Group #49169286 — subgroups

Subgroups — C₂×#34036528 #49169286

Order 37504.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22627616	C₂	1	1	yes	maximal, characteristic
2	#34036528	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22627557	C₂	1	2	yes	maximal
4	C₂×#22627555	C₂	1	2	yes	maximal
5	C₂×C₄×#14556873	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₂₉₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#14556873	C₂²	1	1	yes	characteristic
8	C₂²×#14556873	C₂²	1	1	yes	characteristic
9	#22627616	—	2	2	no	—
10	#22627616	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
12	C₂×#17989781	—	2	2	no	—
13	#22627557	C₂²	1	8	yes	—
14	#22627555	C₂²	1	8	yes	—
15	C₄×#14556873	C₂²	1	4	yes	—
16	#125×C₂₉₃	—	2	2	no	—
17	#125×C₂₉₃	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₂₉₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₂₉₃	—	2	2	no	—
22	C₂×#14556873	Q₈	1	4	yes	—
23	C₂×#14556873	Q₈	1	4	yes	—
24	C₂×#14556873	C₂³	1	2	yes	—
25	C₂×#14556873	C₂³	1	2	yes	—
26	C₄⋊Dic₂₉₃	—	2	2	no	—
27	C₄⋊Dic₂₉₃	C₂³	1	4	yes	—
28	#17989781	—	2	8	no	—
29	C₂²×C₄×C₂₉₃	—	2	2	no	—
30	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
31	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₉₃	—	2	8	no	—
33	C₄⋊C₄×C₂₉₃	—	2	2	no	—
34	C₄⋊C₄×C₂₉₃	C₂³	1	4	yes	—
35	C₄²×C₂₉₃	D₄	1	2	yes	—
36	C₄²×C₂₉₃	C₂³	1	2	yes	—
37	C₂×Dic₂₉₃	—	2	4	no	—
38	C₂×Dic₂₉₃	—	4	4	no	—
39	#14556873	C₂×Q₈	1	4	yes	—
40	#14556873	C₂×Q₈	1	4	yes	—
41	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉₃	—	2	4	no	—
43	C₂×C₄×C₂₉₃	—	2	4	no	—
44	C₂×C₄×C₂₉₃	—	4	4	no	—
45	C₂×C₄×C₂₉₃	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
50	C₂²×C₂₉₃	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₂₉₃	#135	1	4	yes	—
52	C₂²×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₂₉₃	—	2	4	no	—
54	C₄×C₂₉₃	—	4	4	no	—
55	C₄×C₂₉₃	C₄⋊Q₈	1	2	yes	—
56	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
58	Dic₂₉₃	—	4	4	no	—
59	C₂×C₂₉₃	#499	1	4	yes	complemented
60	C₂×C₂₉₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₂₉₃	C₂×#135	1	2	yes	—
62	C₂₉₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
63	C₂×#499	—	293	1	no	maximal, Sylow
64	C₂×#125	—	293	1	no	—
65	C₂×#125	D₂₉₃	1	1	yes	characteristic
66	#499	—	293	8	no	—
67	C₂×#107	—	293	2	no	—
68	C₂×#106	—	293	2	no	—
69	C₂×C₄×C₈	—	293	1	no	—
70	C₂²×C₈	—	293	1	no	—
71	C₂²×C₈	—	293	1	no	—
72	#125	—	2	2	no	—
73	#125	—	293	4	no	—
74	#125	—	586	2	no	—
75	#125	C₂×D₂₉₃	1	4	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	293	2	no	—
78	C₂×C₄⋊C₄	—	586	2	no	—
79	C₂×C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
80	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
81	#107	—	293	8	no	—
82	#106	—	293	8	no	—
83	C₄×C₈	—	293	4	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	586	2	no	—
86	C₂²×C₄	C₂²⋊D₂₉₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
88	C₂×C₈	—	293	4	no	—
89	C₂×C₈	—	293	4	no	—
90	C₂×C₈	—	293	2	no	—
91	C₂×C₈	—	293	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	293	4	no	—
95	C₄⋊C₄	—	586	2	no	—
96	C₄⋊C₄	—	586	8	no	—
97	C₄⋊C₄	C₂²×D₂₉₃	1	4	yes	—
98	C₄²	C₂²⋊D₂₉₃	1	2	yes	—
99	C₄²	C₂²×D₂₉₃	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	586	4	no	—
105	C₂×C₄	—	1172	4	no	—
106	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	4	yes	—
107	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₉₃	1	2	yes	—
109	C₂×C₄	Q₈×D₂₉₃	1	2	yes	—
110	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
111	C₈	—	293	4	no	—
112	C₈	—	293	4	no	—
113	C₂²	#22627622	1	2	yes	—
114	C₂²	#22627644	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₂₉₃	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	1172	4	no	—
119	C₄	#22627622	1	2	yes	—
120	C₄	C₂×Q₈×D₂₉₃	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₂₉₃	1	1	yes	characteristic
122	C₂	#34036528	1	4	yes	direct product factor
123	C₂	C₂×#22627622	1	1	yes	characteristic
124	C₂	C₂×#22627644	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627616

