Group #30640950 — subgroups

Subgroups — #30640950

Order 16256.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22550134	C₂	1	1	yes	maximal, characteristic
2	#22549925	C₂	1	1	yes	maximal, characteristic
3	#22549699	C₂	1	2	yes	maximal, complemented
4	#22549693	C₂	1	2	yes	maximal, complemented
5	#349×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
7	C₂×#14523232	—	2	1	no	—
8	#16861188	—	2	1	no	—
9	#16861145	—	2	1	no	—
10	#16861145	C₂²	1	2	yes	—
11	#16861141	C₂²	1	1	yes	characteristic
12	#16861134	—	2	1	no	—
13	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₂₇	C₂²	1	2	yes	—
15	C₂×Dic₂₅₄	—	2	1	no	—
16	Dic₅₀₈	—	2	2	no	—
17	#14523232	—	2	2	no	—
18	#14523232	—	4	1	no	—
19	C₂×C₁₂₇⋊C₈	—	4	1	no	—
20	C₄⋊Dic₁₂₇	—	2	1	no	—
21	Dic₁₂₇⋊C₄	—	4	1	no	—
22	C₂²⋊Dic₁₂₇	—	4	1	no	—
23	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	M₁₆×C₁₂₇	—	2	2	no	—
25	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₂₇	—	2	2	no	—
28	Dic₂₅₄	—	4	1	no	—
29	Dic₂₅₄	—	4	1	no	—
30	C₂×Dic₁₂₇	—	4	1	no	—
31	C₂×Dic₁₂₇	—	4	1	no	—
32	C₁₂₇⋊C₈	—	4	2	no	—
33	C₂³×C₁₂₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₈×C₁₂₇	#41	1	2	yes	—
40	C₂²×C₁₂₇	—	2	1	no	—
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
43	C₄×C₁₂₇	—	2	1	no	—
44	C₄×C₁₂₇	#123	1	1	yes	characteristic
45	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
46	Dic₁₂₇	—	8	1	no	—
47	Dic₁₂₇	—	8	1	no	—
48	C₂×C₁₂₇	—	2	1	no	—
49	C₂×C₁₂₇	—	4	1	no	—
50	C₂×C₁₂₇	#481	1	1	yes	characteristic
51	#2239	—	127	1	no	maximal, Sylow
52	C₁₂₇	#2239	1	1	yes	characteristic, complemented, Sylow
53	#470	—	127	1	no	—
54	#430	—	127	1	no	—
55	#365	—	127	2	no	—
56	#362	—	127	2	no	—
57	#349	D₁₂₇	1	1	yes	characteristic
58	C₂×Q₁₆	—	127	1	no	—
59	C₂×M₁₆	—	254	1	no	—
60	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂²⋊Q₈	—	254	1	no	—
62	M₃₂	C₂×D₁₂₇	1	2	yes	—
63	#108	—	127	2	no	—
64	#108	—	254	1	no	—
65	#106	—	127	1	no	—
66	#103	—	254	1	no	—
67	C₂×Q₈	—	254	1	no	—
68	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
69	Q₁₆	—	254	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	508	1	no	—
72	M₁₆	—	254	2	no	—
73	C₂×C₈	—	508	1	no	—
74	C₂×C₈	D₅₀₈	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₄⋊C₄	—	508	1	no	—
77	C₄⋊C₄	—	254	1	no	—
78	C₂²⋊C₄	—	508	1	no	—
79	C₁₆	—	2	2	no	—
80	C₂³	#14523237	1	1	yes	characteristic
81	Q₈	—	508	1	no	—
82	Q₈	—	508	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	508	1	no	—
86	C₂×C₄	—	508	1	no	—
87	C₂×C₄	#14523237	1	1	yes	characteristic
88	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	508	2	no	—
91	C₈	#14523247	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂×#14523237	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	1016	1	no	—
97	C₄	—	1016	1	no	—
98	C₄	#16861193	1	1	yes	characteristic
99	C₄	#16861237	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	4	1	no	—
102	C₂	#22550195	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550134

C₂

yes

maximal, characteristic

#22549925

C₂

yes

maximal, characteristic

#22549699

C₂

yes

maximal, complemented

#22549693

C₂

yes

maximal, complemented

#349×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

—

—

—

C₂²

yes

—

#16861141

C₂²

yes

characteristic

#16861134

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

Dic₅₀₈

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₁₆×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#481

yes

characteristic

#2239

—

127

maximal, Sylow

C₁₂₇

#2239

yes

characteristic, complemented, Sylow

#470

—

127

—

#430

—

127

—

#365

—

127

—

#362

—

127

—

#349

D₁₂₇

yes

characteristic

C₂×Q₁₆

—

127

—

C₂×M₁₆

—

254

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²⋊Q₈

—

254

—

M₃₂

C₂×D₁₂₇

yes

—

#108

—

127

—

#108

—

254

—

#106

—

127

—

#103

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

D₅₀₈

yes

characteristic

Q₁₆

—

254

—

M₁₆

—

M₁₆

—

508

—

M₁₆

—

254

—

C₂×C₈

—

508

—

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

508

—

C₁₆

—

C₂³

#14523237

yes

characteristic

Q₈

—

508

—

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

#14523237

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₈

—

C₈

—

508

—

C₈

#14523247

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#14523237

yes

characteristic

C₄

—

C₄

—

1016

—

C₄

—

1016

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#22550195

yes

characteristic

	subgroup	G/H
center	C₂	#22550195
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#349×C₁₂₇	C₂
socle	C₂×C₁₂₇	#481
prime radical	C₂³×C₁₂₇	SD₁₆
hypercenter	#349	D₁₂₇
hypocenter	C₁₂₇	#2239
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#22550195

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#349×C₁₂₇

C₂

socle

C₂×C₁₂₇

#481

prime radical

C₂³×C₁₂₇

SD₁₆

D₁₂₇

C₁₂₇

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
3. #22549699	C₂	semidirect
4. #22549693	C₂	semidirect
52. C₁₂₇	#2239	semidirect

H ≅ G/N

type

3. #22549699

C₂

semidirect

4. #22549693

C₂

semidirect

52. C₁₂₇

#2239

semidirect

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

subgroup	index
#22550134	2
#22549925	2
#22549699	2
#22549693	2
#349×C₁₂₇	2
#2239	127

subgroup

index

#349×C₁₂₇

127

Subgroups — #30640950

Conjugacy classes of subgroups

Complements

Maximal subgroups