Group #30641002 — subgroups

Subgroups — #30641002

Order 16256.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16861155	C₂	1	2	yes	maximal, complemented
2	C₂×#16861153	C₂	1	2	yes	maximal
3	#22550256	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
5	#495×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	—	2	1	no	—
7	C₂×Dic₅₀₈	C₂²	1	2	yes	—
8	#16861209	—	2	1	no	—
9	#16861155	—	2	4	no	—
10	#16861153	—	2	4	no	—
11	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
14	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#124×C₁₂₇	—	2	1	no	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	—	2	2	no	—
18	Dic₅₀₈	—	2	2	no	—
19	Dic₅₀₈	—	2	2	no	—
20	Dic₅₀₈	—	4	2	no	—
21	C₄⋊Dic₁₂₇	—	4	2	no	—
22	C₁₂₇⋊C₁₆	D₄	1	4	yes	—
23	C₂×Q₈×C₁₂₇	—	2	1	no	—
24	C₂×D₄×C₁₂₇	—	2	1	no	—
25	Q₁₆×C₁₂₇	—	2	2	no	—
26	SD₁₆×C₁₂₇	—	4	2	no	—
27	D₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
31	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	Dic₂₅₄	—	4	2	no	—
33	Dic₂₅₄	—	4	2	no	—
34	C₂×Dic₁₂₇	—	4	2	no	—
35	C₂³×C₁₂₇	—	4	1	no	—
36	Q₈×C₁₂₇	—	4	1	no	—
37	Q₈×C₁₂₇	—	4	1	no	—
38	D₄×C₁₂₇	—	4	1	no	—
39	D₄×C₁₂₇	—	4	1	no	—
40	C₂×C₄×C₁₂₇	—	4	1	no	—
41	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₈×C₁₂₇	D₈	1	2	yes	—
44	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₂²×C₁₂₇	—	8	1	no	—
46	C₂²×C₁₂₇	—	8	2	no	—
47	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₁₂₇	—	2	2	no	—
49	C₄×C₁₂₇	—	8	1	no	—
50	C₄×C₁₂₇	#127	1	1	yes	characteristic
51	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	Dic₁₂₇	—	8	2	no	—
53	C₂×C₁₂₇	—	16	1	no	—
54	C₂×C₁₂₇	#493	1	1	yes	characteristic
55	C₂×C₁₂₇	#510	1	2	yes	—
56	#2251	—	127	1	no	maximal, Sylow
57	C₁₂₇	#2251	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	127	2	no	—
59	C₂×SD₃₂	—	127	2	no	—
60	#495	D₁₂₇	1	1	yes	characteristic
61	#494	—	127	1	no	—
62	#346	—	127	1	no	—
63	C₂×Q₁₆	—	127	2	no	—
64	C₂×Q₁₆	—	254	1	no	—
65	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂×SD₁₆	—	2	1	no	—
67	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊Q₈	—	254	1	no	—
69	#124	—	2	1	no	—
70	Q₃₂	—	254	4	no	—
71	SD₃₂	—	254	4	no	—
72	C₂×C₁₆	—	127	2	no	—
73	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	—	254	2	no	—
76	C₂×D₄	—	2	1	no	—
77	Q₁₆	—	2	2	no	—
78	Q₁₆	—	508	2	no	—
79	Q₁₆	—	254	2	no	—
80	Q₁₆	—	254	2	no	—
81	SD₁₆	—	4	2	no	—
82	D₈	—	2	2	no	—
83	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₄⋊C₄	—	508	2	no	—
86	C₂²⋊C₄	—	4	2	no	—
87	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
88	C₁₆	—	127	4	no	—
89	C₂³	—	4	1	no	—
90	Q₈	—	4	1	no	—
91	Q₈	—	4	1	no	—
92	Q₈	—	508	2	no	—
93	Q₈	—	508	2	no	—
94	D₄	—	4	1	no	—
95	D₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	508	2	no	—
98	C₂×C₄	#14523233	1	2	yes	—
99	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
100	C₈	#14523233	1	2	yes	—
101	C₈	D₄×D₁₂₇	1	2	yes	—
102	C₂²	—	8	2	no	—
103	C₂²	—	8	1	no	—
104	C₂²	C₂×#14523233	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	8	1	no	—
107	C₄	—	1016	2	no	—
108	C₄	#16861205	1	1	yes	characteristic
109	C₄	C₂×#14523233	1	1	yes	characteristic
110	C₂	—	16	1	no	—
111	C₂	#22550252	1	1	yes	characteristic
112	C₂	#22550316	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861155

C₂

yes

maximal, complemented

C₂×#16861153

C₂

yes

maximal

#22550256

C₂

yes

maximal, characteristic, complemented

#22549652

C₂

yes

maximal, characteristic, complemented

#495×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

C₂²

yes

—

#16861209

—

#16861155

—

#16861153

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₂×Q₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×SD₁₆×C₁₂₇

—

C₂×D₈×C₁₂₇

C₂²

yes

characteristic

#124×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

Dic₅₀₈

—

Dic₅₀₈

—

Dic₅₀₈

—

C₄⋊Dic₁₂₇

—

C₁₂₇⋊C₁₆

D₄

yes

—

C₂×Q₈×C₁₂₇

—

C₂×D₄×C₁₂₇

—

Q₁₆×C₁₂₇

—

SD₁₆×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

D₈

yes

—

C₈×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#127

yes

characteristic

C₄×C₁₂₇

C₂×D₈

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#493

yes

characteristic

C₂×C₁₂₇

#510

yes

—

#2251

—

127

maximal, Sylow

C₁₂₇

#2251

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

127

—

C₂×SD₃₂

—

127

—

#495

D₁₂₇

yes

characteristic

#494

—

127

—

#346

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

C₂×D₁₂₇

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

254

—

#124

—

Q₃₂

—

254

—

SD₃₂

—

254

—

C₂×C₁₆

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

508

—

Q₁₆

—

254

—

Q₁₆

—

254

—

SD₁₆

—

D₈

—

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

508

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₁₂₇

yes

characteristic

C₁₆

—

127

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

508

—

Q₈

—

508

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

#14523233

yes

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

100

C₈

#14523233

yes

—

101

C₈

D₄×D₁₂₇

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×#14523233

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

1016

—

108

C₄

#16861205

yes

characteristic

109

C₄

C₂×#14523233

yes

characteristic

110

C₂

—

111

C₂

#22550252

yes

characteristic

112

C₂

#22550316

yes

—

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#495×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂×D₈×C₁₂₇	C₂²
hypercenter	#495	D₁₂₇
hypocenter	C₁₂₇	#2251
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#14523233

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#495×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₈

prime radical

C₂×D₈×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₂×#16861155	C₂	semidirect
3. #22550256	C₂	semidirect
4. #22549652	C₂	semidirect
57. C₁₂₇	#2251	semidirect

H ≅ G/N

type

1. C₂×#16861155

C₂

semidirect

3. #22550256

C₂

semidirect

4. #22549652

C₂

semidirect

57. C₁₂₇

#2251

semidirect

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

subgroup	index
C₂×#16861155	2
C₂×#16861153	2
#22550256	2
#22549652	2
#495×C₁₂₇	2
#2251	127

subgroup

index

#495×C₁₂₇

127

Subgroups — #30641002

Conjugacy classes of subgroups

Complements

Maximal subgroups