Group #30639703 — subgroups

Subgroups — #30639703

Order 16256.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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	#22549742	C₂	1	1	yes	maximal, characteristic
3	#22549639	C₂	1	4	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	1	no	—
6	#16861209	—	2	2	no	—
7	#16861209	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₁₂₇⋊#95	—	2	1	no	—
12	C₁₂₇⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	#97×C₁₂₇	—	2	2	no	—
16	C₂×Dic₂₅₄	—	2	4	no	—
17	C₂×Dic₂₅₄	—	4	4	no	—
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	C₂²×Dic₁₂₇	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	4	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	2	4	no	—
23	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	Q₈	1	4	yes	complemented
30	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₄²×C₁₂₇	D₄	1	2	yes	—
32	Dic₂₅₄	—	4	8	no	—
33	C₂×Dic₁₂₇	—	2	4	no	—
34	C₂×Dic₁₂₇	—	4	4	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	#41	1	2	yes	—
47	C₈×C₁₂₇	—	2	4	no	—
48	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₂₇	#124	1	1	yes	characteristic
50	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₂₇	#137	1	2	yes	—
53	C₄×C₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	—	2	2	no	—
55	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
56	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
57	Dic₁₂₇	—	4	4	no	—
58	Dic₁₂₇	—	8	2	no	—
59	C₂×C₁₂₇	#391	1	1	yes	characteristic
60	C₂×C₁₂₇	#418	1	1	yes	characteristic
61	C₂×C₁₂₇	#419	1	1	yes	characteristic
62	C₂×C₁₂₇	#489	1	2	yes	—
63	C₂×C₁₂₇	#501	1	2	yes	—
64	#1993	—	127	1	no	maximal, Sylow
65	C₁₂₇	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	127	1	no	—
67	C₂×#97	D₁₂₇	1	1	yes	characteristic
68	#382	—	127	1	no	—
69	#340	—	127	4	no	—
70	C₂²×Q₈	—	254	1	no	—
71	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
72	C₄⋊Q₈	—	127	4	no	—
73	C₄⋊Q₈	—	254	2	no	—
74	C₂×C₄⋊C₄	—	127	2	no	—
75	C₂×C₄⋊C₄	—	127	2	no	—
76	C₂×C₄⋊C₄	—	254	1	no	—
77	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	254	1	no	—
80	#95	—	254	1	no	—
81	C₂×Q₈	—	508	4	no	—
82	C₂×Q₈	—	254	4	no	—
83	C₂²×C₄	—	254	2	no	—
84	C₂²×C₄	—	254	2	no	—
85	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
86	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₂₅₄	1	4	yes	complemented
91	C₄⋊C₄	—	127	4	no	—
92	C₄⋊C₄	—	254	4	no	—
93	C₄⋊C₄	—	254	4	no	—
94	C₄⋊C₄	—	254	2	no	—
95	C₄²	C₄×D₁₂₇	1	2	yes	—
96	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
97	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
98	Q₈	—	508	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	508	2	no	—
102	C₂×C₄	—	508	2	no	—
103	C₂×C₄	—	508	2	no	—
104	C₂×C₄	—	508	4	no	—
105	C₂×C₄	—	254	4	no	—
106	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
111	C₂×C₄	#14523248	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
114	C₂²	#16861197	1	1	yes	characteristic
115	C₂²	#16861209	1	1	yes	characteristic
116	C₂²	#16861237	1	2	yes	—
117	C₂²	#16861242	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	1016	2	no	—
121	C₄	—	508	4	no	—
122	C₄	C₄×Dic₂₅₄	1	2	yes	—
123	C₄	#16861188	1	2	yes	—
124	C₂	#22549782	1	1	yes	characteristic
125	C₂	#22549882	1	1	yes	characteristic
126	C₂	#22549886	1	1	yes	characteristic
127	C₂	#22550236	1	2	yes	—
128	C₂	#22550285	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

#22549742

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

C₂×#97×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

#16861209

—

#16861209

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₁₂₇⋊#95

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

Q₈

yes

complemented

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

#124

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×Q₈

yes

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#501

yes

—

#1993

—

127

maximal, Sylow

C₁₂₇

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×#97

D₁₂₇

yes

characteristic

#382

—

127

—

#340

—

127

—

C₂²×Q₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#97

—

#95

—

254

—

#95

—

254

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₅₄

yes

complemented

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

508

—

103

C₂×C₄

—

508

—

104

C₂×C₄

—

508

—

105

C₂×C₄

—

254

—

106

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

110

C₂×C₄

C₂×Dic₂₅₄

yes

—

111

C₂×C₄

#14523248

yes

—

112

C₈

—

113

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

114

C₂²

#16861197

yes

characteristic

115

C₂²

#16861209

yes

characteristic

116

C₂²

#16861237

yes

—

117

C₂²

#16861242

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

1016

—

121

C₄

—

508

—

122

C₄

C₄×Dic₂₅₄

yes

—

123

C₄

#16861188

yes

—

124

C₂

#22549782

yes

characteristic

125

C₂

#22549882

yes

characteristic

126

C₂

#22549886

yes

characteristic

127

C₂

#22550236

yes

—

128

C₂

#22550285

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#1993
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#97×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×#97

D₁₂₇

hypocenter

C₁₂₇

#1993

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #16861209	C₄	semidirect
29. C₂×C₈×C₁₂₇	Q₈	semidirect
65. C₁₂₇	#1993	semidirect
90. C₂×C₈	Dic₂₅₄	semidirect

H ≅ G/N

type

7. #16861209

C₄

semidirect

29. C₂×C₈×C₁₂₇

Q₈

semidirect

65. C₁₂₇

#1993

semidirect

90. C₂×C₈

Dic₂₅₄

semidirect

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

subgroup	index
C₂×#16861209	2
#22549742	2
#22549639	2
C₂×#97×C₁₂₇	2
#1993	127

subgroup

index

C₂×#97×C₁₂₇

127

Subgroups — #30639703

Conjugacy classes of subgroups

Complements

Maximal subgroups