Group #49169317 — subgroups

Subgroups — #49169317

Order 37504.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#34036757	C₂	1	2	yes	maximal
2	#34036537	C₂	1	4	yes	maximal, complemented
3	#34036529	C₂	1	2	yes	maximal, complemented
4	#34036526	C₂	1	2	yes	maximal, complemented
5	#34036172	C₂	1	2	yes	maximal
6	C₂×#22627556	C₂	1	1	yes	maximal, characteristic
7	#34036084	C₂	1	1	yes	maximal, characteristic
8	#427×C₂₉₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#17989786	C₂²	1	1	yes	characteristic
10	#22627624	—	2	2	no	—
11	#22627623	C₂²	1	4	yes	—
12	#22627616	—	2	2	no	—
13	#22627615	C₂²	1	4	yes	—
14	#22627601	—	2	4	no	—
15	#22627583	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
17	#22627557	C₂²	1	4	yes	—
18	#22627556	C₂²	1	2	yes	—
19	#22627556	C₂²	1	2	yes	—
20	#22627555	C₂²	1	4	yes	—
21	#22627534	C₂²	1	2	yes	—
22	C₈×Dic₂₉₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
24	#117×C₂₉₃	C₂²	1	2	yes	—
25	#107×C₂₉₃	C₂²	1	2	yes	—
26	#106×C₂₉₃	C₂²	1	1	yes	characteristic
27	#106×C₂₉₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₅₈₆	—	2	4	no	—
29	C₂²×Dic₂₉₃	—	2	2	no	—
30	#17989786	Q₈	1	4	yes	complemented
31	C₂×#14556873	C₂³	1	2	yes	—
32	C₄⋊Dic₂₉₃	—	2	4	no	—
33	C₄⋊Dic₂₉₃	—	2	2	no	—
34	C₄⋊Dic₂₉₃	D₄	1	2	yes	—
35	C₄⋊Dic₂₉₃	C₂³	1	4	yes	—
36	#17989781	—	2	4	no	—
37	#17989781	—	2	4	no	—
38	C₂²⋊Dic₂₉₃	D₄	1	2	yes	—
39	C₄×Dic₂₉₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₂₉₃	Q₈	1	4	yes	complemented
42	C₂×C₈×C₂₉₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₂₉₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₉₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₉₃	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₂₉₃	—	2	2	no	—
47	C₄²×C₂₉₃	—	2	2	no	—
48	Dic₅₈₆	—	8	4	no	—
49	C₂×Dic₂₉₃	—	2	4	no	—
50	C₂×Dic₂₉₃	—	8	2	no	—
51	C₂×Dic₂₉₃	C₂×D₄	1	4	yes	—
52	#14556873	C₂×Q₈	1	4	yes	—
53	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉₃	—	2	2	no	—
55	C₂×C₄×C₂₉₃	—	2	2	no	—
56	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
60	C₈×C₂₉₃	C₂×Q₈	1	2	yes	—
61	C₈×C₂₉₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₂₉₃	—	2	2	no	—
63	C₂²×C₂₉₃	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₂₉₃	—	4	2	no	—
66	C₄×C₂₉₃	—	4	2	no	—
67	C₄×C₂₉₃	C₄⋊Q₈	1	2	yes	—
68	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₂₉₃	—	2	4	no	—
