Group #30640900 — subgroups

Subgroups — #30640900

Order 16256.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22550273	C₂	1	1	yes	maximal, characteristic, complemented
2	#22549992	C₂	1	1	yes	maximal, characteristic
3	#22549700	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549696	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549687	C₂	1	1	yes	maximal, characteristic
6	#22549685	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16861231	C₂²	1	1	yes	characteristic
9	#16861217	—	2	1	no	—
10	#16861208	—	2	1	no	—
11	#16861149	C₂²	1	1	yes	characteristic
12	#16861141	C₂²	1	1	yes	characteristic
13	#16861140	—	2	1	no	—
14	#16861135	—	2	1	no	—
15	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
17	#108×C₁₂₇	C₂²	1	1	yes	characteristic
18	#106×C₁₂₇	C₂²	1	1	yes	characteristic
19	#14523248	—	2	1	no	—
20	C₂×Dic₂₅₄	—	4	1	no	—
21	Dic₅₀₈	Q₈	1	1	yes	characteristic, complemented
22	#14523237	—	2	1	no	—
23	D₁₀₁₆	Q₈	1	1	yes	characteristic, complemented
24	#14523231	—	4	1	no	—
25	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₁₂₇	—	4	1	no	—
27	C₄⋊Dic₁₂₇	—	2	1	no	—
28	Dic₁₂₇⋊C₄	—	4	1	no	—
29	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₁₂₇⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₂₇	—	2	1	no	—
32	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	—	2	1	no	—
34	C₁₆×C₁₂₇	—	2	1	no	—
35	Dic₂₅₄	—	2	1	no	—
36	Dic₂₅₄	—	8	2	no	—
37	D₅₀₈	—	2	1	no	—
38	C₂²⋊D₁₂₇	—	4	1	no	—
39	C₂×Dic₁₂₇	—	2	1	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₄×D₁₂₇	—	4	1	no	—
42	C₁₂₇⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₁₂₇⋊C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	—	4	1	no	—
45	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₂₇	—	4	1	no	—
47	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₁₂₇	#41	1	1	yes	characteristic
49	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₂₇	—	4	1	no	—
51	C₄×C₁₂₇	—	8	1	no	—
52	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₂₇	—	2	2	no	—
55	Dic₁₂₇	—	4	1	no	—
56	Dic₁₂₇	—	8	1	no	—
57	C₂×C₁₂₇	—	2	1	no	—
58	C₂×C₁₂₇	#476	1	1	yes	characteristic
59	D₁₂₇	—	8	1	no	—
60	#2230	—	127	1	no	maximal, Sylow
61	C₁₂₇	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	127	1	no	—
63	#443	—	127	1	no	—
64	#365	—	127	1	no	—
65	#365	D₁₂₇	1	1	yes	characteristic, complemented
66	#364	—	127	1	no	—
67	#360	—	127	1	no	—
68	#360	—	127	1	no	—
69	#135	—	127	1	no	—
70	#131	—	254	1	no	—
71	C₄⋊Q₈	—	254	1	no	—
72	M₃₂	—	127	1	no	—
73	M₃₂	C₂×D₁₂₇	1	1	yes	characteristic
74	#108	C₂×D₁₂₇	1	1	yes	characteristic
75	#106	—	127	1	no	—
76	#106	—	254	1	no	—
77	#106	C₂×D₁₂₇	1	1	yes	characteristic
78	#104	—	254	1	no	—
79	C₄×C₈	—	127	1	no	—
80	#41	—	254	1	no	—
81	C₂×Q₈	—	508	1	no	—
82	Q₁₆	—	127	1	no	—
83	SD₁₆	—	254	1	no	—
84	D₈	—	127	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	508	1	no	—
87	C₂×C₈	—	127	1	no	—
88	C₂×C₈	—	508	1	no	—
89	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	508	1	no	—
92	C₄⋊C₄	—	254	1	no	—
93	C₄²	—	127	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	254	1	no	—
96	Q₈	—	1016	2	no	—
97	Q₈	—	254	1	no	—
98	D₄	—	508	1	no	—
99	D₄	—	254	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	508	1	no	—
102	C₂×C₄	—	508	1	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	127	1	no	—
107	C₈	—	127	1	no	—
108	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
109	C₈	#14523248	1	1	yes	characteristic
110	C₂²	—	508	1	no	—
111	C₂²	#16861236	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	1016	1	no	—
114	C₄	—	508	1	no	—
115	C₄	—	254	2	no	—
116	C₄	#16861187	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₂₇	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	1016	1	no	—
120	C₂	#22550165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550273

