Group #30638865 — subgroups

Subgroups — #30638865

Order 16256.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861120	C₂	1	1	yes	maximal, characteristic
2	#22549742	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal
4	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
7	#16861120	C₄	1	4	yes	complemented
8	C₁₂₇⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₂₇	—	2	2	no	—
13	C₂×C₁₂₇⋊C₈	—	2	2	no	—
14	C₂×C₁₂₇⋊C₈	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₂₇	—	2	4	no	—
17	C₂²×C₄×C₁₂₇	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	4	no	—
21	C₄⋊C₄×C₁₂₇	—	4	4	no	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₂₇	Q₈	1	2	yes	—
24	C₂×Dic₁₂₇	—	4	2	no	—
25	C₂×Dic₁₂₇	—	4	2	no	—
26	C₂×Dic₁₂₇	—	4	2	no	—
27	C₁₂₇⋊C₈	—	2	4	no	—
28	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	2	no	—
31	C₂×C₄×C₁₂₇	—	4	2	no	—
32	C₂×C₄×C₁₂₇	—	4	4	no	—
33	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	#41	1	2	yes	—
37	C₂×C₄×C₁₂₇	#41	1	2	yes	—
38	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₂₇	#124	1	1	yes	characteristic
40	C₂²×C₁₂₇	#124	1	1	yes	characteristic
41	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₂₇	#137	1	2	yes	—
43	C₄×C₁₂₇	—	2	4	no	—
44	C₄×C₁₂₇	—	8	2	no	—
45	C₄×C₁₂₇	#117	1	2	yes	—
46	C₄×C₁₂₇	#125	1	2	yes	—
47	Dic₁₂₇	—	8	2	no	—
48	C₂×C₁₂₇	#382	1	1	yes	characteristic
49	C₂×C₁₂₇	#428	1	2	yes	—
50	C₂×C₁₂₇	#489	1	2	yes	—
51	C₂×C₁₂₇	#490	1	1	yes	characteristic
52	C₂×C₁₂₇	#491	1	1	yes	characteristic
53	#1836	—	127	1	no	maximal, Sylow
54	C₁₂₇	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	127	1	no	—
56	#384	D₁₂₇	1	1	yes	characteristic
57	#382	—	127	1	no	—
58	#340	—	127	4	no	—
59	C₂²×C₈	—	127	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	127	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
63	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
64	#97	—	127	4	no	—
65	#95	—	254	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	254	2	no	—
68	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₂×C₈	—	127	4	no	—
71	C₂×C₈	—	254	2	no	—
72	C₂×C₈	—	254	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	254	4	no	—
76	C₄²	C₄×D₁₂₇	1	2	yes	—
77	C₄²	Dic₂₅₄	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	508	2	no	—
84	C₂×C₄	—	508	2	no	—
85	C₂×C₄	—	508	2	no	—
86	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
89	C₂×C₄	#14523247	1	2	yes	—
90	C₂×C₄	#14523249	1	2	yes	—
91	C₈	—	254	4	no	—
92	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
93	C₂²	#16861196	1	1	yes	characteristic
94	C₂²	#16861198	1	1	yes	characteristic
95	C₂²	#16861235	1	2	yes	—
96	C₂²	#16861240	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1016	2	no	—
100	C₄	#16861167	1	2	yes	—
101	C₄	#16861200	1	2	yes	—
102	C₂	#22549741	1	1	yes	characteristic
103	C₂	#22549916	1	2	yes	—
104	C₂	#22550238	1	2	yes	—
105	C₂	#22550240	1	1	yes	characteristic
106	C₂	#22550244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861120

C₂

yes

maximal, characteristic

#22549742

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

#384×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861120

C₄

yes

complemented

C₁₂₇⋊#95

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

—

C₄×C₁₂₇

#125

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#382

yes

characteristic

C₂×C₁₂₇

#428

yes

—

C₂×C₁₂₇

#489

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1836

yes

characteristic, complemented, Sylow

C₂×#97

—

127

—

#384

D₁₂₇

yes

characteristic

#382

—

127

—

#340

—

127

—

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#97

—

127

—

#95

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄²

C₄×D₁₂₇

yes

—

C₄²

Dic₂₅₄

yes

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

C₂×C₄

#14523247

yes

—

C₂×C₄

#14523249

yes

—

C₈

—

254

—

C₂²

C₂×Dic₁₂₇⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16861240

yes

—

C₄

—

C₄

—

C₄

—

1016

—

100

C₄

#16861167

yes

—

101

C₄

#16861200

yes

—

102

C₂

#22549741

yes

characteristic

103

C₂

#22549916

yes

—

104

C₂

#22550238

yes

—

105

C₂

#22550240

yes

characteristic

106

C₂

#22550244

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#384×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#384	D₁₂₇
hypocenter	C₁₂₇	#1836
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#384×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #16861120	C₄	semidirect
54. C₁₂₇	#1836	semidirect

H ≅ G/N

type

7. #16861120

C₄

semidirect

54. C₁₂₇

#1836

semidirect

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

subgroup	index
C₂×#16861120	2
#22549742	2
#22549638	2
#384×C₁₂₇	2
#1836	127

subgroup

index

#384×C₁₂₇

127

Subgroups — #30638865

Conjugacy classes of subgroups

Complements

Maximal subgroups