Group #44507572 — subgroups

Subgroups — #44507572

Order 30848.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×#14515349	C₂	1	1	yes	maximal, characteristic
2	#29507587	C₂	1	1	yes	maximal, characteristic
3	#29507479	C₂	1	4	yes	maximal
4	#384×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14515349	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₄₁	—	2	1	no	—
7	C₂×C₄⋊Dic₂₄₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₄₁	C₂²	1	2	yes	—
9	C₄×#14515349	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₂₄₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₂₄₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₄₁	—	2	2	no	—
15	C₂×#14515349	—	2	2	no	—
16	C₂×#14515349	—	2	2	no	—
17	C₂×#14515349	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₂₄₁	—	2	4	no	—
19	C₄⋊Dic₂₄₁	—	4	2	no	—
20	C₄⋊Dic₂₄₁	—	4	2	no	—
21	C₂²×C₄×C₂₄₁	—	2	2	no	—
22	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₂₄₁	—	2	4	no	—
26	C₄⋊C₄×C₂₄₁	—	4	2	no	—
27	C₄⋊C₄×C₂₄₁	—	4	2	no	—
28	C₄²×C₂₄₁	C₂×C₄	1	2	yes	—
29	C₄²×C₂₄₁	Q₈	1	2	yes	—
30	C₂×Dic₂₄₁	—	4	2	no	—
31	C₂×Dic₂₄₁	—	4	2	no	—
32	C₂×Dic₂₄₁	—	4	2	no	—
33	#14515349	—	2	4	no	—
34	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₄₁	—	4	2	no	—
36	C₂×C₄×C₂₄₁	—	4	2	no	—
37	C₂×C₄×C₂₄₁	—	4	2	no	—
38	C₂×C₄×C₂₄₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₄₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₄₁	D₈	1	2	yes	—
41	C₂×C₄×C₂₄₁	SD₁₆	1	2	yes	—
42	C₂×C₄×C₂₄₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₄₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₄₁	#41	1	2	yes	—
47	C₂×C₄×C₂₄₁	#41	1	2	yes	—
48	C₂²×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₄₁	#124	1	1	yes	characteristic
50	C₂²×C₂₄₁	#124	1	1	yes	characteristic
51	C₂²×C₂₄₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₂₄₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₂₄₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₂₄₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₂₄₁	—	8	2	no	—
56	C₄×C₂₄₁	#106	1	4	yes	—
57	C₄×C₂₄₁	#107	1	4	yes	—
58	C₄×C₂₄₁	#117	1	2	yes	—
59	C₄×C₂₄₁	#125	1	2	yes	—
60	Dic₂₄₁	—	8	2	no	—
61	C₂×C₂₄₁	#382	1	1	yes	characteristic
62	C₂×C₂₄₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₂₄₁	C₂×#107	1	1	yes	characteristic
64	C₂×C₂₄₁	#486	1	1	yes	characteristic
65	C₂×C₂₄₁	#486	1	1	yes	characteristic
66	C₂×C₂₄₁	#487	1	1	yes	characteristic
67	C₂×C₂₄₁	#487	1	1	yes	characteristic
68	C₂₄₁	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	241	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	241	1	no	—
71	#384	—	241	1	no	—
72	#384	D₂₄₁	1	1	yes	characteristic
73	#340	—	241	4	no	—
74	C₂²×C₈	—	241	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	482	1	no	—
78	C₂×C₄⋊C₄	—	482	1	no	—
79	C₂×C₄⋊C₄	—	241	2	no	—
80	C₂×C₄⋊C₄	C₂×D₂₄₁	1	2	yes	—
81	C₂×C₄²	C₂×D₂₄₁	1	1	yes	characteristic
82	C₄×C₈	—	241	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	482	2	no	—
85	C₂²×C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
88	C₂×C₈	—	482	2	no	—
89	C₂×C₈	—	482	2	no	—
90	C₂×C₈	—	241	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	964	2	no	—
95	C₄⋊C₄	—	964	2	no	—
96	C₄⋊C₄	—	482	4	no	—
97	C₄²	C₄×D₂₄₁	1	2	yes	—
98	C₄²	Dic₄₈₂	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	964	2	no	—
104	C₂×C₄	—	964	2	no	—
105	C₂×C₄	—	964	2	no	—
106	C₂×C₄	#16838749	1	2	yes	—
107	C₂×C₄	#16838749	1	2	yes	—
108	C₂×C₄	#16838755	1	2	yes	—
109	C₂×C₄	#16838757	1	2	yes	—
110	C₂×C₄	#16838758	1	2	yes	—
111	C₂×C₄	#16838760	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₄₈₂	1	1	yes	characteristic
114	C₂×C₄	#16838769	1	2	yes	—
115	C₂×C₄	#16838771	1	2	yes	—
116	C₈	—	482	4	no	—
117	C₂²	C₂×#16838749	1	1	yes	characteristic
118	C₂²	#21440629	1	1	yes	characteristic
119	C₂²	#21440631	1	1	yes	characteristic
120	C₂²	C₂×#16838755	1	1	yes	characteristic
121	C₂²	C₂×#16838757	1	1	yes	characteristic
122	C₂²	C₂×#16838758	1	1	yes	characteristic
123	C₂²	C₂×#16838760	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1928	2	no	—
126	C₄	#21440573	1	4	yes	—
127	C₄	#21440575	1	4	yes	—
128	C₄	#21440600	1	2	yes	—
129	C₄	#21440633	1	2	yes	—
130	C₂	#29507582	1	1	yes	characteristic
131	C₂	C₂×#21440573	1	1	yes	characteristic
132	C₂	C₂×#21440575	1	1	yes	characteristic
133	C₂	#29508063	1	1	yes	characteristic
134	C₂	#29508065	1	1	yes	characteristic
135	C₂	#29508067	1	1	yes	characteristic
136	C₂	#29508069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#14515349

