Group #44506526 — subgroups

Subgroups — #44506526

Order 30848.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#29508433	C₂	1	1	yes	maximal, characteristic, complemented
2	#29507909	C₂	1	2	yes	maximal, complemented
3	#29507466	C₂	1	1	yes	maximal, characteristic, complemented
4	#29507446	C₂	1	2	yes	maximal, complemented
5	#492×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16838755	—	2	2	no	—
7	#21440631	—	2	2	no	—
8	#21440617	—	2	2	no	—
9	C₄×D₉₆₄	C₂²	1	2	yes	—
10	#21440600	—	2	2	no	—
11	C₄⋊#14515349	C₂²	1	2	yes	—
12	#21440563	—	2	2	no	—
13	C₂×SD₁₆×C₂₄₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	#127×C₂₄₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
17	C₂×D₉₆₄	—	2	2	no	—
18	C₂×D₉₆₄	D₄	1	2	yes	—
19	C₂×C₄×D₂₄₁	—	2	2	no	—
20	#16838755	—	4	4	no	—
21	C₂×#14515349	—	2	2	no	—
22	C₄⋊Dic₂₄₁	D₄	1	2	yes	—
23	#16838747	—	2	4	no	—
24	#16838747	—	2	2	no	—
25	C₄×Dic₂₄₁	—	2	2	no	—
26	C₂×Q₈×C₂₄₁	D₄	1	2	yes	—
27	C₂×D₄×C₂₄₁	—	2	1	no	—
28	C₂×D₄×C₂₄₁	—	2	2	no	—
29	SD₁₆×C₂₄₁	—	4	4	no	—
30	C₂×C₈×C₂₄₁	—	2	1	no	—
31	C₄⋊C₄×C₂₄₁	—	2	2	no	—
32	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
33	C₂²×D₂₄₁	—	2	2	no	—
34	D₉₆₄	—	2	2	no	—
35	D₉₆₄	—	2	2	no	—
36	D₉₆₄	—	8	2	no	—
37	C₂×Dic₂₄₁	—	2	2	no	—
38	C₄×D₂₄₁	—	4	2	no	—
39	C₄×D₂₄₁	—	4	2	no	—
40	#14515349	—	4	2	no	—
41	C₂³×C₂₄₁	—	4	1	no	—
42	Q₈×C₂₄₁	—	4	2	no	—
43	D₄×C₂₄₁	—	4	2	no	—
44	D₄×C₂₄₁	—	4	2	no	—
45	D₄×C₂₄₁	—	4	2	no	—
46	C₂×C₄×C₂₄₁	—	2	2	no	—
47	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₄₁	—	2	2	no	—
50	C₂²×C₂₄₁	—	8	2	no	—
51	C₂²×C₂₄₁	—	8	1	no	—
52	C₂²×C₂₄₁	#120	1	1	yes	characteristic
53	C₂×D₂₄₁	—	4	2	no	—
54	C₂×D₂₄₁	—	4	2	no	—
55	C₂×D₂₄₁	—	4	2	no	—
56	C₄×C₂₄₁	—	2	1	no	—
57	C₄×C₂₄₁	—	4	2	no	—
58	C₄×C₂₄₁	#135	1	2	yes	—
59	C₄×C₂₄₁	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₂₄₁	—	4	2	no	—
61	C₂×C₂₄₁	—	16	1	no	—
62	C₂×C₂₄₁	#449	1	2	yes	—
63	C₂×C₂₄₁	#453	1	1	yes	characteristic
64	D₂₄₁	—	8	2	no	—
65	C₂₄₁	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	241	1	no	maximal, Sylow
67	#555	—	241	1	no	—
68	#492	D₂₄₁	1	1	yes	characteristic, complemented
69	#459	—	241	2	no	—
70	#333	—	241	1	no	—
71	D₄⋊C₈	—	241	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	482	2	no	—
74	C₄⋊Q₈	C₂×D₂₄₁	1	1	yes	characteristic
75	#127	C₂×D₂₄₁	1	1	yes	characteristic
76	#124	—	482	2	no	—
77	#121	—	482	2	no	—
78	C₄×D₄	—	241	2	no	—
79	#117	—	482	2	no	—
80	C₄⋊C₈	—	241	2	no	—
81	#102	—	482	2	no	—
82	C₄×C₈	C₂×D₂₄₁	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₂₄₁	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	482	2	no	—
87	C₂×D₄	—	241	2	no	—
88	C₂²×C₄	—	482	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	964	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	482	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	241	2	no	—
95	C₂²⋊C₄	—	482	4	no	—
96	C₂²⋊C₄	—	482	2	no	—
97	C₄²	—	482	2	no	—
98	C₄²	C₂²×D₂₄₁	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	482	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	1928	2	no	—
106	D₄	—	482	2	no	—
107	D₄	—	482	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	964	2	no	—
110	C₂×C₄	—	964	2	no	—
111	C₂×C₄	—	482	2	no	—
112	C₂×C₄	D₄×D₂₄₁	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	964	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	964	2	no	—
119	C₂²	—	964	2	no	—
120	C₂²	—	964	2	no	—
121	C₂²	#21440610	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	964	2	no	—
125	C₄	#21440663	1	2	yes	—
126	C₄	#21440669	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1928	2	no	—
129	C₂	#29507861	1	2	yes	—
130	C₂	#29507884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29508433

