Group #30640323 — subgroups

Subgroups — #30640323

Order 16256.

196 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16861148	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
9	#16861148	C₄	1	8	yes	complemented
10	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₄	1	1	yes	characteristic
13	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₈	1	2	yes	—
17	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₂₇	C₈	1	4	yes	—
21	C₈×D₁₂₇	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₂₇	C₈	1	2	yes	—
23	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₁₂₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
25	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
29	C₂²×D₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×Dic₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
32	C₄×D₁₂₇	—	2	4	no	—
33	C₄×D₁₂₇	C₄²	1	2	yes	—
34	C₄×D₁₂₇	C₂×C₈	1	2	yes	—
35	C₁₂₇⋊C₈	C₄²	1	2	yes	—
36	C₁₂₇⋊C₈	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂×C₈	1	2	yes	—
40	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
42	C₂×D₁₂₇	—	2	2	no	—
43	C₂×D₁₂₇	C₄×C₈	1	2	yes	—
44	C₄×C₁₂₇	—	2	2	no	—
45	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
47	Dic₁₂₇	C₄×C₈	1	2	yes	—
48	Dic₁₂₇	M₃₂	1	4	yes	complemented
49	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×M₃₂	1	2	yes	—
51	D₁₂₇	—	2	2	no	—
52	C₄×M₃₂	—	127	1	no	maximal, Sylow
53	C₁₂₇	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₃₂	—	127	2	no	—
55	C₂×C₄×C₈	—	127	1	no	—
56	#346	—	127	1	no	—
57	#346	D₁₂₇	1	1	yes	characteristic, complemented
58	C₄×C₁₆	—	127	2	no	—
59	C₂²×C₈	—	127	1	no	—
60	C₂²×C₈	—	127	1	no	—
61	C₂×C₄²	—	127	1	no	—
62	M₃₂	—	127	8	no	—
63	C₂×C₁₆	—	127	2	no	—
64	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
65	C₄×C₈	—	127	1	no	—
66	C₄×C₈	—	127	1	no	—
67	C₄×C₈	—	127	1	no	—
68	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂²×C₄	—	127	1	no	—
70	C₂²×C₄	—	127	2	no	—
71	C₂×C₈	—	127	1	no	—
72	C₂×C₈	—	127	1	no	—
73	C₂×C₈	—	127	4	no	—
74	C₂×C₈	—	127	4	no	—
75	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
76	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
77	C₄²	—	127	2	no	—
78	C₄²	—	127	1	no	—
79	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
80	C₁₆	—	127	4	no	—
81	C₁₆	C₄×D₁₂₇	1	4	yes	complemented
82	C₂³	—	127	1	no	—
83	C₂×C₄	—	127	2	no	—
84	C₂×C₄	—	127	1	no	—
85	C₂×C₄	—	127	2	no	—
86	C₂×C₄	—	127	2	no	—
87	C₂×C₄	—	254	4	no	—
88	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
90	C₈	—	127	2	no	—
91	C₈	—	127	2	no	—
92	C₈	C₈×D₁₂₇	1	2	yes	—
93	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
94	C₂²	—	127	2	no	—
95	C₂²	—	254	2	no	—
96	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	127	4	no	—
99	C₄	—	127	2	no	—
100	C₄	C₄²×D₁₂₇	1	1	yes	characteristic
101	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
102	C₂	—	254	2	no	—
103	C₂	C₄×C₈×D₁₂₇	1	1	yes	characteristic
104	C₂	M₃₂×D₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861148