C₂

yes

maximal, characteristic

#29507587

C₂

yes

maximal, characteristic

#29507479

C₂

yes

maximal

#384×C₂₄₁

C₂

yes

maximal, characteristic

C₂²×#14515349

C₂²

yes

—

C₂×C₄⋊Dic₂₄₁

—

C₂×C₄⋊Dic₂₄₁

—

C₂×C₄⋊Dic₂₄₁

C₂²

yes

—

C₄×#14515349

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₄₁

—

C₂×C₄⋊C₄×C₂₄₁

—

C₂×C₄⋊C₄×C₂₄₁

C₂²

yes

—

C₂×C₄²×C₂₄₁

C₂²

yes

characteristic

C₂²×Dic₂₄₁

—

C₂×#14515349

—

C₂×#14515349

—

C₂×#14515349

C₂×C₄

yes

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₂²×C₄×C₂₄₁

—

C₂²×C₄×C₂₄₁

D₄

yes

characteristic

C₂²×C₄×C₂₄₁

D₄

yes

characteristic

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

C₂×C₄

yes

—

C₄²×C₂₄₁

Q₈

yes

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

#14515349

—

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

D₈

yes

—

C₂×C₄×C₂₄₁

SD₁₆

yes

—

C₂×C₄×C₂₄₁

SD₁₆

yes

—

C₂×C₄×C₂₄₁

Q₁₆

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

—

C₂×C₄×C₂₄₁

#41

yes

—

C₂²×C₂₄₁

C₂×C₄⋊C₄

yes

C₂²×C₂₄₁

yes

C₂²×C₂₄₁

yes

C₂²×C₂₄₁

C₂×D₈

yes

characteristic

C₂²×C₂₄₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₄₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₄₁

C₂×Q₁₆

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

#106

yes

—

C₄×C₂₄₁

#107

yes

—

C₄×C₂₄₁

#117

yes

—

C₄×C₂₄₁

#125

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂₄₁

yes

characteristic, complemented, Sylow

#1834

—

241

maximal, Sylow

C₂×C₄×C₈

—

241

—

#384

—

241

—

#384

D₂₄₁

yes

characteristic

#340

—

241

—

C₂²×C₈

—

241

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

482

—

C₂×C₄⋊C₄

—

482

—

C₂×C₄⋊C₄

—

241

—

C₂×C₄⋊C₄

C₂×D₂₄₁

yes

—

C₂×C₄²

C₂×D₂₄₁

yes

characteristic

C₄×C₈

—

241

—

C₂²×C₄

—

C₂²×C₄

—

482

—

C₂²×C₄

C₂²⋊D₂₄₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₄₁

yes

characteristic

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

C₂×C₈

—

482

—

C₂×C₈

—

482

—

C₂×C₈

—

241

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

964

—

C₄⋊C₄

—

964

—

C₄⋊C₄

—

482

—

C₄²

C₄×D₂₄₁

yes

—

C₄²

Dic₄₈₂

yes

—

C₂³

C₂×C₂²⋊D₂₄₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

964

—

104

C₂×C₄

—

964

—

105

C₂×C₄

—

964

—

106

C₂×C₄

#16838749

yes

—

107

C₂×C₄

#16838749

yes

—

108

C₂×C₄

#16838755

yes

—

109

C₂×C₄

#16838757

yes

—

110

C₂×C₄

#16838758

yes

—

111

C₂×C₄

#16838760

yes

—

112

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

113

C₂×C₄

C₂×Dic₄₈₂

yes

characteristic

114

C₂×C₄

#16838769

yes

—

115

C₂×C₄

#16838771

yes

—

116

C₈

—

482

—

117

C₂²

C₂×#16838749

yes

characteristic

118

C₂²

#21440629

yes

characteristic

119

C₂²

#21440631

yes

characteristic

120

C₂²

C₂×#16838755

yes

characteristic

121

C₂²

C₂×#16838757

yes

characteristic

122

C₂²

C₂×#16838758

yes

characteristic

123

C₂²

C₂×#16838760

yes

characteristic

124

C₄

—

125

C₄

—

1928

—

126

C₄

#21440573

yes

—

127

C₄

#21440575

yes

—

128

C₄

#21440600

yes

—

129

C₄

#21440633

yes

—

130

C₂

#29507582

yes

characteristic

131

C₂

C₂×#21440573

yes

characteristic

132

C₂

C₂×#21440575

yes

characteristic

133

C₂

#29508063

yes

characteristic

134

C₂

#29508065

yes

characteristic

135

C₂

#29508067

yes

characteristic

136

C₂

#29508069

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₄₁
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	#384×C₂₄₁	C₂
socle, prime radical	C₂³×C₂₄₁	C₂×D₄
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	#384	D₂₄₁
hypocenter	C₂₄₁	#1834
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₄₁

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

#384×C₂₄₁

C₂

socle, prime radical

C₂³×C₂₄₁

C₂×D₄

cosocle

C₂²×C₄×C₂₄₁

C₂³

D₂₄₁

C₂₄₁

G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#14515349	C₄	semidirect
68. C₂₄₁	#1834	semidirect

H ≅ G/N

type

9. C₄×#14515349

C₄

semidirect

68. C₂₄₁

#1834

semidirect

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

subgroup	index
C₂×C₄×#14515349	2
#29507587	2
#29507479	2
#384×C₂₄₁	2
#1834	241

subgroup

index

#384×C₂₄₁

241

Subgroups — #44507572

Conjugacy classes of subgroups

Complements

Maximal subgroups