C₂

yes

maximal, characteristic, complemented

#29507909

C₂

yes

maximal, complemented

#29507466

C₂

yes

maximal, characteristic, complemented

#29507446

C₂

yes

maximal, complemented

#492×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₂×#16838755

—

#21440631

—

#21440617

—

C₄×D₉₆₄

C₂²

yes

—

#21440600

—

C₄⋊#14515349

C₂²

yes

—

#21440563

—

C₂×SD₁₆×C₂₄₁

—

C₄⋊Q₈×C₂₄₁

C₂²

yes

characteristic

#127×C₂₄₁

C₂²

yes

characteristic

C₄×C₈×C₂₄₁

C₂²

yes

characteristic

C₂×D₉₆₄

—

C₂×D₉₆₄

D₄

yes

—

C₂×C₄×D₂₄₁

—

#16838755

—

C₂×#14515349

—

C₄⋊Dic₂₄₁

D₄

yes

—

#16838747

—

#16838747

—

C₄×Dic₂₄₁

—

C₂×Q₈×C₂₄₁

D₄

yes

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

—

SD₁₆×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

C₂³

yes

characteristic

C₂²×D₂₄₁

—

D₉₆₄

—

D₉₆₄

—

D₉₆₄

—

C₂×Dic₂₄₁

—

C₄×D₂₄₁

—

C₄×D₂₄₁

—

#14515349

—

C₂³×C₂₄₁

—

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₈×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

#120

yes

characteristic

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

#135

yes

—

C₄×C₂₄₁

AGL(1,ℤ/8)

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#449

yes

—

C₂×C₂₄₁

#453

yes

characteristic

D₂₄₁

—

C₂₄₁

#1624

yes

characteristic, complemented, Sylow

#1624

—

241

maximal, Sylow

#555

—

241

—

#492

D₂₄₁

yes

characteristic, complemented

#459

—

241

—

#333

—

241

—

D₄⋊C₈

—

241

—

C₂×SD₁₆

—

C₂×D₈

—

482

—

C₄⋊Q₈

C₂×D₂₄₁

yes

characteristic

#127

C₂×D₂₄₁

yes

characteristic

#124

—

482

—

#121

—

482

—

C₄×D₄

—

241

—

#117

—

482

—

C₄⋊C₈

—

241

—

#102

—

482

—

C₄×C₈

C₂×D₂₄₁

yes

characteristic

C₂×Q₈

C₂²⋊D₂₄₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

482

—

C₂×D₄

—

241

—

C₂²×C₄

—

482

—

SD₁₆

—

D₈

—

964

—

C₂×C₈

—

C₂×C₈

—

482

—

C₄⋊C₄

—

C₄⋊C₄

—

241

—

C₂²⋊C₄

—

482

—

C₂²⋊C₄

—

482

—

C₄²

—

482

—

C₄²

C₂²×D₂₄₁

yes

characteristic

C₂³

—

100

C₂³

—

482

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

1928

—

106

D₄

—

482

—

107

D₄

—

482

—

108

C₂×C₄

—

109

C₂×C₄

—

964

—

110

C₂×C₄

—

964

—

111

C₂×C₄

—

482

—

112

C₂×C₄

D₄×D₂₄₁

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

characteristic

114

C₈

—

115

C₈

—

964

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

964

—

119

C₂²

—

964

—

120

C₂²

—

964

—

121

C₂²

#21440610

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

964

—

125

C₄

#21440663

yes

—

126

C₄

#21440669

yes

—

127

C₂

—

128

C₂

—

1928

—

129

C₂

#29507861

yes

—

130

C₂

#29507884

yes

characteristic

	subgroup	G/H
center	C₂²	#21440610
commutator, cosocle	C₄²×C₂₄₁	C₂³
Frattini	C₄²	C₂²×D₂₄₁
Fitting	#492×C₂₄₁	C₂
socle	C₂²×C₂₄₁	#120
hypercenter	#492	D₂₄₁
hypocenter	C₂₄₁	#1624
LCS	G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#21440610

commutator, cosocle

C₄²×C₂₄₁

C₂³

Frattini

C₄²

C₂²×D₂₄₁

Fitting

#492×C₂₄₁

C₂

C₂²×C₂₄₁

D₂₄₁

C₂₄₁

G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #29508433	C₂	semidirect
2. #29507909	C₂	semidirect
3. #29507466	C₂	semidirect
4. #29507446	C₂	semidirect
5. #492×C₂₄₁	C₂	semidirect
65. C₂₄₁	#1624	semidirect
68. #492	D₂₄₁	semidirect

H ≅ G/N

type

1. #29508433

C₂

semidirect

2. #29507909

C₂

semidirect

3. #29507466

C₂

semidirect

4. #29507446

C₂

semidirect

5. #492×C₂₄₁

C₂

semidirect

65. C₂₄₁

#1624

semidirect

68. #492

D₂₄₁

semidirect

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

subgroup	index
#29508433	2
#29507909	2
#29507466	2
#29507446	2
#492×C₂₄₁	2
#1624	241

subgroup

index

#492×C₂₄₁

241

Subgroups — #44506526

Conjugacy classes of subgroups

Complements

Maximal subgroups