Group #30637519 — subgroups

Subgroups — #30637519

Order 16256.

158 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22549929	C₂	1	1	yes	maximal, characteristic
2	#22549900	C₂	1	1	yes	maximal, characteristic
3	#22549629	C₂	1	2	yes	maximal, complemented
4	#22549627	C₂	1	2	yes	maximal, complemented
5	#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	—	2	1	no	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₁₂₇⋊C₈)	—	2	1	no	—
12	C₄×C₁₂₇⋊C₈	—	2	1	no	—
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₂₇	C₄	1	2	yes	—
15	C₂²⋊C₈×C₁₂₇	C₄	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
17	#14523232	—	4	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	—	2	1	no	—
20	C₂×C₁₂₇⋊C₈	—	2	1	no	—
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
27	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
28	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₁₂₇⋊C₈	—	4	1	no	—
30	C₁₂₇⋊C₈	—	4	2	no	—
31	C₁₂₇⋊C₈	—	4	1	no	—
32	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	—	2	2	no	—
34	C₂×C₄×C₁₂₇	—	2	2	no	—
35	C₂×C₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₂₇	—	2	4	no	—
44	C₂²×C₁₂₇	—	2	1	no	—
45	C₂²×C₁₂₇	—	2	1	no	—
46	C₂²×C₁₂₇	—	2	1	no	—
47	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₂₇	—	2	1	no	—
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	—	2	1	no	—
51	C₄×C₁₂₇	#108	1	2	yes	—
52	C₄×C₁₂₇	#131	1	2	yes	—
53	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₁₂₇	#137	1	1	yes	characteristic
55	C₂×C₁₂₇	—	4	1	no	—
56	C₂×C₁₂₇	#424	1	1	yes	characteristic
57	C₂×C₁₂₇	#428	1	1	yes	characteristic
58	C₂×C₁₂₇	C₂×#108	1	1	yes	characteristic
59	#1574	—	127	1	no	maximal, Sylow
60	C₁₂₇	#1574	1	1	yes	characteristic, complemented, Sylow
61	#431	—	127	1	no	—
62	#431	D₁₂₇	1	1	yes	characteristic
63	#423	—	127	1	no	—
64	#335	—	127	2	no	—
65	#334	—	127	2	no	—
66	C₂×M₁₆	—	254	1	no	—
67	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₈	—	127	1	no	—
69	C₄⋊C₈	—	127	2	no	—
70	C₄⋊C₈	—	127	1	no	—
71	C₄⋊C₈	—	254	1	no	—
72	C₄⋊C₈	Dic₁₂₇	1	2	yes	—
73	C₂²⋊C₈	—	254	1	no	—
74	C₂²⋊C₈	Dic₁₂₇	1	2	yes	—
75	C₄×C₈	—	254	1	no	—
76	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
77	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
78	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
79	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
80	M₁₆	—	508	2	no	—
81	C₂×C₈	—	254	1	no	—
82	C₂×C₈	—	254	1	no	—
83	C₂×C₈	—	254	2	no	—
84	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
85	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
86	C₄²	D₅₀₈	1	1	yes	characteristic
87	C₄²	Dic₂₅₄	1	1	yes	characteristic
88	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₁₂₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	508	2	no	—
102	C₈	—	508	1	no	—
103	C₈	—	508	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#16861145	1	2	yes	—
112	C₄	#16861216	1	2	yes	—
113	C₄	#16861237	1	1	yes	characteristic
114	C₄	#16861242	1	1	yes	characteristic
115	C₂	—	4	1	no	—
116	C₂	#22549903	1	1	yes	characteristic
117	C₂	#22549918	1	1	yes	characteristic
118	C₂	C₂×#16861145	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549929

C₂

yes

maximal, characteristic

#22549900

C₂

yes

maximal, characteristic

#22549629

C₂

yes

maximal, complemented

#22549627

C₂

yes

maximal, complemented

#431×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₁₂₇⋊C₈)

—

C₄×C₁₂₇⋊C₈

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₄

yes

—

C₂²⋊C₈×C₁₂₇

C₄

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

Q₈

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#108

yes

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1574

yes

characteristic, complemented, Sylow

#431

—

127

—

#431

D₁₂₇

yes

characteristic

#423

—

127

—

#335

—

127

—

#334

—

127

—

C₂×M₁₆

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

Dic₁₂₇

yes

—

C₂²⋊C₈

—

254

—

C₂²⋊C₈

Dic₁₂₇

yes

—

C₄×C₈

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

M₁₆

—

508

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₄²

D₅₀₈

yes

characteristic

C₄²

Dic₂₅₄

yes

characteristic

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

100

C₈

—

101

C₈

—

508

—

102

C₈

—

508

—

103

C₈

—

508

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#16861145

yes

—

112

C₄

#16861216

yes

—

113

C₄

#16861237

yes

characteristic

114

C₄

#16861242

yes

characteristic

115

C₂

—

116

C₂

#22549903

yes

characteristic

117

C₂

#22549918

yes

characteristic

118

C₂

C₂×#16861145

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#431×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₄⋊C₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#431	D₁₂₇
hypocenter	C₁₂₇	#1574
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#431×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₂₇

C₄⋊C₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #22549629	C₂	semidirect
4. #22549627	C₂	semidirect
60. C₁₂₇	#1574	semidirect

H ≅ G/N

type

3. #22549629

C₂

semidirect

4. #22549627

C₂

semidirect

60. C₁₂₇

#1574

semidirect

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

subgroup	index
#22549929	2
#22549900	2
#22549629	2
#22549627	2
#431×C₁₂₇	2
#1574	127

subgroup

index

#431×C₁₂₇

127

Subgroups — #30637519

Conjugacy classes of subgroups

Complements

Maximal subgroups