Group #44450403 — subgroups

Subgroups — #44450403

Order 29824.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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	#28416991	C₂	1	1	yes	maximal, characteristic
2	#28416387	C₂	1	2	yes	maximal
3	#28416177	C₂	1	1	yes	maximal, characteristic, complemented
4	#28416176	C₂	1	2	yes	maximal, complemented
5	#330×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16836577	—	2	2	no	—
7	#21418534	C₄	1	2	yes	—
8	#21418527	C₂²	1	1	yes	characteristic
9	#21418513	—	2	2	no	—
10	#21418500	—	2	2	no	—
11	#21418493	C₄	1	2	yes	—
12	C₄²×D₂₃₃	C₂²	1	1	yes	characteristic
13	#21418446	C₂²	1	2	yes	—
14	C₈×Dic₂₃₃	—	2	2	no	—
15	#125×C₂₃₃	C₂²	1	1	yes	characteristic
16	#97×C₂₃₃	C₂²	1	2	yes	—
17	C₂×Dic₄₆₆	—	2	2	no	—
18	C₂×C₄×D₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₂₃₃	D₄	1	2	yes	—
20	#16836577	—	2	8	no	—
21	C₂×#14272338	—	2	2	no	—
22	C₄⋊Dic₂₃₃	C₂×C₄	1	2	yes	—
23	#16836573	—	2	2	no	—
24	#16836573	—	2	2	no	—
25	#16836571	—	2	2	no	—
26	C₄×Dic₂₃₃	—	2	2	no	—
27	C₄×Dic₂₃₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₂₃₃	D₄	1	2	yes	—
29	C₂×C₈×C₂₃₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃₃	—	2	2	no	—
31	C₄⋊C₄×C₂₃₃	C₂×C₄	1	2	yes	—
32	C₄²×C₂₃₃	C₂³	1	1	yes	characteristic
33	C₂²×D₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₄₆₆	—	4	4	no	—
35	C₂×Dic₂₃₃	—	2	2	no	—
36	C₂×Dic₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₂₃₃	C₂²⋊C₄	1	2	yes	—
38	C₄×D₂₃₃	—	2	2	no	—
39	C₄×D₂₃₃	—	2	4	no	—
40	#14272338	—	2	4	no	—
41	C₂×C₄×C₂₃₃	—	2	2	no	—
42	C₂×C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
44	C₈×C₂₃₃	—	2	4	no	—
45	C₂²×C₂₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×D₂₃₃	—	2	1	no	—
47	C₂×D₂₃₃	—	2	2	no	—
48	C₄×C₂₃₃	—	2	1	no	—
49	C₄×C₂₃₃	—	2	2	no	—
50	C₄×C₂₃₃	—	4	2	no	—
51	Dic₂₃₃	—	4	2	no	—
52	Dic₂₃₃	#101	1	2	yes	—
53	Dic₂₃₃	#104	1	4	yes	—
54	C₂×C₂₃₃	C₂×#101	1	1	yes	characteristic
55	C₂×C₂₃₃	C₂×#104	1	2	yes	—
56	D₂₃₃	—	4	1	no	—
57	C₂₃₃	#1526	1	1	yes	characteristic, complemented, Sylow
58	#1526	—	233	1	no	maximal, Sylow
59	#533	—	233	1	no	—
60	C₄×M₁₆	—	233	2	no	—
61	#330	—	233	1	no	—
62	#330	—	233	2	no	—
63	#330	D₂₃₃	1	1	yes	characteristic, complemented
64	C₂×M₁₆	—	466	2	no	—
65	C₄⋊Q₈	—	233	2	no	—
66	#125	—	233	1	no	—
67	#125	C₂×D₂₃₃	1	1	yes	characteristic
68	C₂²⋊Q₈	—	466	2	no	—
69	C₄×Q₈	—	466	2	no	—
70	#117	—	233	2	no	—
71	C₂×C₄²	—	233	1	no	—
72	#97	—	233	2	no	—
73	#97	C₂×D₂₃₃	1	2	yes	—
74	C₄×C₈	—	466	2	no	—
75	C₂×Q₈	—	466	2	no	—
76	C₂²×C₄	—	233	1	no	—
77	C₂²×C₄	—	233	2	no	—
78	M₁₆	—	466	8	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	466	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	466	2	no	—
83	C₄⋊C₄	—	466	2	no	—
84	C₄⋊C₄	—	233	2	no	—
85	C₄⋊C₄	C₄×D₂₃₃	1	2	yes	—
86	C₂²⋊C₄	—	466	2	no	—
87	C₄²	—	466	2	no	—
88	C₄²	—	233	1	no	—
89	C₄²	—	233	2	no	—
90	C₄²	C₂²×D₂₃₃	1	1	yes	characteristic
91	C₂³	—	233	1	no	—
92	Q₈	—	932	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	466	4	no	—
95	C₂×C₄	—	466	2	no	—
96	C₂×C₄	—	466	2	no	—
97	C₂×C₄	—	233	1	no	—
98	C₂×C₄	—	233	2	no	—
99	C₂×C₄	C₂×C₄×D₂₃₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₃₃	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	466	4	no	—
103	C₂²	—	466	1	no	—
104	C₂²	—	466	2	no	—
105	C₂²	C₂²⋊C₄×D₂₃₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	932	2	no	—
110	C₄	—	233	2	no	—
111	C₄	—	233	4	no	—
112	C₂	—	932	1	no	—
113	C₂	#101×D₂₃₃	1	1	yes	characteristic
114	C₂	#104×D₂₃₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#28416991