C₂

yes

maximal, characteristic

#34036528

C₂

yes

maximal, direct product factor

C₂×#22627557

C₂

yes

maximal

C₂×#22627555

C₂

yes

maximal

C₂×C₄×#14556873

C₂

yes

maximal, characteristic

C₂×#125×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×#14556873

C₂²

yes

characteristic

C₂²×#14556873

C₂²

yes

characteristic

#22627616

—

#22627616

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×#17989781

—

#22627557

C₂²

yes

—

#22627555

C₂²

yes

—

C₄×#14556873

C₂²

yes

—

#125×C₂₉₃

—

#125×C₂₉₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉₃

—

C₂×C₄⋊C₄×C₂₉₃

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

C₂²×Dic₂₉₃

—

C₂×#14556873

Q₈

yes

—

C₂×#14556873

Q₈

yes

—

C₂×#14556873

C₂³

yes

—

C₂×#14556873

C₂³

yes

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂³

yes

—

#17989781

—

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

C₂³

yes

—

C₄²×C₂₉₃

D₄

yes

—

C₄²×C₂₉₃

C₂³

yes

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

C₂×Q₈

yes

—

#14556873

C₂×Q₈

yes

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×Q₈

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₂²×C₂₉₃

C₄⋊Q₈

yes

—

C₂²×C₂₉₃

#135

yes

—

C₂²×C₂₉₃

C₂²×D₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄⋊Q₈

yes

—

C₄×C₂₉₃

C₂²×Q₈

yes

characteristic

C₄×C₂₉₃

C₂²×Q₈

yes

characteristic

Dic₂₉₃

—

C₂×C₂₉₃

#499

yes

complemented

C₂×C₂₉₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉₃

C₂×#135

yes

—

C₂₉₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

293

maximal, Sylow

C₂×#125

—

293

—

C₂×#125

D₂₉₃

yes

characteristic

#499

—

293

—

C₂×#107

—

293

—

C₂×#106

—

293

—

C₂×C₄×C₈

—

293

—

C₂²×C₈

—

293

—

C₂²×C₈

—

293

—

#125

—

#125

—

293

—

#125

—

586

—

#125

C₂×D₂₉₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄⋊C₄

C₂×D₂₉₃

yes

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#107

—

293

—

#106

—

293

—

C₄×C₈

—

293

—

C₂²×C₄

—

C₂²×C₄

—

586

—

C₂²×C₄

C₂²⋊D₂₉₃

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄⋊C₄

C₂²×D₂₉₃

yes

—

C₄²

C₂²⋊D₂₉₃

yes

—

C₄²

C₂²×D₂₉₃

yes

—

100

C₂³

C₂×C₂²⋊D₂₉₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

586

—

105

C₂×C₄

—

1172

—

106

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

107

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

108

C₂×C₄

Q₈×D₂₉₃

yes

—

109

C₂×C₄

Q₈×D₂₉₃

yes

—

110

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

111

C₈

—

293

—

112

C₈

—

293

—

113

C₂²

#22627622

yes

—

114

C₂²

#22627644

yes

—

115

C₂²

C₂²×C₂²⋊D₂₉₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

1172

—

119

C₄

#22627622

yes

—

120

C₄

C₂×Q₈×D₂₉₃

yes

characteristic

121

C₄

C₂×Q₈×D₂₉₃

yes

characteristic

122

C₂

#34036528

yes

direct product factor

123

C₂

C₂×#22627622

yes

characteristic

124

C₂

C₂×#22627644

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₉₃
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	C₂×#125×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	C₂×#125	D₂₉₃
hypocenter	C₂₉₃	C₂×#499
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₉₃

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

C₂×#125×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #34036528	C₂	direct (N × H)
59. C₂×C₂₉₃	#499	semidirect
62. C₂₉₃	C₂×#499	semidirect
122. C₂	#34036528	direct (N × H)

H ≅ G/N

type

2. #34036528

C₂

direct (N × H)

59. C₂×C₂₉₃

#499

semidirect

62. C₂₉₃

C₂×#499

semidirect

122. C₂

#34036528

direct (N × H)

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

subgroup	index
C₂×#22627616	2
#34036528	2
C₂×#22627557	2
C₂×#22627555	2
C₂×C₄×#14556873	2
C₂×#125×C₂₉₃	2
C₂×#499	293

subgroup

index

C₂×#125×C₂₉₃

293

Subgroups — C₂×#34036528 #49169286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#34036528 #49169286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#34036528 #49169286