Group #30637640 — subgroups

Subgroups — #30637640

Order 16256.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22550006	C₂	1	2	yes	maximal
2	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
3	#22549627	C₂	1	1	yes	maximal, characteristic
4	#22549608	C₂	1	2	yes	maximal
5	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₄	1	1	yes	characteristic
7	C₄×Dic₂₅₄	C₂²	1	2	yes	—
8	#16861143	C₄	1	1	yes	characteristic
9	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
10	Dic₁₂₇⋊C₈	—	2	2	no	—
11	#16861134	C₄	1	2	yes	—
12	#16861119	—	2	2	no	—
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
17	Dic₅₀₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
20	Dic₁₂₇⋊C₄	—	2	2	no	—
21	C₄×Dic₁₂₇	—	2	2	no	—
22	C₂×C₈×C₁₂₇	—	2	1	no	—
23	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	Dic₂₅₄	—	2	2	no	—
28	Dic₂₅₄	—	2	2	no	—
29	C₂×Dic₁₂₇	—	2	2	no	—
30	C₂×Dic₁₂₇	—	2	2	no	—
31	C₁₂₇⋊C₈	—	4	2	no	—
32	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
35	C₈×C₁₂₇	—	2	1	no	—
36	C₈×C₁₂₇	—	2	2	no	—
37	C₈×C₁₂₇	—	4	1	no	—
38	C₈×C₁₂₇	M₁₆	1	2	yes	—
39	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₂₇	#131	1	1	yes	characteristic
43	C₄×C₁₂₇	#137	1	2	yes	—
44	Dic₁₂₇	—	4	2	no	—
45	Dic₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	#436	1	1	yes	characteristic
47	C₂×C₁₂₇	#441	1	1	yes	characteristic
48	C₂×C₁₂₇	#444	1	1	yes	characteristic
49	#1593	—	127	1	no	maximal, Sylow
50	C₁₂₇	#1593	1	1	yes	characteristic, complemented, Sylow
51	#446	—	127	2	no	—
52	C₄×Q₁₆	—	127	1	no	—
53	#334	—	127	1	no	—
54	Q₈⋊C₈	—	127	2	no	—
55	#322	D₁₂₇	1	1	yes	characteristic
56	C₂×Q₁₆	—	127	1	no	—
57	C₄×Q₈	—	127	2	no	—
58	#107	—	127	1	no	—
59	C₄⋊C₈	—	127	2	no	—
60	C₄⋊C₈	—	254	2	no	—
61	#103	—	127	2	no	—
62	#97	—	254	2	no	—
63	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂×Q₈	—	127	2	no	—
67	Q₁₆	—	254	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	254	2	no	—
70	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
71	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
72	C₂×C₈	D₅₀₈	1	2	yes	—
73	C₄⋊C₄	—	127	2	no	—
74	C₄⋊C₄	—	254	2	no	—
75	C₄²	—	254	2	no	—
76	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
77	Q₈	—	254	2	no	—
78	Q₈	—	254	2	no	—
79	C₂×C₄	—	254	2	no	—
80	C₂×C₄	—	254	2	no	—
81	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
82	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
83	C₂×C₄	#14523248	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	508	2	no	—
88	C₈	#14523231	1	2	yes	—
89	C₂²	C₄×D₅₀₈	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	508	2	no	—
92	C₄	—	508	2	no	—
93	C₄	C₂×#14523231	1	1	yes	characteristic
94	C₄	#16861215	1	1	yes	characteristic
95	C₄	#16861242	1	2	yes	—
96	C₂	#22549955	1	1	yes	characteristic
97	C₂	#22549984	1	1	yes	characteristic
98	C₂	#22549999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550006

C₂

yes

maximal

C₄×Dic₅₀₈

C₂

yes

maximal, characteristic

#22549627

C₂

yes

maximal, characteristic

#22549608

C₂

yes

maximal

#322×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₄

yes

characteristic

C₄×Dic₂₅₄

C₂²

yes

—

#16861143

C₄

yes

characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

Dic₁₂₇⋊C₈

—

#16861134

C₄

yes

—

#16861119

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

C₂×C₄

yes

—

Dic₅₀₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

M₁₆

yes

—

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×M₁₆

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1593

yes

characteristic, complemented, Sylow

#446

—

127

—

C₄×Q₁₆

—

127

—

#334

—

127

—

Q₈⋊C₈

—

127

—

#322

D₁₂₇

yes

characteristic

C₂×Q₁₆

—

127

—

C₄×Q₈

—

127

—

#107

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

#103

—

127

—

#97

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

Q₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

Q₈

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

#14523248

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

508

—

C₈

#14523231

yes

—

C₂²

C₄×D₅₀₈

yes

characteristic

C₄

—

C₄

—

508

—

C₄

—

508

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#322×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₄×D₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	#1593
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#322×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₄×D₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
50. C₁₂₇	#1593	semidirect

H ≅ G/N

type

50. C₁₂₇

#1593

semidirect

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

subgroup	index
#22550006	2
C₄×Dic₅₀₈	2
#22549627	2
#22549608	2
#322×C₁₂₇	2
#1593	127

subgroup

index

C₄×Dic₅₀₈

#322×C₁₂₇

127

Subgroups — #30637640

Conjugacy classes of subgroups

Complements

Maximal subgroups