Group #30650534 — subgroups

Subgroups — #30650534

Order 16256.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22550689	C₂	1	4	yes	maximal, complemented
2	#22550148	C₂	1	4	yes	maximal, complemented
3	#22550058	C₂	1	4	yes	maximal, complemented
4	#22549994	C₂	1	2	yes	maximal, complemented
5	#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861257	C₂²	1	4	yes	—
7	#16861241	—	2	8	no	—
8	#16861240	C₂²	1	8	yes	—
9	#16861232	—	2	4	no	—
10	#16861231	C₂²	1	2	yes	—
11	Q₁₆×D₁₂₇	—	2	4	no	—
12	#16861215	C₂²	1	4	yes	—
13	#16861145	C₂²	1	2	yes	—
14	#16861137	C₂²	1	4	yes	—
15	#16861128	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₂₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#14523249	—	2	8	no	—
20	#14523248	—	4	4	no	—
21	#14523248	C₂³	1	4	yes	—
22	Q₈×D₁₂₇	—	2	8	no	—
23	Dic₅₀₈	D₄	1	2	yes	—
24	#14523238	—	2	8	no	—
25	#14523237	D₄	1	4	yes	—
26	#14523235	—	2	8	no	—
27	D₁₀₁₆	D₄	1	2	yes	—
28	#14523232	C₂³	1	4	yes	—
29	#14523231	—	2	4	no	—
30	C₈×D₁₂₇	—	2	4	no	—
31	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
32	Q₁₆×C₁₂₇	—	2	4	no	—
33	Q₁₆×C₁₂₇	—	2	4	no	—
34	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₂₇	—	4	2	no	—
36	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
37	Dic₂₅₄	—	4	4	no	—
38	Dic₂₅₄	C₂×D₄	1	4	yes	—
39	D₅₀₈	—	4	4	no	—
40	D₅₀₈	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₂₇	—	2	4	no	—
42	C₄×D₁₂₇	—	2	4	no	—
43	C₄×D₁₂₇	—	4	8	no	—
44	C₁₂₇⋊C₈	—	2	4	no	—
45	Q₈×C₁₂₇	—	2	8	no	—
46	C₂×C₄×C₁₂₇	—	2	1	no	—
47	C₂×C₄×C₁₂₇	—	2	4	no	—
48	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₂₇	—	2	4	no	—
53	C₄×C₁₂₇	—	2	2	no	—
54	C₄×C₁₂₇	—	4	4	no	—
55	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
56	Dic₁₂₇	—	2	4	no	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
59	D₁₂₇	—	4	4	no	—
60	#3293	—	127	1	no	maximal, Sylow
61	C₁₂₇	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	127	4	no	—
63	#494	D₁₂₇	1	1	yes	characteristic, complemented
64	#473	—	127	4	no	—
65	#456	—	127	4	no	—
66	#443	—	127	2	no	—
67	2₋¹⁺⁴	—	127	4	no	—
68	#137	—	127	8	no	—
69	#137	—	254	8	no	—
70	#135	—	127	2	no	—
71	#135	—	254	4	no	—
72	C₂×Q₁₆	—	254	4	no	—
73	C₂×Q₁₆	C₂×D₁₂₇	1	4	yes	—
74	#131	—	127	4	no	—
75	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
76	#108	—	127	2	no	—
77	#104	—	127	4	no	—
78	#101	—	127	4	no	—
79	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
80	#41	—	127	4	no	—
81	#41	—	508	4	no	—
82	#41	—	254	8	no	—
83	C₂×Q₈	—	254	8	no	—
84	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	127	2	no	—
88	Q₁₆	—	254	8	no	—
89	SD₁₆	—	127	4	no	—
90	SD₁₆	—	254	8	no	—
91	D₈	—	127	2	no	—
92	M₁₆	—	127	4	no	—
93	M₁₆	—	254	4	no	—
94	C₂×C₈	—	254	4	no	—
95	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	127	4	no	—
100	Q₈	—	508	4	no	—
101	D₄	—	127	4	no	—
102	D₄	—	508	4	no	—
103	D₄	—	254	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	508	8	no	—
107	C₂×C₄	—	254	4	no	—
108	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
109	C₈	—	254	4	no	—
110	C₈	D₄×D₁₂₇	1	2	yes	—
111	C₈	D₄×D₁₂₇	1	2	yes	—
112	C₂²	—	254	4	no	—
113	C₂²	#16861254	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	254	4	no	—
117	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	508	4	no	—
120	C₂	#22550511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550689

