Group #49158684 — subgroups

Subgroups — #49158684

Order 37504.

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

normal subgroups only

This group has 272 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×#22627556	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627534	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₂₉₃	C₂	1	2	yes	maximal
4	#34035891	C₂	1	2	yes	maximal
5	C₂×#107×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14556873	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₉₃	—	2	2	no	—
9	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
10	#22627556	C₄	1	4	yes	complemented
11	#22627534	C₄	1	4	yes	complemented
12	#22627531	—	2	2	no	—
13	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
15	#107×C₂₉₃	C₄	1	4	yes	complemented
16	C₂²×Dic₂₉₃	—	2	2	no	—
17	C₂²×Dic₂₉₃	D₄	1	2	yes	—
18	C₂×#14556873	—	2	1	no	—
19	C₂×#14556873	—	2	1	no	—
20	C₂×#14556873	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₂₉₃	—	2	2	no	—
22	C₄⋊Dic₂₉₃	C₂×C₄	1	4	yes	—
23	C₄×Dic₂₉₃	—	4	4	no	—
24	C₄×Dic₂₉₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₂₉₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₉₃	—	2	2	no	—
27	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₉₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₉₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₉₃	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₂₉₃	—	2	2	no	—
32	C₄⋊C₄×C₂₉₃	C₂×C₄	1	4	yes	—
33	C₂×Dic₂₉₃	—	2	2	no	—
34	C₂×Dic₂₉₃	—	2	2	no	—
35	C₂×Dic₂₉₃	—	2	4	no	—
36	C₂×Dic₂₉₃	—	4	2	no	—
37	C₂×Dic₂₉₃	—	4	2	no	—
38	C₂×Dic₂₉₃	C₄⋊C₄	1	4	yes	—
39	#14556873	—	2	2	no	—
40	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉₃	—	2	4	no	—
42	C₂×C₄×C₂₉₃	—	4	2	no	—
43	C₂×C₄×C₂₉₃	—	4	2	no	—
44	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
50	C₈×C₂₉₃	C₄²	1	4	yes	complemented
51	C₂²×C₂₉₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₉₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₉₃	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₂₉₃	#137	1	2	yes	—
56	C₄×C₂₉₃	—	4	4	no	—
57	C₄×C₂₉₃	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₂₉₃	#117	1	1	yes	characteristic
59	C₄×C₂₉₃	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₂₉₃	C₄×Q₈	1	1	yes	characteristic
61	Dic₂₉₃	—	2	4	no	—
62	Dic₂₉₃	—	4	4	no	—
63	C₂×C₂₉₃	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₂₉₃	#428	1	2	yes	—
65	C₂×C₂₉₃	#440	1	2	yes	—
66	C₂×C₂₉₃	#441	1	1	yes	characteristic
67	C₂×C₂₉₃	#442	1	1	yes	characteristic
68	C₂₉₃	#1776	1	1	yes	characteristic, complemented, Sylow
69	#1776	—	293	1	no	maximal, Sylow
70	C₂×#107	D₂₉₃	1	1	yes	characteristic
71	C₂×#106	—	293	1	no	—
72	C₂×#97	—	293	1	no	—
73	C₄×C₄⋊C₄	—	293	2	no	—
74	#340	—	293	2	no	—
75	C₂²×C₈	—	293	1	no	—
76	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	293	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
79	C₂×C₄²	—	293	1	no	—
80	C₂×C₄²	—	586	2	no	—
81	#107	Dic₂₉₃	1	4	yes	complemented
82	#106	—	293	4	no	—
83	#97	—	293	4	no	—
84	#95	—	586	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	293	2	no	—
87	C₂²×C₄	—	586	2	no	—
88	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
89	C₂×C₈	—	293	2	no	—
90	C₂×C₈	—	586	1	no	—
91	C₂×C₈	—	586	1	no	—
92	C₂×C₈	C₄×D₂₉₃	1	2	yes	—
93	C₂×C₈	C₄×D₂₉₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₂₉₃	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	293	4	no	—
97	C₄⋊C₄	—	586	2	no	—
98	C₄⋊C₄	C₂×Dic₂₉₃	1	4	yes	—
99	C₄²	—	293	2	no	—
100	C₄²	—	293	2	no	—
101	C₄²	—	1172	4	no	—
102	C₂³	D₄×D₂₉₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	293	4	no	—
107	C₂×C₄	—	586	2	no	—
108	C₂×C₄	—	586	2	no	—
109	C₂×C₄	—	586	4	no	—
110	C₂×C₄	—	1172	2	no	—
111	C₂×C₄	—	1172	2	no	—
112	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₂₉₃	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₂₉₃	1	1	yes	characteristic
116	C₂×C₄	#17989801	1	1	yes	characteristic
117	C₂×C₄	#17989803	1	1	yes	characteristic
118	C₈	—	586	2	no	—
119	C₈	C₄×Dic₂₉₃	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₂₉₃	1	1	yes	characteristic
121	C₂²	#22627586	1	1	yes	characteristic
122	C₂²	D₄×Dic₂₉₃	1	1	yes	characteristic
123	C₂²	#22627648	1	2	yes	—
124	C₂²	#22627656	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	586	4	no	—
127	C₄	—	1172	4	no	—
128	C₄	C₂×C₄×Dic₂₉₃	1	1	yes	characteristic
129	C₄	#22627582	1	1	yes	characteristic
130	C₄	#22627589	1	1	yes	characteristic
131	C₄	Q₈×Dic₂₉₃	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₂₉₃	1	1	yes	characteristic
133	C₂	#34036170	1	2	yes	—
134	C₂	#34036228	1	2	yes	—
135	C₂	#34036236	1	1	yes	characteristic
136	C₂	#34036241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627556