C₂

yes

maximal, characteristic

#28416387

C₂

yes

maximal

#28416177

C₂

yes

maximal, characteristic, complemented

#28416176

C₂

yes

maximal, complemented

#330×C₂₃₃

C₂

yes

maximal, characteristic, complemented

C₂×#16836577

—

#21418534

C₄

yes

—

#21418527

C₂²

yes

—

—

C₄

yes

—

C₄²×D₂₃₃

C₂²

yes

characteristic

#21418446

C₂²

yes

—

C₈×Dic₂₃₃

—

#125×C₂₃₃

C₂²

yes

characteristic

#97×C₂₃₃

C₂²

yes

—

C₂×Dic₄₆₆

—

C₂×C₄×D₂₃₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃₃

D₄

yes

—

#16836577

—

C₂×#14272338

—

C₄⋊Dic₂₃₃

C₂×C₄

yes

—

#16836573

—

#16836573

—

#16836571

—

C₄×Dic₂₃₃

—

C₄×Dic₂₃₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃₃

D₄

yes

—

C₂×C₈×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

C₂×C₄

yes

—

C₄²×C₂₃₃

C₂³

yes

characteristic

C₂²×D₂₃₃

C₂²⋊C₄

yes

characteristic

Dic₄₆₆

—

C₂×Dic₂₃₃

—

C₂×Dic₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃₃

C₂²⋊C₄

yes

—

C₄×D₂₃₃

—

C₄×D₂₃₃

—

#14272338

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂×D₄

yes

—

C₈×C₂₃₃

—

C₂²×C₂₃₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

Dic₂₃₃

—

Dic₂₃₃

#101

yes

—

Dic₂₃₃

#104

yes

—

C₂×C₂₃₃

C₂×#101

yes

characteristic

C₂×C₂₃₃

C₂×#104

yes

—

D₂₃₃

—

C₂₃₃

#1526

yes

characteristic, complemented, Sylow

#1526

—

233

maximal, Sylow

#533

—

233

—

C₄×M₁₆

—

233

—

#330

—

233

—

#330

—

233

—

#330

D₂₃₃

yes

characteristic, complemented

C₂×M₁₆

—

466

—

C₄⋊Q₈

—

233

—

#125

—

233

—

#125

C₂×D₂₃₃

yes

characteristic

C₂²⋊Q₈

—

466

—

C₄×Q₈

—

466

—

#117

—

233

—

C₂×C₄²

—

233

—

#97

—

233

—

#97

C₂×D₂₃₃

yes

—

C₄×C₈

—

466

—

C₂×Q₈

—

466

—

C₂²×C₄

—

233

—

C₂²×C₄

—

233

—

M₁₆

—

466

—

C₂×C₈

—

C₂×C₈

—

466

—

C₄⋊C₄

—

C₄⋊C₄

—

466

—

C₄⋊C₄

—

466

—

C₄⋊C₄

—

233

—

C₄⋊C₄

C₄×D₂₃₃

yes

—

C₂²⋊C₄

—

466

—

C₄²

—

466

—

C₄²

—

233

—

C₄²

—

233

—

C₄²

C₂²×D₂₃₃

yes

characteristic

C₂³

—

233

—

Q₈

—

932

—

C₂×C₄

—

C₂×C₄

—

466

—

C₂×C₄

—

466

—

C₂×C₄

—

466

—

C₂×C₄

—

233

—

C₂×C₄

—

233

—

C₂×C₄

C₂×C₄×D₂₃₃

yes

characteristic

100

C₂×C₄

D₄×D₂₃₃

yes

—

101

C₈

—

102

C₈

—

466

—

103

C₂²

—

466

—

104

C₂²

—

466

—

105

C₂²

C₂²⋊C₄×D₂₃₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

932

—

110

C₄

—

233

—

111

C₄

—

233

—

112

C₂

—

932

—

113

C₂

#101×D₂₃₃

yes

characteristic

114

C₂

#104×D₂₃₃

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃₃
commutator	C₂×C₄×C₂₃₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃₃
Fitting	#330×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₃₃	C₂²⋊C₄
cosocle	C₄²×C₂₃₃	C₂³
hypercenter	#330	D₂₃₃
hypocenter	C₂₃₃	#1526
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃₃

commutator

C₂×C₄×C₂₃₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃₃

Fitting

#330×C₂₃₃

C₂

socle

C₂²×C₂₃₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₃₃

C₂²⋊C₄

cosocle

C₄²×C₂₃₃

C₂³

D₂₃₃

C₂₃₃

G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #28416177	C₂	semidirect
4. #28416176	C₂	semidirect
5. #330×C₂₃₃	C₂	semidirect
57. C₂₃₃	#1526	semidirect
63. #330	D₂₃₃	semidirect

H ≅ G/N

type

3. #28416177

C₂

semidirect

4. #28416176

C₂

semidirect

5. #330×C₂₃₃

C₂

semidirect

57. C₂₃₃

#1526

semidirect

63. #330

D₂₃₃

semidirect

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

subgroup	index
#28416991	2
#28416387	2
#28416177	2
#28416176	2
#330×C₂₃₃	2
#1526	233

subgroup

index

#330×C₂₃₃

233

Subgroups — #44450403

Conjugacy classes of subgroups

Complements

Maximal subgroups