Group #44550268 — subgroups

Subgroups — C₄×#22548650 #44550268

Order 32128.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#22548650	C₂	1	2	yes	maximal
2	C₄×#16860794	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#30631451	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16860794	C₂²	1	2	yes	—
7	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
9	#22548650	C₄	1	8	yes	direct product factor
10	#22548626	—	2	1	no	—
11	C₄×C₂₅₁⋊C₈	—	2	1	no	—
12	C₄×C₂₅₁⋊C₈	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	#97×C₂₅₁	—	2	1	no	—
16	C₄×C₈×C₂₅₁	—	2	1	no	—
17	#16860794	—	2	2	no	—
18	#16860794	C₂×C₄	1	4	yes	—
19	C₂×C₂₅₁⋊C₈	—	2	2	no	—
20	C₂×C₂₅₁⋊C₈	—	2	2	no	—
21	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₅₁	—	2	2	no	—
27	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₅₁	—	2	2	no	—
29	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
31	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
32	C₂₅₁⋊C₈	—	2	2	no	—
33	C₂₅₁⋊C₈	—	2	4	no	—
34	C₂₅₁⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₁	—	2	4	no	—
37	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	#41	1	2	yes	—
43	C₈×C₂₅₁	—	2	4	no	—
44	C₂²×C₂₅₁	—	2	2	no	—
45	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
47	C₄×C₂₅₁	—	2	2	no	—
48	C₄×C₂₅₁	#108	1	4	yes	complemented
49	C₄×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₅₁	#117	1	1	yes	characteristic
51	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
52	C₂×C₂₅₁	—	2	2	no	—
53	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₅₁	C₂×#108	1	2	yes	—
55	C₂₅₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
56	C₄×#108	—	251	1	no	maximal, Sylow
57	C₂×#108	—	251	2	no	—
58	C₄×M₁₆	—	251	1	no	—
59	C₄×M₁₆	D₂₅₁	1	1	yes	characteristic
60	C₂×C₄×C₈	—	251	1	no	—
61	#341	—	251	2	no	—
62	C₂×M₁₆	—	251	2	no	—
63	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
64	C₂²×C₈	—	251	1	no	—
65	C₂²×C₈	—	251	1	no	—
66	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
67	#108	—	251	8	no	—
68	#97	—	2	1	no	—
69	#97	—	502	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	502	1	no	—
72	C₄×C₈	—	251	4	no	—
73	C₂²×C₄	Dic₅₀₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	502	2	no	—
77	M₁₆	—	251	4	no	—
78	M₁₆	C₄×D₂₅₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	502	2	no	—
81	C₂×C₈	—	502	2	no	—
82	C₂×C₈	—	251	2	no	—
83	C₂×C₈	—	251	2	no	—
84	C₂×C₈	—	251	4	no	—
85	C₄²	C₄×D₂₅₁	1	2	yes	—
86	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
95	C₂×C₄	#16860810	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	502	2	no	—
98	C₈	—	502	4	no	—
99	C₈	—	251	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₅₀₂	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#22548650	1	4	yes	direct product factor
105	C₄	C₄²×D₂₅₁	1	1	yes	characteristic
106	C₄	#22548672	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₅₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₂₅₁⋊C₄	1	1	yes	characteristic
110	C₂	C₂×#22548650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548650

C₂

yes

maximal

C₄×#16860794

C₂

yes

maximal, characteristic

C₂×C₄×C₂₅₁⋊C₈

C₂

yes

maximal, characteristic

#30631451

C₂

yes

maximal

C₄×M₁₆×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

#22548650

C₄

yes

direct product factor

#22548626

—

C₄×C₂₅₁⋊C₈

—

C₄×C₂₅₁⋊C₈

C₄

yes

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#97×C₂₅₁

—

C₄×C₈×C₂₅₁

—

#16860794

—

#16860794

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

Q₈

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

C₄²

yes

—

C₂³×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×Q₈

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#108

yes

complemented

C₄×C₂₅₁

C₂×C₄²

yes

characteristic

C₄×C₂₅₁

#117

yes

characteristic

C₄×C₂₅₁

C₄×D₄

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₁

C₂×#108

yes

—

C₂₅₁

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

251

maximal, Sylow

C₂×#108

—

251

—

C₄×M₁₆

—

251

—

C₄×M₁₆

D₂₅₁

yes

characteristic

C₂×C₄×C₈

—

251

—

#341

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₂²×C₈

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#108

—

251

—

#97

—

#97

—

502

—

C₄×C₈

—

C₄×C₈

—

502

—

C₄×C₈

—

251

—

C₂²×C₄

Dic₅₀₂

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

—

502

—

M₁₆

—

251

—

M₁₆

C₄×D₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₄²

C₄×D₂₅₁

yes

—

C₄²

C₂²⋊D₂₅₁

yes

characteristic

C₄²

C₂²⋊D₂₅₁

yes

characteristic

C₂³

C₂×Dic₅₀₂

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₂₅₁⋊C₄

yes

—

C₂×C₄

Dic₂₅₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₂×C₄

#16860810

yes

—

C₈

—

C₈

—

502

—

C₈

—

502

—

C₈

—

251

—

100

C₂²

—

101

C₂²

C₂×Dic₂₅₁⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₅₀₂

yes

—

103

C₄

—

104

C₄

#22548650

yes

direct product factor

105

C₄

C₄²×D₂₅₁

yes

characteristic

106

C₄

#22548672

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₅₁

yes

—

108

C₂

—

109

C₂

C₄×Dic₂₅₁⋊C₄

yes

characteristic

110

C₂

C₂×#22548650

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₂₅₁
commutator	C₄×C₂₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₄×M₁₆×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₄×M₁₆	D₂₅₁
hypocenter	C₂₅₁	C₄×#108
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₂₅₁

commutator

C₄×C₂₅₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₄×M₁₆×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₅₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₄×M₁₆

D₂₅₁

hypocenter

C₂₅₁

C₄×#108

LCS

G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #22548650	C₄	direct (N × H)
48. C₄×C₂₅₁	#108	semidirect
55. C₂₅₁	C₄×#108	semidirect
104. C₄	#22548650	direct (N × H)

H ≅ G/N

type

9. #22548650

C₄

direct (N × H)

48. C₄×C₂₅₁

#108

semidirect

55. C₂₅₁

C₄×#108

semidirect

104. C₄

#22548650

direct (N × H)

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

subgroup	index
C₂×#22548650	2
C₄×#16860794	2
C₂×C₄×C₂₅₁⋊C₈	2
#30631451	2
C₄×M₁₆×C₂₅₁	2
C₄×#108	251

subgroup

index

C₂×C₄×C₂₅₁⋊C₈

C₄×M₁₆×C₂₅₁

251

Subgroups — C₄×#22548650 #44550268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#22548650 #44550268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22548650 #44550268