Group #30640567 — subgroups

Subgroups — #30640567

Order 16256.

146 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22550293	C₂	1	2	yes	maximal, complemented
2	#22549697	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549660	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#16861149	—	2	2	no	—
7	#16861149	C₂²	1	2	yes	—
8	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#135×C₁₂₇	C₄	1	1	yes	characteristic
12	#131×C₁₂₇	C₂²	1	2	yes	—
13	#108×C₁₂₇	C₄	1	1	yes	characteristic
14	#104×C₁₂₇	C₄	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₁₂₇⋊C₁₆	—	2	2	no	—
17	C₁₂₇⋊C₁₆	—	2	1	no	—
18	C₁₂₇⋊C₁₆	D₄	1	1	yes	characteristic
19	C₁₂₇⋊C₁₆	D₄	1	1	yes	characteristic
20	#41×C₁₂₇	C₂×C₄	1	2	yes	—
21	Q₁₆×C₁₂₇	C₈	1	1	yes	characteristic
22	SD₁₆×C₁₂₇	C₈	1	2	yes	—
23	D₈×C₁₂₇	C₈	1	1	yes	characteristic
24	M₁₆×C₁₂₇	—	2	2	no	—
25	M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
31	D₄×C₁₂₇	—	2	2	no	—
32	D₄×C₁₂₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄×C₁₂₇	—	2	2	no	—
35	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₂₇	—	2	2	no	—
37	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₂₇	#41	1	1	yes	characteristic
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	#131	1	1	yes	characteristic
43	C₄×C₁₂₇	—	2	2	no	—
44	C₄×C₁₂₇	—	2	2	no	—
45	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
46	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₁₂₇	—	2	1	no	—
48	C₂×C₁₂₇	—	4	2	no	—
49	C₂×C₁₂₇	C₈×D₄	1	1	yes	characteristic
50	#2170	—	127	1	no	maximal, Sylow
51	C₁₂₇	#2170	1	1	yes	characteristic, complemented, Sylow
52	#504	—	127	2	no	—
53	#443	D₁₂₇	1	1	yes	characteristic
54	#364	—	127	1	no	—
55	#350	—	127	2	no	—
56	C₄×C₁₆	—	127	1	no	—
57	#135	Dic₁₂₇	1	1	yes	characteristic
58	#131	C₂×D₁₂₇	1	2	yes	—
59	M₃₂	—	127	2	no	—
60	M₃₂	—	254	2	no	—
61	C₂×C₁₆	—	127	1	no	—
62	C₂×C₁₆	—	127	1	no	—
63	C₂×C₁₆	—	254	2	no	—
64	#108	Dic₁₂₇	1	1	yes	characteristic
65	#104	Dic₁₂₇	1	2	yes	—
66	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
67	#41	C₂×Dic₁₂₇	1	2	yes	—
68	Q₁₆	C₁₂₇⋊C₈	1	1	yes	characteristic
69	SD₁₆	C₁₂₇⋊C₈	1	2	yes	—
70	D₈	C₁₂₇⋊C₈	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	C₂×Dic₁₂₇	1	2	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
77	C₁₆	—	127	1	no	—
78	C₁₆	—	127	1	no	—
79	C₁₆	—	254	2	no	—
80	C₁₆	—	254	1	no	—
81	Q₈	C₂×C₁₂₇⋊C₈	1	2	yes	—
82	D₄	—	2	2	no	—
83	D₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
89	C₈	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
90	C₈	D₄×D₁₂₇	1	1	yes	characteristic
91	C₈	#14523247	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	#16861216	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
97	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	4	2	no	—
100	C₂	D₄×C₁₂₇⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550293

C₂

yes

maximal, complemented

#22549697

C₂

yes

maximal, characteristic, complemented

#22549660

C₂

yes

maximal, complemented

C₄×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#443×C₁₂₇

C₂

yes

maximal, characteristic

#16861149

—

#16861149

C₂²

yes

—

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#135×C₁₂₇

C₄

yes

characteristic

#131×C₁₂₇

C₂²

yes

—

#108×C₁₂₇

C₄

yes

characteristic

#104×C₁₂₇

C₄

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₁₂₇⋊C₁₆

—

C₁₂₇⋊C₁₆

—

C₁₂₇⋊C₁₆

D₄

yes

characteristic

C₁₂₇⋊C₁₆

D₄

yes

characteristic

#41×C₁₂₇

C₂×C₄

yes

—

Q₁₆×C₁₂₇

C₈

yes

characteristic

SD₁₆×C₁₂₇

C₈

yes

—

D₈×C₁₂₇

C₈

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

Q₈×C₁₂₇

C₂×C₈

yes

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×C₈

yes

characteristic

C₈×C₁₂₇

C₂×C₈

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#131

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₈×D₄

yes

characteristic

#2170

—

127

maximal, Sylow

C₁₂₇

#2170

yes

characteristic, complemented, Sylow

#504

—

127

—

#443

D₁₂₇

yes

characteristic

#364

—

127

—

#350

—

127

—

C₄×C₁₆

—

127

—

#135

Dic₁₂₇

yes

characteristic

#131

C₂×D₁₂₇

yes

—

M₃₂

—

127

—

M₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

#108

Dic₁₂₇

yes

characteristic

#104

Dic₁₂₇

yes

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

#41

C₂×Dic₁₂₇

yes

—

Q₁₆

C₁₂₇⋊C₈

yes

characteristic

SD₁₆

C₁₂₇⋊C₈

yes

—

D₈

C₁₂₇⋊C₈

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₁₂₇

yes

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₁₆

—

127

—

C₁₆

—

127

—

C₁₆

—

254

—

C₁₆

—

254

—

Q₈

C₂×C₁₂₇⋊C₈

yes

—

D₄

—

D₄

C₂×C₁₂₇⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₈

—

C₈

C₂×C₁₂₇⋊C₈

yes

characteristic

C₈

C₂×C₁₂₇⋊C₈

yes

characteristic

C₈

D₄×D₁₂₇

yes

characteristic

C₈

#14523247

yes

characteristic

C₂²

—

C₂²

#16861216

yes

characteristic

C₄

—

C₄

—

C₄

D₄×Dic₁₂₇

yes

characteristic

C₄

C₂²×C₁₂₇⋊C₈

yes

characteristic

C₂

—

C₂

—

100

C₂

D₄×C₁₂₇⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#443×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₈×D₄
prime radical	#135×C₁₂₇	C₄
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#443	D₁₂₇
hypocenter	C₁₂₇	#2170
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#443×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₈×D₄

prime radical

#135×C₁₂₇

C₄

cosocle

C₂×C₈×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #22550293	C₂	semidirect
2. #22549697	C₂	semidirect
3. #22549660	C₂	semidirect
4. C₄×C₁₂₇⋊C₁₆	C₂	semidirect
51. C₁₂₇	#2170	semidirect

H ≅ G/N

type

1. #22550293

C₂

semidirect

2. #22549697

C₂

semidirect

3. #22549660

C₂

semidirect

4. C₄×C₁₂₇⋊C₁₆

C₂

semidirect

51. C₁₂₇

#2170

semidirect

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

subgroup	index
#22550293	2
#22549697	2
#22549660	2
C₄×C₁₂₇⋊C₁₆	2
#443×C₁₂₇	2
#2170	127

subgroup

index

#22550293

#22549697

#22549660

C₄×C₁₂₇⋊C₁₆

#443×C₁₂₇

#2170

127

Subgroups — #30640567

Conjugacy classes of subgroups

Complements

Maximal subgroups