C₂

yes

maximal, characteristic, complemented

#22549992

C₂

yes

maximal, characteristic

#22549700

C₂

yes

maximal, characteristic, complemented

#22549696

C₂

yes

maximal, characteristic, complemented

#22549687

C₂

yes

maximal, characteristic

#22549685

C₂

yes

maximal, characteristic

#365×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861231

C₂²

yes

—

—

C₂²

yes

characteristic

#16861141

C₂²

yes

characteristic

#16861140

—

#16861135

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

C₂²

yes

characteristic

#106×C₁₂₇

C₂²

yes

characteristic

#14523248

—

C₂×Dic₂₅₄

—

Dic₅₀₈

Q₈

yes

characteristic, complemented

#14523237

—

D₁₀₁₆

Q₈

yes

characteristic, complemented

#14523231

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₈×D₁₂₇

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

SD₁₆

yes

characteristic

C₁₂₇⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×Q₈

yes

C₈×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₂₇

C₂×SD₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#476

yes

characteristic

D₁₂₇

—

#2230

—

127

maximal, Sylow

C₁₂₇

#2230

yes

characteristic, complemented, Sylow

#498

—

127

—

#443

—

127

—

#365

—

127

—

#365

D₁₂₇

yes

characteristic, complemented

#364

—

127

—

#360

—

127

—

#360

—

127

—

#135

—

127

—

#131

—

254

—

C₄⋊Q₈

—

254

—

M₃₂

—

127

—

M₃₂

C₂×D₁₂₇

yes

characteristic

#108

C₂×D₁₂₇

yes

characteristic

#106

—

127

—

#106

—

254

—

#106

C₂×D₁₂₇

yes

characteristic

#104

—

254

—

C₄×C₈

—

127

—

#41

—

254

—

C₂×Q₈

—

508

—

Q₁₆

—

127

—

SD₁₆

—

254

—

D₈

—

127

—

M₁₆

—

M₁₆

—

508

—

C₂×C₈

—

127

—

C₂×C₈

—

508

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄²

—

127

—

C₁₆

—

C₁₆

—

254

—

Q₈

—

1016

—

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

100

C₂×C₄

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

508

—

103

C₂×C₄

—

254

—

104

C₂×C₄

D₄×D₁₂₇

yes

characteristic

105

C₈

—

106

C₈

—

127

—

107

C₈

—

127

—

108

C₈

Q₈×D₁₂₇

yes

characteristic

109

C₈

#14523248

yes

characteristic

110

C₂²

—

508

—

111

C₂²

#16861236

yes

characteristic

112

C₄

—

113

C₄

—

1016

—

114

C₄

—

508

—

115

C₄

—

254

—

116

C₄

#16861187

yes

characteristic

117

C₄

SD₁₆×D₁₂₇

yes

characteristic

118

C₂

—

119

C₂

—

1016

—

120

C₂

#22550165

yes

characteristic

	subgroup	G/H
center	C₂	#22550165
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#365×C₁₂₇	C₂
socle	C₂×C₁₂₇	#476
prime radical	#16861231	C₂²
hypercenter	#365	D₁₂₇
hypocenter	C₁₂₇	#2230
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#22550165

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#365×C₁₂₇

C₂

C₂×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #22550273	C₂	semidirect
3. #22549700	C₂	semidirect
4. #22549696	C₂	semidirect
7. #365×C₁₂₇	C₂	semidirect
21. Dic₅₀₈	Q₈	semidirect
23. D₁₀₁₆	Q₈	semidirect
61. C₁₂₇	#2230	semidirect
65. #365	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550273

C₂

semidirect

3. #22549700

C₂

semidirect

4. #22549696

C₂

semidirect

7. #365×C₁₂₇

C₂

semidirect

21. Dic₅₀₈

Q₈

semidirect

23. D₁₀₁₆

Q₈

semidirect

61. C₁₂₇

#2230

semidirect

65. #365

D₁₂₇

semidirect

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

subgroup	index
#22550273	2
#22549992	2
#22549700	2
#22549696	2
#22549687	2
#22549685	2
#365×C₁₂₇	2
#2230	127

subgroup

index

#365×C₁₂₇

127

Subgroups — #30640900

Conjugacy classes of subgroups

Complements

Maximal subgroups