Group #49157469 — subgroups

Subgroups — #49157469

Order 37504.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#22627624	C₂	1	1	yes	maximal, characteristic
2	#34036153	C₂	1	1	yes	maximal, characteristic
3	#34035876	C₂	1	4	yes	maximal, complemented
4	#431×C₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈₆	—	2	1	no	—
6	C₂×#17989786	—	2	1	no	—
7	#22627624	—	2	1	no	—
8	#22627624	—	2	1	no	—
9	#22627624	C₄	1	2	yes	—
10	#22627624	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₉₃	—	2	1	no	—
12	C₂×C₄⋊Dic₂₉₃	C₄	1	2	yes	—
13	C₄⋊#14556873	—	2	1	no	—
14	C₄⋊#14556873	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₉₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₂₉₃	—	2	1	no	—
18	C₄×C₈×C₂₉₃	—	2	1	no	—
19	C₂×Dic₅₈₆	—	2	2	no	—
20	C₂×Dic₅₈₆	—	2	2	no	—
21	C₂×Dic₅₈₆	—	4	4	no	—
22	C₂²×Dic₂₉₃	—	2	2	no	—
23	#17989786	—	4	2	no	—
24	C₂×#14556873	—	2	2	no	—
25	C₄⋊Dic₂₉₃	—	2	2	no	—
26	C₄⋊Dic₂₉₃	—	2	2	no	—
27	C₄⋊Dic₂₉₃	—	2	2	no	—
28	C₄⋊Dic₂₉₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₉₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₉₃	—	2	2	no	—
33	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₉₃	D₄	1	1	yes	characteristic
35	C₄²×C₂₉₃	D₄	1	1	yes	characteristic
36	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
37	Dic₅₈₆	—	4	4	no	—
38	Dic₅₈₆	—	4	4	no	—
39	C₂×Dic₂₉₃	—	2	4	no	—
40	C₂×Dic₂₉₃	—	4	4	no	—
41	#14556873	—	4	2	no	—
42	C₂³×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉₃	—	2	2	no	—
44	C₂×C₄×C₂₉₃	—	2	2	no	—
45	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₉₃	Q₁₆	1	2	yes	—
50	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
53	C₈×C₂₉₃	—	4	2	no	—
54	C₂²×C₂₉₃	—	2	1	no	—
55	C₂²×C₂₉₃	—	2	1	no	—
56	C₂²×C₂₉₃	#103	1	2	yes	—
57	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₉₃	—	2	1	no	—
59	C₄×C₂₉₃	—	2	1	no	—
60	C₄×C₂₉₃	#101	1	2	yes	—
61	C₄×C₂₉₃	#103	1	2	yes	—
62	C₄×C₂₉₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
65	Dic₂₉₃	—	4	4	no	—
66	C₂×C₂₉₃	—	2	2	no	—
67	C₂×C₂₉₃	C₂×#101	1	1	yes	characteristic
68	C₂×C₂₉₃	C₂×#103	1	1	yes	characteristic
69	C₂×C₂₉₃	#418	1	1	yes	characteristic
70	C₂₉₃	#1549	1	1	yes	characteristic, complemented, Sylow
71	#1549	—	293	1	no	maximal, Sylow
72	C₂×C₄⋊Q₈	—	293	1	no	—
73	#431	D₂₉₃	1	1	yes	characteristic
74	#423	—	293	1	no	—
75	#332	—	293	4	no	—
76	C₂²×Q₈	—	586	1	no	—
77	C₂×M₁₆	—	586	1	no	—
78	C₄⋊Q₈	—	293	2	no	—
79	C₄⋊Q₈	—	293	2	no	—
80	C₄⋊Q₈	—	586	1	no	—
81	C₄⋊Q₈	—	586	1	no	—
82	C₂×C₄⋊C₄	—	293	2	no	—
83	C₂×C₄⋊C₄	—	586	1	no	—
84	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
85	C₄⋊C₈	—	293	2	no	—
86	C₄⋊C₈	—	586	1	no	—
87	C₄⋊C₈	C₂×D₂₉₃	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₂×Q₈	—	586	2	no	—
91	C₂×Q₈	—	586	2	no	—
92	C₂×Q₈	—	1172	4	no	—
93	C₂²×C₄	—	586	2	no	—
94	C₂²×C₄	C₄×D₂₉₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₂₉₃	1	1	yes	characteristic
96	C₂²×C₄	D₁₁₇₂	1	1	yes	characteristic
97	M₁₆	—	1172	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	586	2	no	—
100	C₄⋊C₄	—	293	4	no	—
101	C₄⋊C₄	—	586	2	no	—
102	C₄⋊C₄	—	586	2	no	—
103	C₄⋊C₄	—	586	2	no	—
104	C₄²	C₄×D₂₉₃	1	1	yes	characteristic
105	C₄²	C₂²⋊D₂₉₃	1	1	yes	characteristic
106	C₄²	D₁₁₇₂	1	1	yes	characteristic
107	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
108	C₂³	#17989779	1	1	yes	characteristic
109	Q₈	—	1172	4	no	—
110	Q₈	—	1172	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	586	4	no	—
114	C₂×C₄	—	1172	4	no	—
115	C₂×C₄	#17989779	1	1	yes	characteristic
116	C₂×C₄	#17989779	1	1	yes	characteristic
117	C₂×C₄	#17989779	1	1	yes	characteristic
118	C₂×C₄	#17989791	1	2	yes	—
119	C₂×C₄	Dic₁₁₇₂	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	1172	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#22627549	1	2	yes	—
128	C₂²	C₂×#17989779	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	1172	4	no	—
132	C₄	#22627542	1	2	yes	—
133	C₄	#22627549	1	2	yes	—
134	C₄	C₂×#17989791	1	1	yes	characteristic
135	C₄	C₂×Dic₁₁₇₂	1	1	yes	characteristic
136	C₄	#22627649	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#22627542	1	1	yes	characteristic
139	C₂	C₂×#22627549	1	1	yes	characteristic
140	C₂	#34036133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627624