71	Dic₂₉₃	—	4	4	no	—
72	C₂×C₂₉₃	—	2	2	no	—
73	C₂×C₂₉₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₂₉₃	#577	1	2	yes	—
75	C₂₉₃	#3163	1	1	yes	characteristic, complemented, Sylow
76	#3163	—	293	1	no	maximal, Sylow
77	#544	—	293	2	no	—
78	#501	—	293	4	no	—
79	#499	—	293	2	no	—
80	#498	—	293	2	no	—
81	#428	—	293	2	no	—
82	#427	D₂₉₃	1	1	yes	characteristic
83	C₂×#106	—	293	1	no	—
84	#405	—	293	1	no	—
85	C₂×M₁₆	—	293	1	no	—
86	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
87	C₄⋊Q₈	—	293	4	no	—
88	C₄⋊Q₈	—	586	2	no	—
89	#125	—	293	4	no	—
90	#125	—	586	2	no	—
91	C₂²⋊Q₈	—	586	4	no	—
92	#117	—	293	1	no	—
93	#117	C₂×D₂₉₃	1	2	yes	—
94	C₂×C₄⋊C₄	—	293	2	no	—
95	#107	—	293	4	no	—
96	#107	C₂×D₂₉₃	1	2	yes	—
97	#106	—	293	2	no	—
98	#106	—	293	2	no	—
99	#106	—	293	4	no	—
100	#106	C₂×D₂₉₃	1	1	yes	characteristic
101	#106	C₂×D₂₉₃	1	1	yes	characteristic
102	#97	—	293	2	no	—
103	C₄×C₈	—	293	2	no	—
104	C₂×Q₈	—	586	4	no	—
105	C₂²×C₄	—	586	2	no	—
106	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
107	M₁₆	—	293	4	no	—
108	C₂×C₈	—	293	2	no	—
109	C₂×C₈	Dic₅₈₆	1	4	yes	complemented
110	C₂×C₈	C₂²×D₂₉₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₂₉₃	1	1	yes	characteristic
112	C₄⋊C₄	—	293	2	no	—
113	C₄⋊C₄	—	293	4	no	—
114	C₄⋊C₄	—	586	4	no	—
115	C₄⋊C₄	—	586	4	no	—
116	C₄⋊C₄	—	586	4	no	—
117	C₄⋊C₄	—	586	2	no	—
118	C₄⋊C₄	C₂²×D₂₉₃	1	2	yes	—
119	C₄⋊C₄	C₂²×D₂₉₃	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	293	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	293	2	no	—
124	C₂³	D₄×D₂₉₃	1	1	yes	characteristic
125	Q₈	—	2344	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	293	4	no	—
129	C₂×C₄	—	2344	2	no	—
130	C₂×C₄	—	586	4	no	—
131	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₂₉₃	1	2	yes	—
133	C₂×C₄	C₂×Dic₅₈₆	1	2	yes	—
134	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
135	C₈	—	293	4	no	—
136	C₈	C₂×Dic₅₈₆	1	2	yes	—
137	C₈	C₂×Dic₅₈₆	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#22627623	1	2	yes	—
140	C₂²	C₂×D₄×D₂₉₃	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	586	4	no	—
144	C₄	—	1172	4	no	—
145	C₄	#22627623	1	2	yes	—
146	C₄	C₂×Q₈×D₂₉₃	1	1	yes	characteristic
147	C₄	C₂²×Dic₅₈₆	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#22627623	1	1	yes	characteristic
150	C₂	#34036936	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#34036757

