Group #30640485 — subgroups

Subgroups — #30640485

Order 16256.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861149	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	Dic₁₂₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
7	#16861149	—	2	2	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₂₇	C₄	1	2	yes	—
10	C₈×Dic₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₂₇	—	2	2	no	—
13	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₂₇	C₈	1	2	yes	—
15	C₂×C₁₂₇⋊C₈	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
18	C₄×Dic₁₂₇	C₈	1	1	yes	characteristic
19	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
20	C₁₂₇⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
25	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₂₇	—	2	2	no	—
27	C₂×Dic₁₂₇	—	2	4	no	—
28	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
30	C₁₂₇⋊C₈	—	2	2	no	—
31	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
37	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
43	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₂₇	—	2	2	no	—
46	Dic₁₂₇	M₃₂	1	4	yes	complemented
47	C₂×C₁₂₇	—	2	2	no	—
48	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₁₂₇	C₂×M₃₂	1	2	yes	—
50	#2150	—	127	1	no	maximal, Sylow
51	C₁₂₇	#2150	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₃₂	—	127	1	no	—
53	C₂×M₃₂	D₁₂₇	1	1	yes	characteristic
54	C₂×C₄×C₈	—	127	1	no	—
55	C₄⋊C₁₆	—	127	4	no	—
56	C₂²×C₈	—	127	1	no	—
57	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
58	C₂×C₄²	—	127	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	254	2	no	—
61	C₂×C₁₆	—	127	2	no	—
62	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
63	C₄×C₈	—	127	2	no	—
64	C₄×C₈	—	127	2	no	—
65	C₂²×C₄	—	127	2	no	—
66	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
67	C₂×C₈	—	127	2	no	—
68	C₂×C₈	—	254	2	no	—
69	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
71	C₂×C₈	Dic₂₅₄	1	2	yes	—
72	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₄²	—	127	1	no	—
74	C₄²	—	127	1	no	—
75	C₄²	—	127	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	254	2	no	—
78	C₂³	C₈×D₁₂₇	1	1	yes	characteristic
79	C₂×C₄	—	127	2	no	—
80	C₂×C₄	—	127	2	no	—
81	C₂×C₄	—	254	4	no	—
82	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
83	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
84	C₂×C₄	#14523231	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
86	C₈	—	254	2	no	—
87	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
88	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₁₂₇⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
93	C₄	—	127	4	no	—
94	C₄	—	254	2	no	—
95	C₄	Dic₁₂₇⋊C₈	1	2	yes	—
96	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
97	C₄	C₂×#14523231	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₁₂₇⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861149

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₂₇

C₄

yes

characteristic

#16861149

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₂₇

C₄

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₈

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₈

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

Q₈

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊C₈

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₄×C₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

M₃₂

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

—

#2150

—

127

maximal, Sylow

C₁₂₇

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

C₂×M₃₂

D₁₂₇

yes

characteristic

C₂×C₄×C₈

—

127

—

C₄⋊C₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

—

127

—

M₃₂

—

M₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₂×C₈

Dic₂₅₄

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₁₆

—

C₁₆

—

254

—

C₂³

C₈×D₁₂₇

yes

characteristic

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₈×D₁₂₇

yes

characteristic

C₂×C₄

#14523231

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

Dic₁₂₇⋊C₄

yes

—

C₈

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₈

C₂×Dic₂₅₄

yes

characteristic

C₂²

—

C₂²

Dic₁₂₇⋊C₈

yes

—

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

C₄

—

127

—

C₄

—

254

—

C₄

Dic₁₂₇⋊C₈

yes

—

C₄

C₂×Dic₁₂₇⋊C₄

yes

characteristic

C₄

C₂×#14523231

yes

characteristic

C₂

—

C₂

C₂×Dic₁₂₇⋊C₈

yes

characteristic

100

C₂

M₃₂×D₁₂₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₂×M₃₂×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×M₃₂	D₁₂₇
hypocenter	C₁₂₇	#2150
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₈

Fitting

C₂×M₃₂×C₁₂₇

C₂

prime radical

C₂³×C₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

C₂×M₃₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₁₂₇⋊C₁₆	C₂	semidirect
46. Dic₁₂₇	M₃₂	semidirect
51. C₁₂₇	#2150	semidirect

H ≅ G/N

type

3. Dic₁₂₇⋊C₁₆

C₂

semidirect

46. Dic₁₂₇

M₃₂

semidirect

51. C₁₂₇

#2150

semidirect

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

subgroup	index
C₂×#16861149	2
C₂×C₈×Dic₁₂₇	2
Dic₁₂₇⋊C₁₆	2
C₂×M₃₂×C₁₂₇	2
#2150	127

subgroup

index

C₂×#16861149

C₂×C₈×Dic₁₂₇

Dic₁₂₇⋊C₁₆

C₂×M₃₂×C₁₂₇

#2150

127

Subgroups — #30640485

Conjugacy classes of subgroups

Complements

Maximal subgroups