Group #30636613 — subgroups

Subgroups — #30636613

Order 16256.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	—	2	4	no	—
10	C₂×C₁₂₇⋊C₈	—	4	2	no	—
11	C₂×C₁₂₇⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₂₇	—	2	2	no	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₂₇	—	4	2	no	—
16	C₄⋊C₄×C₁₂₇	—	4	2	no	—
17	C₄⋊C₄×C₁₂₇	—	4	2	no	—
18	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
21	C₁₂₇⋊C₈	—	4	4	no	—
22	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	—	2	1	no	—
25	C₂×C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	—	4	4	no	—
27	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
32	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
34	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
35	C₂²×C₁₂₇	#95	1	1	yes	characteristic
36	C₂²×C₁₂₇	#100	1	1	yes	characteristic
37	C₂²×C₁₂₇	#101	1	1	yes	characteristic
38	C₂²×C₁₂₇	#102	1	2	yes	—
39	C₂²×C₁₂₇	#103	1	2	yes	—
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	—	2	1	no	—
42	C₄×C₁₂₇	—	8	2	no	—
43	C₄×C₁₂₇	#102	1	2	yes	—
44	C₄×C₁₂₇	#103	1	2	yes	—
45	C₄×C₁₂₇	#106	1	2	yes	—
46	C₄×C₁₂₇	#107	1	2	yes	—
47	C₂×C₁₂₇	#331	1	1	yes	characteristic
48	C₂×C₁₂₇	#332	1	2	yes	—
49	C₂×C₁₂₇	#333	1	1	yes	characteristic
50	C₂×C₁₂₇	#340	1	2	yes	—
51	C₂×C₁₂₇	#343	1	1	yes	characteristic
52	#1296	—	127	1	no	maximal, Sylow
53	C₁₂₇	#1296	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊C₈	—	127	2	no	—
55	#384	D₁₂₇	1	1	yes	characteristic
56	C₂²×C₈	—	254	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
59	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
60	C₄⋊C₈	—	127	4	no	—
61	C₄⋊C₈	—	254	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
65	C₂×C₈	—	508	2	no	—
66	C₂×C₈	—	508	2	no	—
67	C₂×C₈	—	254	4	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₁₂₇	1	2	yes	—
72	C₄²	D₅₀₈	1	1	yes	characteristic
73	C₄²	Dic₂₅₄	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
80	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
83	C₂×C₄	#14523233	1	2	yes	—
84	C₂×C₄	#14523235	1	2	yes	—
85	C₂×C₄	#14523236	1	2	yes	—
86	C₂×C₄	#14523238	1	2	yes	—
87	C₈	—	508	4	no	—
88	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
89	C₂²	#16861126	1	1	yes	characteristic
90	C₂²	#16861128	1	1	yes	characteristic
91	C₂²	#16861130	1	2	yes	—
92	C₂²	#16861133	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#16861129	1	2	yes	—
97	C₄	#16861132	1	2	yes	—
98	C₄	#16861140	1	2	yes	—
99	C₄	#16861142	1	2	yes	—
100	C₂	#22549620	1	1	yes	characteristic
101	C₂	#22549622	1	2	yes	—
102	C₂	#22549625	1	1	yes	characteristic
103	C₂	#22549638	1	2	yes	—
104	C₂	#22549645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal

#384×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

—

C₄⋊(C₁₂₇⋊C₈)

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

Q₈

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂²×C₁₂₇

#103

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#102

yes

—

C₄×C₁₂₇

#103

yes

—

C₄×C₁₂₇

#106

yes

—

C₄×C₁₂₇

#107

yes

—

C₂×C₁₂₇

#331

yes

characteristic

C₂×C₁₂₇

#332

yes

—

C₂×C₁₂₇

#333

yes

characteristic

C₂×C₁₂₇

#340

yes

—

C₂×C₁₂₇

#343

yes

characteristic

#1296

—

127

maximal, Sylow

C₁₂₇

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

127

—

#384

D₁₂₇

yes

characteristic

C₂²×C₈

—

254

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

254

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₈

—

508

—

C₂×C₈

—

508

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

D₅₀₈

yes

characteristic

C₄²

Dic₂₅₄

yes

characteristic

C₂³

C₂²⋊Dic₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₂₇

yes

characteristic

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

#14523233

yes

—

C₂×C₄

#14523235

yes

—

C₂×C₄

#14523236

yes

—

C₂×C₄

#14523238

yes

—

C₈

—

508

—

C₂²

C₁₂₇⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16861133

yes

—

C₄

—

C₄

—

C₄

—

C₄

#16861129

yes

—

C₄

#16861132

yes

—

C₄

#16861140

yes

—

C₄

#16861142

yes

—

100

C₂

#22549620

yes

characteristic

101

C₂

#22549622

yes

—

102

C₂

#22549625

yes

characteristic

103

C₂

#22549638

yes

—

104

C₂

#22549645

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	#384×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	#384	D₁₂₇
hypocenter	C₁₂₇	#1296
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₂₇

commutator

C₂×C₄×C₁₂₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₂₇

Fitting

#384×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₁₂₇⋊C₈)	C₄	semidirect
53. C₁₂₇	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₁₂₇⋊C₈)

C₄

semidirect

53. C₁₂₇

#1296

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#384×C₁₂₇	2
#1296	127

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

#384×C₁₂₇

#1296

127

Subgroups — #30636613

Conjugacy classes of subgroups

Complements

Maximal subgroups