C₂

yes

maximal

#34036537

C₂

yes

maximal, complemented

#34036529

C₂

yes

maximal, complemented

#34036526

C₂

yes

maximal, complemented

#34036172

C₂

yes

maximal

C₂×#22627556

C₂

yes

maximal, characteristic

#34036084

C₂

yes

maximal, characteristic

#427×C₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989786

C₂²

yes

characteristic

#22627624

—

#22627623

C₂²

yes

—

#22627616

—

#22627615

C₂²

yes

—

#22627601

—

#22627583

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

#22627557

C₂²

yes

—

#22627556

C₂²

yes

—

#22627556

C₂²

yes

—

#22627555

C₂²

yes

—

#22627534

C₂²

yes

—

C₈×Dic₂₉₃

C₂²

yes

—

C₂²×C₈×C₂₉₃

C₂²

yes

characteristic

#117×C₂₉₃

C₂²

yes

—

#107×C₂₉₃

C₂²

yes

—

#106×C₂₉₃

C₂²

yes

characteristic

#106×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂²×Dic₂₉₃

—

#17989786

Q₈

yes

complemented

C₂×#14556873

C₂³

yes

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

—

C₄⋊Dic₂₉₃

C₂³

yes

—

#17989781

—

#17989781

—

C₂²⋊Dic₂₉₃

D₄

yes

—

C₄×Dic₂₉₃

C₂³

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

Q₈

yes

complemented

C₂×C₈×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₉₃

C₂³

yes

—

C₄⋊C₄×C₂₉₃

C₂³

yes

—

C₂²⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂×D₄

yes

—

#14556873

C₂×Q₈

yes

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

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

C₈×C₂₉₃

C₂×Q₈

yes

—

C₈×C₂₉₃

C₂×Q₈

yes

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₄⋊Q₈

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

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉₃

#577

yes

—

C₂₉₃

#3163

yes

characteristic, complemented, Sylow

#3163

—

293

maximal, Sylow

#544

—

293

—

#501

—

293

—

#499

—

293

—

#498

—

293

—

#428

—

293

—

#427

D₂₉₃

yes

characteristic

C₂×#106

—

293

—

#405

—

293

—

C₂×M₁₆

—

293

—

C₂²×C₈

C₂×D₂₉₃

yes

characteristic

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

#125

—

293

—

#125

—

586

—

C₂²⋊Q₈

—

586

—

#117

—

293

—

#117

C₂×D₂₉₃

yes

—

C₂×C₄⋊C₄

—

293

—

#107

—

293

—

#107

C₂×D₂₉₃

yes

—

#106

—

293

—

#106

—

293

—

#106

—

293

—

100

#106

C₂×D₂₉₃

yes

characteristic

101

#106

C₂×D₂₉₃

yes

characteristic

102

#97

—

293

—

103

C₄×C₈

—

293

—

104

C₂×Q₈

—

586

—

105

C₂²×C₄

—

586

—

106

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

107

M₁₆

—

293

—

108

C₂×C₈

—

293

—

109

C₂×C₈

Dic₅₈₆

yes

complemented

110

C₂×C₈

C₂²×D₂₉₃

yes

characteristic

111

C₂×C₈

C₂²×D₂₉₃

yes

characteristic

112

C₄⋊C₄

—

293

—

113

C₄⋊C₄

—

293

—

114

C₄⋊C₄

—

586

—

115

C₄⋊C₄

—

586

—

116

C₄⋊C₄

—

586

—

117

C₄⋊C₄

—

586

—

118

C₄⋊C₄

C₂²×D₂₉₃

yes

—

119

C₄⋊C₄

C₂²×D₂₉₃

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

293

—

122

C₄²

—

123

C₄²

—

293

—

124

C₂³

D₄×D₂₉₃

yes

characteristic

125

Q₈

—

2344

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

293

—

129

C₂×C₄

—

2344

—

130

C₂×C₄

—

586

—

131

C₂×C₄

D₄×D₂₉₃

yes

characteristic

132

C₂×C₄

Q₈×D₂₉₃

yes

—

133

C₂×C₄

C₂×Dic₅₈₆

yes

—

134

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

135

C₈

—

293

—

136

C₈

C₂×Dic₅₈₆

yes

—

137

C₈

C₂×Dic₅₈₆

yes

—

138

C₂²

—

139

C₂²

#22627623

yes

—

140

C₂²

C₂×D₄×D₂₉₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

586

—

144

C₄

—

1172

—

145

C₄

#22627623

yes

—

146

C₄

C₂×Q₈×D₂₉₃

yes

characteristic

147

C₄

C₂²×Dic₅₈₆

yes

characteristic

148

C₂

—

149

C₂

C₂×#22627623

yes

characteristic

150

C₂

#34036936

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₉₃
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	#427×C₂₉₃	C₂
socle	C₂²×C₂₉₃	C₂²×D₄
prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	#427	D₂₉₃
hypocenter	C₂₉₃	#3163
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₂₉₃

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

#427×C₂₉₃

C₂

socle

C₂²×C₂₉₃

C₂²×D₄

prime radical

C₂³×C₂₉₃

C₂×D₄

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #34036537	C₂	semidirect
3. #34036529	C₂	semidirect
4. #34036526	C₂	semidirect
30. #17989786	Q₈	semidirect
41. C₂×C₈×C₂₉₃	Q₈	semidirect
75. C₂₉₃	#3163	semidirect
109. C₂×C₈	Dic₅₈₆	semidirect

H ≅ G/N

type

2. #34036537

C₂

semidirect

3. #34036529

C₂

semidirect

4. #34036526

C₂

semidirect

30. #17989786

Q₈

semidirect

41. C₂×C₈×C₂₉₃

Q₈

semidirect

75. C₂₉₃

#3163

semidirect

109. C₂×C₈

Dic₅₈₆

semidirect

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

subgroup	index
#34036757	2
#34036537	2
#34036529	2
#34036526	2
#34036172	2
C₂×#22627556	2
#34036084	2
#427×C₂₉₃	2
#3163	293

subgroup

index

#427×C₂₉₃

293

Subgroups — #49169317

Conjugacy classes of subgroups

Complements

Maximal subgroups