C₂

yes

maximal, complemented

#22550148

C₂

yes

maximal, complemented

#22550058

C₂

yes

maximal, complemented

#22549994

C₂

yes

maximal, complemented

#494×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861257

C₂²

yes

—

#16861241

—

#16861240

C₂²

yes

—

#16861232

—

#16861231

C₂²

yes

—

Q₁₆×D₁₂₇

—

#16861215

C₂²

yes

—

#16861145

C₂²

yes

—

#16861137

C₂²

yes

—

#16861128

C₂²

yes

—

C₂×Q₁₆×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₂₇

—

Dic₅₀₈

D₄

yes

—

#14523238

—

#14523237

D₄

yes

—

#14523235

—

D₁₀₁₆

D₄

yes

—

#14523232

C₂³

yes

—

#14523231

—

C₈×D₁₂₇

—

C₂×Q₈×C₁₂₇

C₂³

yes

—

Q₁₆×C₁₂₇

—

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

C₂×D₄

yes

—

D₅₀₈

—

D₅₀₈

C₂×D₄

yes

—

C₂²⋊D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×D₄

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

D₄²

yes

characteristic

D₁₂₇

—

#3293

—

127

maximal, Sylow

C₁₂₇

#3293

yes

characteristic, complemented, Sylow

#578

—

127

—

#494

D₁₂₇

yes

characteristic, complemented

#473

—

127

—

#456

—

127

—

#443

—

127

—

2₋¹⁺⁴

—

127

—

#137

—

127

—

#137

—

254

—

#135

—

127

—

#135

—

254

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

C₂×D₁₂₇

yes

—

#131

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#108

—

127

—

#104

—

127

—

#101

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

#41

—

127

—

#41

—

508

—

#41

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂²×D₁₂₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

127

—

Q₁₆

—

254

—

SD₁₆

—

127

—

SD₁₆

—

254

—

D₈

—

127

—

M₁₆

—

127

—

M₁₆

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

127

—

100

Q₈

—

508

—

101

D₄

—

127

—

102

D₄

—

508

—

103

D₄

—

254

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

508

—

107

C₂×C₄

—

254

—

108

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

109

C₈

—

254

—

110

C₈

D₄×D₁₂₇

yes

—

111

C₈

D₄×D₁₂₇

yes

—

112

C₂²

—

254

—

113

C₂²

#16861254

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

254

—

117

C₄

C₂×D₄×D₁₂₇

yes

—

118

C₂

—

119

C₂

—

508

—

120

C₂

#22550511

yes

characteristic

	subgroup	G/H
center	C₂	#22550511
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#494×C₁₂₇	C₂
socle	C₂×C₁₂₇	D₄²
hypercenter	#494	D₁₂₇
hypocenter	C₁₂₇	#3293
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#22550511

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#494×C₁₂₇

C₂

socle

C₂×C₁₂₇

D₄²

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #22550689	C₂	semidirect
2. #22550148	C₂	semidirect
3. #22550058	C₂	semidirect
4. #22549994	C₂	semidirect
5. #494×C₁₂₇	C₂	semidirect
61. C₁₂₇	#3293	semidirect
63. #494	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550689

C₂

semidirect

2. #22550148

C₂

semidirect

3. #22550058

C₂

semidirect

4. #22549994

C₂

semidirect

5. #494×C₁₂₇

C₂

semidirect

61. C₁₂₇

#3293

semidirect

63. #494

D₁₂₇

semidirect

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

subgroup	index
#22550689	2
#22550148	2
#22550058	2
#22549994	2
#494×C₁₂₇	2
#3293	127

subgroup

index

#494×C₁₂₇

127

Subgroups — #30650534

Conjugacy classes of subgroups

Complements

Maximal subgroups