C₂

yes

maximal, characteristic

#34036153

C₂

yes

maximal, characteristic

#34035876

C₂

yes

maximal, complemented

#431×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

—

—

—

—

C₄

yes

—

#22627624

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

C₄

yes

—

C₄⋊#14556873

—

C₄⋊#14556873

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₉₃

C₂²

yes

—

C₂²⋊C₈×C₂₉₃

—

C₄×C₈×C₂₉₃

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂²×Dic₂₉₃

—

#17989786

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉₃

D₄

yes

characteristic

C₂²×C₄×C₂₉₃

D₄

yes

characteristic

C₂×C₈×C₂₉₃

—

C₄²×C₂₉₃

C₂×C₄

yes

characteristic

C₄²×C₂₉₃

D₄

yes

characteristic

C₄²×C₂₉₃

D₄

yes

characteristic

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂³×C₂₉₃

C₂²⋊C₄

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄×C₂₉₃

Q₁₆

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

#103

yes

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#101

yes

—

C₄×C₂₉₃

#103

yes

—

C₄×C₂₉₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₉₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₉₃

AGL(1,ℤ/8)

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#1549

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

#431

D₂₉₃

yes

characteristic

#423

—

293

—

#332

—

293

—

C₂²×Q₈

—

586

—

C₂×M₁₆

—

586

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

C₄⋊C₈

—

293

—

C₄⋊C₈

—

586

—

C₄⋊C₈

C₂×D₂₉₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

1172

—

C₂²×C₄

—

586

—

C₂²×C₄

C₄×D₂₉₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉₃

yes

characteristic

C₂²×C₄

D₁₁₇₂

yes

characteristic

M₁₆

—

1172

—

C₂×C₈

—

C₂×C₈

—

586

—

100

C₄⋊C₄

—

293

—

101

C₄⋊C₄

—

586

—

102

C₄⋊C₄

—

586

—

103

C₄⋊C₄

—

586

—

104

C₄²

C₄×D₂₉₃

yes

characteristic

105

C₄²

C₂²⋊D₂₉₃

yes

characteristic

106

C₄²

D₁₁₇₂

yes

characteristic

107

C₄²

C₂²×D₂₉₃

yes

characteristic

108

C₂³

#17989779

yes

characteristic

109

Q₈

—

1172

—

110

Q₈

—

1172

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

586

—

114

C₂×C₄

—

1172

—

115

C₂×C₄

#17989779

yes

characteristic

116

C₂×C₄

#17989779

yes

characteristic

117

C₂×C₄

#17989779

yes

characteristic

118

C₂×C₄

#17989791

yes

—

119

C₂×C₄

Dic₁₁₇₂

yes

—

120

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

122

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

123

C₈

—

124

C₈

—

1172

—

125

C₂²

—

126

C₂²

—

127

C₂²

#22627549

yes

—

128

C₂²

C₂×#17989779

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1172

—

132

C₄

#22627542

yes

—

133

C₄

#22627549

yes

—

134

C₄

C₂×#17989791

yes

characteristic

135

C₄

C₂×Dic₁₁₇₂

yes

characteristic

136

C₄

#22627649

yes

—

137

C₂

—

138

C₂

C₂×#22627542

yes

characteristic

139

C₂

C₂×#22627549

yes

characteristic

140

C₂

#34036133

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#17989779
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#431×C₂₉₃	C₂
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉₃	C₂²⋊C₄
cosocle	C₄²×C₂₉₃	C₂³
hypercenter	#431	D₂₉₃
hypocenter	C₂₉₃	#1549
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#17989779

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#431×C₂₉₃

C₂

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉₃

C₂²⋊C₄

cosocle

C₄²×C₂₉₃

C₂³

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #34035876	C₂	semidirect
70. C₂₉₃	#1549	semidirect

H ≅ G/N

type

3. #34035876

C₂

semidirect

70. C₂₉₃

#1549

semidirect

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

subgroup	index
C₂×#22627624	2
#34036153	2
#34035876	2
#431×C₂₉₃	2
#1549	293

subgroup

index

#431×C₂₉₃

293

Subgroups — #49157469

Conjugacy classes of subgroups

Complements

Maximal subgroups