁₂₇	#492	1	1	yes	characteristic
61	C₂×C₁₂₇	#494	1	1	yes	characteristic
62	#1956	—	127	1	no	maximal, Sylow
63	C₁₂₇	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	127	1	no	—
65	C₂×#103	—	127	4	no	—
66	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
67	#384	—	127	1	no	—
68	C₂²×Q₈	—	127	2	no	—
69	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
70	C₄⋊Q₈	—	127	4	no	—
71	C₄⋊Q₈	—	254	2	no	—
72	C₂×C₄⋊C₄	—	127	2	no	—
73	C₂×C₄⋊C₄	—	254	2	no	—
74	C₂×C₄⋊C₄	—	254	2	no	—
75	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
76	#103	—	254	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	127	4	no	—
79	C₂×Q₈	—	254	4	no	—
80	C₂×Q₈	—	254	4	no	—
81	C₂×Q₈	—	254	2	no	—
82	C₂²×C₄	—	254	2	no	—
83	C₂²×C₄	—	254	2	no	—
84	C₂²×C₄	D₅₀₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₅₀₈	1	4	yes	—
89	C₄⋊C₄	—	508	4	no	—
90	C₄⋊C₄	—	254	4	no	—
91	C₄⋊C₄	—	254	8	no	—
92	C₄²	C₄×D₁₂₇	1	2	yes	—
93	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
94	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
95	Q₈	—	508	4	no	—
96	Q₈	—	254	8	no	—
97	C₂×C₄	—	508	2	no	—
98	C₂×C₄	—	508	4	no	—
99	C₂×C₄	—	508	2	no	—
100	C₂×C₄	—	508	2	no	—
101	C₂×C₄	—	254	4	no	—
102	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
103	C₂×C₄	#14523237	1	4	yes	—
104	C₂×C₄	Dic₅₀₈	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
108	C₂×C₄	#14523248	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
111	C₂²	#16861197	1	1	yes	characteristic
112	C₂²	#16861206	1	1	yes	characteristic
113	C₂²	C₂×#14523237	1	2	yes	—
114	C₂²	C₂×Dic₅₀₈	1	2	yes	—
115	C₄	—	1016	2	no	—
116	C₄	—	508	4	no	—
117	C₄	#16861134	1	8	yes	complemented
118	C₄	C₄×D₅₀₈	1	2	yes	—
119	C₄	#16861183	1	2	yes	—
120	C₂	#22549778	1	1	yes	characteristic
121	C₂	C₂×#16861134	1	2	yes	—
122	C₂	#22550227	1	2	yes	—
123	C₂	#22550249	1	1	yes	characteristic
124	C₂	#22550256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₂×#16861134

C₂

yes

maximal

#22549746

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

#16861209

—

#16861209

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861134

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×SD₁₆

yes

—

C₂²×C₁₂₇

C₂×Q₁₆

yes

—

C₄×C₁₂₇

#103

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

#121

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

C₂×#103

yes

—

C₂×C₁₂₇

#487

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1956

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×#103

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

#384

—

127

—

C₂²×Q₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

C₄×C₈

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

100

C₂×C₄

—

508

—

101

C₂×C₄

—

254

—

102

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

103

C₂×C₄

#14523237

yes

—

104

C₂×C₄

Dic₅₀₈

yes

—

105

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

107

C₂×C₄

C₂×D₅₀₈

yes

—

108

C₂×C₄

#14523248

yes

—

109

C₈

—

110

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

111

C₂²

#16861197

yes

characteristic

112

C₂²

#16861206

yes

characteristic

113

C₂²

C₂×#14523237

yes

—

114

C₂²

C₂×Dic₅₀₈

yes

—

115

C₄

—

1016

—

116

C₄

—

508

—

117

C₄

#16861134

yes

complemented

118

C₄

C₄×D₅₀₈

yes

—

119

C₄

#16861183

yes

—

120

C₂

#22549778

yes

characteristic

121

C₂

C₂×#16861134

yes

—

122

C₂

#22550227

yes

—

123

C₂

#22550249

yes

characteristic

124

C₂

#22550256

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1956
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

#1956

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #16861209	C₄	semidirect
52. C₄×C₁₂₇	#103	semidirect
63. C₁₂₇	#1956	semidirect
117. C₄	#16861134	semidirect

H ≅ G/N

type

7. #16861209

C₄

semidirect

52. C₄×C₁₂₇

#103

semidirect

63. C₁₂₇

#1956

semidirect

117. C₄

#16861134

semidirect

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

subgroup	index
C₂×#16861209	2
C₂×#16861134	2
#22549746	2
C₂×C₄×C₈×C₁₂₇	2
#1956	127

subgroup

index

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30639532

Conjugacy classes of subgroups

Complements

Maximal subgroups