C₂

yes

maximal, characteristic

C₂×#22627534

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₂₉₃

C₂

yes

maximal

#34035891

C₂

yes

maximal

C₂×#107×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×#14556873

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×C₄×Dic₂₉₃

—

C₂×C₄×Dic₂₉₃

C₂²

yes

characteristic

#22627556

C₄

yes

complemented

#22627534

C₄

yes

complemented

#22627531

—

C₂²×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉₃

C₂²

yes

—

#107×C₂₉₃

C₄

yes

complemented

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

D₄

yes

—

C₂×#14556873

—

C₂×#14556873

—

C₂×#14556873

C₂×C₄

yes

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

C₂×C₄

yes

—

C₄×Dic₂₉₃

Q₈

yes

—

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

C₂×C₄

yes

—

C₂×C₈×C₂₉₃

C₂×C₄

yes

—

C₂×C₈×C₂₉₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

C₂×C₄

yes

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₄⋊C₄

yes

—

#14556873

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×Q₈

yes

C₂×C₄×C₂₉₃

yes

C₂×C₄×C₂₉₃

yes

C₈×C₂₉₃

C₄²

yes

complemented

C₂²×C₂₉₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₉₃

C₄×D₄

yes

characteristic

C₂²×C₂₉₃

C₄×D₄

yes

characteristic

C₂²×C₂₉₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₉₃

#137

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×C₄²

yes

characteristic

C₄×C₂₉₃

#117

yes

characteristic

C₄×C₂₉₃

C₄×Q₈

yes

characteristic

C₄×C₂₉₃

C₄×Q₈

yes

characteristic

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₉₃

#428

yes

—

C₂×C₂₉₃

#440

yes

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#1776

—

293

maximal, Sylow

C₂×#107

D₂₉₃

yes

characteristic

C₂×#106

—

293

—

C₂×#97

—

293

—

C₄×C₄⋊C₄

—

293

—

#340

—

293

—

C₂²×C₈

—

293

—

C₂²×C₈

C₂×D₂₉₃

yes

characteristic

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

C₂×D₂₉₃

yes

—

C₂×C₄²

—

293

—

C₂×C₄²

—

586

—

#107

Dic₂₉₃

yes

complemented

#106

—

293

—

#97

—

293

—

#95

—

586

—

C₂²×C₄

—

C₂²×C₄

—

293

—

C₂²×C₄

—

586

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₂×C₈

C₄×D₂₉₃

yes

—

C₂×C₈

C₄×D₂₉₃

yes

—

C₂×C₈

C₂×Dic₂₉₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

C₂×Dic₂₉₃

yes

—

C₄²

—

293

—

100

C₄²

—

293

—

101

C₄²

—

1172

—

102

C₂³

D₄×D₂₉₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

293

—

107

C₂×C₄

—

586

—

108

C₂×C₄

—

586

—

109

C₂×C₄

—

586

—

110

C₂×C₄

—

1172

—

111

C₂×C₄

—

1172

—

112

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

113

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

114

C₂×C₄

C₂²×Dic₂₉₃

yes

characteristic

115

C₂×C₄

Q₈×D₂₉₃

yes

characteristic

116

C₂×C₄

#17989801

yes

characteristic

117

C₂×C₄

#17989803

yes

characteristic

118

C₈

—

586

—

119

C₈

C₄×Dic₂₉₃

yes

complemented

120

C₂²

C₄⋊C₄×D₂₉₃

yes

characteristic

121

C₂²

#22627586

yes

characteristic

122

C₂²

D₄×Dic₂₉₃

yes

characteristic

123

C₂²

#22627648

yes

—

124

C₂²

#22627656

yes

—

125

C₄

—

126

C₄

—

586

—

127

C₄

—

1172

—

128

C₄

C₂×C₄×Dic₂₉₃

yes

characteristic

129

C₄

#22627582

yes

characteristic

130

C₄

#22627589

yes

characteristic

131

C₄

Q₈×Dic₂₉₃

yes

characteristic

132

C₂

C₄⋊C₄×Dic₂₉₃

yes

characteristic

133

C₂

#34036170

yes

—

134

C₂

#34036228

yes

—

135

C₂

#34036236

yes

characteristic

136

C₂

#34036241

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₉₃
commutator	C₄×C₂₉₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	C₂×#107×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
cosocle	C₂²×C₄×C₂₉₃	C₂³
hypercenter	C₂×#107	D₂₉₃
hypocenter	C₂₉₃	#1776
LCS	G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₉₃

commutator

C₄×C₂₉₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

C₂×#107×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

cosocle

C₂²×C₄×C₂₉₃

C₂³

D₂₉₃

C₂₉₃

G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #22627556	C₄	semidirect
11. #22627534	C₄	semidirect
15. #107×C₂₉₃	C₄	semidirect
50. C₈×C₂₉₃	C₄²	semidirect
68. C₂₉₃	#1776	semidirect
81. #107	Dic₂₉₃	semidirect
119. C₈	C₄×Dic₂₉₃	semidirect

H ≅ G/N

type

10. #22627556

C₄

semidirect

11. #22627534

C₄

semidirect

15. #107×C₂₉₃

C₄

semidirect

50. C₈×C₂₉₃

C₄²

semidirect

68. C₂₉₃

#1776

semidirect

81. #107

Dic₂₉₃

semidirect

119. C₈

C₄×Dic₂₉₃

semidirect

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

subgroup	index
C₂×#22627556	2
C₂×#22627534	2
C₄⋊C₄×Dic₂₉₃	2
#34035891	2
C₂×#107×C₂₉₃	2
#1776	293

subgroup

index

C₄⋊C₄×Dic₂₉₃

C₂×#107×C₂₉₃

293

Subgroups — #49158684

Conjugacy classes of subgroups

Complements

Maximal subgroups