C₂

yes

maximal

C₄×C₈×D₁₂₇

C₂

yes

maximal, characteristic

#22549652

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₁₂₇

C₂

yes

maximal, complemented

#346×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₄

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

C₄²×D₁₂₇

C₄

yes

characteristic

#16861148

C₄

yes

complemented

C₂×C₁₂₇⋊C₁₆

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₄

yes

characteristic

C₈×Dic₁₂₇

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₈

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₂₇

C₈

yes

—

C₈×D₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₈

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

C₂×C₄

yes

complemented

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₁₆×C₁₂₇

C₂×C₄

yes

complemented

C₂²×D₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

characteristic

C₂×Dic₁₂₇

C₂×C₈

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₄²

yes

—

C₄×D₁₂₇

C₂×C₈

yes

—

C₁₂₇⋊C₈

C₄²

yes

—

C₁₂₇⋊C₈

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂×C₈

yes

—

C₈×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄×C₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄²

yes

characteristic

C₄×C₁₂₇

C₂²×C₈

yes

characteristic

Dic₁₂₇

C₄×C₈

yes

—

Dic₁₂₇

M₃₂

yes

complemented

C₂×C₁₂₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₂₇

C₂×M₃₂

yes

—

D₁₂₇

—

C₄×M₃₂

—

127

maximal, Sylow

C₁₂₇

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

C₂×C₄×C₈

—

127

—

#346

—

127

—

#346

D₁₂₇

yes

characteristic, complemented

C₄×C₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄²

—

127

—

M₃₂

—

127

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₁₆

—

127

—

C₁₆

C₄×D₁₂₇

yes

complemented

C₂³

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₈×D₁₂₇

yes

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

127

—

C₈

C₈×D₁₂₇

yes

—

C₈

C₂×C₄×D₁₂₇

yes

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

C₂×C₈×D₁₂₇

yes

characteristic

C₄

—

C₄

—

127

—

C₄

—

127

—

100

C₄

C₄²×D₁₂₇

yes

characteristic

101

C₄

C₂×C₈×D₁₂₇

yes

characteristic

102

C₂

—

254

—

103

C₂

C₄×C₈×D₁₂₇

yes

characteristic

104

C₂

M₃₂×D₁₂₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Fitting	#346×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₈
prime radical	C₂²×D₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	C₄×M₃₂
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₂₇

commutator

C₂×C₁₂₇

C₂×C₄×C₈

Fitting

#346×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×C₈

prime radical

C₂²×D₁₂₇

C₂×C₈

cosocle

C₂×C₈×C₁₂₇

C₂³

hypercenter

#346

D₁₂₇

hypocenter

C₁₂₇

C₄×M₃₂

LCS

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #22549652	C₂	semidirect
4. C₁₆×Dic₁₂₇	C₂	semidirect
5. #346×C₁₂₇	C₂	semidirect
9. #16861148	C₄	semidirect
24. C₁₂₇⋊C₁₆	C₂×C₄	semidirect
28. C₁₆×C₁₂₇	C₂×C₄	semidirect
48. Dic₁₂₇	M₃₂	semidirect
53. C₁₂₇	C₄×M₃₂	semidirect
57. #346	D₁₂₇	semidirect
81. C₁₆	C₄×D₁₂₇	semidirect

H ≅ G/N

type

3. #22549652

C₂

semidirect

4. C₁₆×Dic₁₂₇

C₂

semidirect

5. #346×C₁₂₇

C₂

semidirect

9. #16861148

C₄

semidirect

24. C₁₂₇⋊C₁₆

C₂×C₄

semidirect

28. C₁₆×C₁₂₇

C₂×C₄

semidirect

48. Dic₁₂₇

M₃₂

semidirect

53. C₁₂₇

C₄×M₃₂

semidirect

57. #346

D₁₂₇

semidirect

81. C₁₆

C₄×D₁₂₇

semidirect

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

subgroup	index
C₂×#16861148	2
C₄×C₈×D₁₂₇	2
#22549652	2
C₁₆×Dic₁₂₇	2
#346×C₁₂₇	2
C₄×M₃₂	127

subgroup

index

C₂×#16861148

C₄×C₈×D₁₂₇

#22549652

C₁₆×Dic₁₂₇

#346×C₁₂₇

C₄×M₃₂

127

Subgroups — #30640323

Conjugacy classes of subgroups

Complements

Maximal subgroups