Group #30639476 — subgroups

Subgroups — #30639476

Order 16256.

208 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22549914	C₂	1	1	yes	maximal, characteristic
2	#22549904	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549637	C₂	1	2	yes	maximal
5	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	#16861168	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	#16861143	C₄	1	2	yes	complemented
10	#16861141	C₄	1	2	yes	complemented
11	Dic₁₂₇⋊C₈	—	2	4	no	—
12	C₈×Dic₁₂₇	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#108×C₁₂₇	—	2	2	no	—
16	C₂²×Dic₁₂₇	—	2	1	no	—
17	C₂×C₁₂₇⋊C₈	—	2	1	no	—
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
20	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₁₂₇	—	2	2	no	—
22	C₄×Dic₁₂₇	—	2	1	no	—
23	C₄×Dic₁₂₇	—	2	2	no	—
24	C₄×Dic₁₂₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₂₇	—	2	2	no	—
27	M₁₆×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
33	C₂×Dic₁₂₇	—	2	4	no	—
34	C₂×Dic₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	2	4	no	—
36	C₁₂₇⋊C₈	—	2	2	no	—
37	C₂³×C₁₂₇	#41	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	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
44	C₈×C₁₂₇	—	4	2	no	—
45	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₂₇	#125	1	1	yes	characteristic
51	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₁₂₇	#121	1	1	yes	characteristic
54	C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₁₂₇	—	2	4	no	—
56	Dic₁₂₇	—	4	4	no	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	—	2	1	no	—
59	C₂×C₁₂₇	#389	1	1	yes	characteristic
60	C₂×C₁₂₇	#443	1	2	yes	—
61	#1945	—	127	1	no	maximal, Sylow
62	C₁₂₇	#1945	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	D₁₂₇	1	1	yes	characteristic
64	#427	—	127	1	no	—
65	#424	—	127	2	no	—
66	C₂×C₄×C₈	—	127	1	no	—
67	#339	—	127	2	no	—
68	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
69	C₂²×C₈	—	127	1	no	—
70	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
71	#117	—	127	2	no	—
72	C₂×C₄²	—	127	1	no	—
73	#108	—	2	2	no	—
74	#107	—	127	2	no	—
75	#106	—	127	2	no	—
76	C₄⋊C₈	—	254	4	no	—
77	C₄×C₈	—	254	2	no	—
78	C₂²×C₄	—	254	1	no	—
79	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	127	2	no	—
84	C₂×C₈	—	254	1	no	—
85	C₂×C₈	—	254	1	no	—
86	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
87	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
89	C₂×C₈	Dic₂₅₄	1	2	yes	—
90	C₄⋊C₄	—	127	4	no	—
91	C₂²⋊C₄	—	254	2	no	—
92	C₄²	—	127	2	no	—
93	C₄²	—	254	2	no	—
94	C₄²	—	254	1	no	—
95	C₂³	#14523249	1	1	yes	characteristic
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	—	254	4	no	—
98	C₂×C₄	—	254	2	no	—
99	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
104	C₂×C₄	#14523247	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	254	2	no	—
107	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
108	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#16861171	1	1	yes	characteristic
111	C₂²	#16861185	1	1	yes	characteristic
112	C₂²	#16861200	1	1	yes	characteristic
113	C₄	—	508	4	no	—
114	C₄	—	254	4	no	—
115	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
116	C₄	#16861174	1	1	yes	characteristic
117	C₄	#16861180	1	1	yes	characteristic
118	C₄	#16861208	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#22549772	1	1	yes	characteristic
122	C₂	#22549992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549914

C₂

yes

maximal, characteristic

#22549904

C₂

yes

maximal

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

#22549637

C₂

yes

maximal

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861143

C₄

yes

complemented

#16861141

C₄

yes

complemented

Dic₁₂₇⋊C₈

—

C₈×Dic₁₂₇

—

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

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

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#125

yes

characteristic

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

characteristic

C₄×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

C₄⋊Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#389

yes

characteristic

C₂×C₁₂₇

#443

yes

—

#1945

—

127

maximal, Sylow

C₁₂₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

D₁₂₇

yes

characteristic

#427

—

127

—

#424

—

127

—

C₂×C₄×C₈

—

127

—

#339

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

C₂×C₄²

—

127

—

#108

—

#107

—

127

—

#106

—

127

—

C₄⋊C₈

—

254

—

C₄×C₈

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₂×C₈

Dic₂₅₄

yes

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

—

254

—

C₂³

#14523249

yes

characteristic

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

100

C₂×C₄

D₄×D₁₂₇

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

103

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

104

C₂×C₄

#14523247

yes

characteristic

105

C₈

—

106

C₈

—

254

—

107

C₈

Dic₁₂₇⋊C₄

yes

—

108

C₈

Dic₁₂₇⋊C₄

yes

—

109

C₂²

—

110

C₂²

#16861171

yes

characteristic

111

C₂²

#16861185

yes

characteristic

112

C₂²

#16861200

yes

characteristic

113

C₄

—

508

—

114

C₄

—

254

—

115

C₄

C₂×Dic₁₂₇⋊C₄

yes

characteristic

116

C₄

#16861174

yes

characteristic

117

C₄

#16861180

yes

characteristic

118

C₄

#16861208

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#22549772

yes

characteristic

122

C₂

#22549992

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄×D₄
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#1945
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#108×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₄×D₄

prime radical

C₂³×C₁₂₇

#41

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #16861143	C₄	semidirect
10. #16861141	C₄	semidirect
62. C₁₂₇	#1945	semidirect

H ≅ G/N

type

9. #16861143

C₄

semidirect

10. #16861141

C₄

semidirect

62. C₁₂₇

#1945

semidirect

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

subgroup	index
#22549914	2
#22549904	2
C₂×C₈×Dic₁₂₇	2
#22549637	2
C₂×#108×C₁₂₇	2
#1945	127

subgroup

index

C₂×C₈×Dic₁₂₇

C₂×#108×C₁₂₇

127

Subgroups — #30639476

Conjugacy classes of subgroups

Complements

Maximal subgroups