Group #49167437 — subgroups

Subgroups — #49167437

Order 37504.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#17989802	C₂	1	1	yes	maximal, characteristic
2	#34036188	C₂	1	4	yes	maximal, complemented
3	#34036184	C₂	1	4	yes	maximal, complemented
4	#34036157	C₂	1	2	yes	maximal, complemented
5	#34036097	C₂	1	2	yes	maximal
6	C₂×#22627535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#17989802	C₂²	1	1	yes	characteristic
9	C₂²×#14556873	C₂²	1	2	yes	—
10	C₄×Dic₅₈₆	C₄	1	2	yes	—
11	C₄×D₁₁₇₂	C₄	1	2	yes	—
12	C₄×C₂²⋊D₂₉₃	C₄	1	4	yes	—
13	#22627583	C₂²	1	1	yes	characteristic
14	#22627581	C₂²	1	2	yes	—
15	C₄²×D₂₉₃	C₂²	1	2	yes	—
16	#22627553	C₂²	1	8	yes	—
17	#22627536	C₂²	1	8	yes	—
18	#22627535	C₂²	1	2	yes	—
19	#22627535	C₂²	1	2	yes	—
20	C₂×M₁₆×C₂₉₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
22	#97×C₂₉₃	C₂²	1	2	yes	—
23	C₄×C₈×C₂₉₃	C₂²	1	2	yes	—
24	#17989802	—	2	4	no	—
25	C₂×Dic₅₈₆	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₁₇₂	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₂₉₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₂₉₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₂₉₃	C₂³	1	2	yes	—
30	C₂×#14556873	—	2	4	no	—
31	C₂×#14556873	C₂³	1	4	yes	—
32	C₄⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
33	#17989781	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
35	#17989779	C₂×C₄	1	4	yes	—
36	C₄×Dic₂₉₃	C₂×C₄	1	4	yes	—
37	C₄×Dic₂₉₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
40	M₁₆×C₂₉₃	—	2	4	no	—
41	C₂×C₈×C₂₉₃	C₂³	1	4	yes	—
42	C₄²×C₂₉₃	C₂×C₄	1	2	yes	—
43	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
44	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
45	C₂²×D₂₉₃	C₂²×C₄	1	2	yes	—
46	Dic₅₈₆	—	2	2	no	—
47	D₁₁₇₂	—	2	2	no	—
48	C₂²⋊D₂₉₃	—	2	4	no	—
49	C₂×Dic₂₉₃	C₂²×C₄	1	2	yes	—
50	C₂×Dic₂₉₃	C₂²×C₄	1	4	yes	—
51	C₄×D₂₉₃	—	2	4	no	—
52	C₄×D₂₉₃	—	4	4	no	—
53	#14556873	—	2	4	no	—
54	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉₃	—	4	2	no	—
56	C₂×C₄×C₂₉₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₉₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₉₃	#41	1	4	yes	—
60	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
61	C₈×C₂₉₃	—	2	4	no	—
62	C₂²×C₂₉₃	—	2	2	no	—
63	C₂²×C₂₉₃	#117	1	2	yes	—
64	C₂²×C₂₉₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₂₉₃	—	2	2	no	—
66	C₂×D₂₉₃	—	2	4	no	—
67	C₄×C₂₉₃	—	2	2	no	—
68	C₄×C₂₉₃	—	2	2	no	—
69	C₄×C₂₉₃	#117	1	2	yes	—
70	C₄×C₂₉₃	C₂×#41	1	2	yes	—
71	Dic₂₉₃	—	2	2	no	—
72	Dic₂₉₃	—	2	4	no	—
73	C₂×C₂₉₃	—	2	2	no	—
74	C₂×C₂₉₃	C₂×#117	1	1	yes	characteristic
75	C₂×C₂₉₃	#568	1	2	yes	—
76	D₂₉₃	—	4	2	no	—
77	C₂₉₃	#2923	1	1	yes	characteristic, complemented, Sylow
78	#2923	—	293	1	no	maximal, Sylow
79	C₄×#41	—	293	1	no	—
80	#433	—	293	4	no	—
81	#432	—	293	4	no	—
82	#424	—	293	2	no	—
83	#408	—	293	2	no	—
84	C₄×M₁₆	D₂₉₃	1	1	yes	characteristic, complemented
85	C₂×#97	—	293	1	no	—
86	C₂×#41	—	293	1	no	—
87	C₂×M₁₆	C₂×D₂₉₃	1	2	yes	—
88	C₂²×C₈	—	293	2	no	—
89	C₄×Q₈	—	293	2	no	—
90	C₄×D₄	—	293	4	no	—
91	C₄×D₄	—	293	2	no	—
92	#117	—	293	1	no	—
93	#117	—	293	2	no	—
94	C₂×C₄²	—	293	2	no	—
95	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
96	C₄⋊C₈	—	293	8	no	—
97	C₂²⋊C₈	—	293	8	no	—
98	#97	—	293	2	no	—
99	#97	—	293	2	no	—
100	#97	C₂×D₂₉₃	1	2	yes	—
101	C₄×C₈	C₂×D₂₉₃	1	2	yes	—
102	#41	—	586	4	no	—
103	C₂×Q₈	—	293	1	no	—
104	C₂×D₄	—	293	2	no	—
105	C₂×D₄	—	293	1	no	—
106	C₂²×C₄	—	293	2	no	—
107	C₂²×C₄	—	293	4	no	—
108	C₂²×C₄	C₄×D₂₉₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	293	4	no	—
112	C₂×C₈	—	586	4	no	—
113	C₂×C₈	C₂²×D₂₉₃	1	4	yes	—
114	C₄⋊C₄	—	293	4	no	—
115	C₄⋊C₄	—	293	2	no	—
116	C₂²⋊C₄	—	293	2	no	—
117	C₂²⋊C₄	—	293	4	no	—
118	C₄²	—	293	4	no	—
119	C₄²	—	293	2	no	—
120	C₄²	C₄×D₂₉₃	1	2	yes	—
121	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
122	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
123	C₂³	—	293	2	no	—
124	C₂³	C₂×C₄×D₂₉₃	1	1	yes	characteristic
125	Q₈	—	586	2	no	—
126	D₄	—	586	4	no	—
127	D₄	—	586	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	293	4	no	—
130	C₂×C₄	—	293	2	no	—
131	C₂×C₄	—	586	4	no	—
132	C₂×C₄	—	1172	4	no	—
133	C₂×C₄	C₂×C₄×D₂₉₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₂₉₃	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
136	C₂×C₄	#17989802	1	4	yes	—
137	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	586	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	586	2	no	—
142	C₂²	—	586	4	no	—
143	C₂²	#22627581	1	2	yes	—
144	C₂²	C₂²×C₄×D₂₉₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	586	2	no	—
148	C₄	—	586	4	no	—
149	C₄	#22627581	1	2	yes	—
150	C₄	C₂×#17989802	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	1172	2	no	—
153	C₂	C₂×#22627581	1	1	yes	characteristic
154	C₂	#34036895	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#17989802

C₂

yes

maximal, characteristic

#34036188

C₂

yes

maximal, complemented

#34036184

C₂

yes

maximal, complemented

#34036157

C₂

yes

maximal, complemented

#34036097

C₂

yes

maximal

C₂×#22627535

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₂₉₃

C₂

yes

maximal, characteristic, complemented

C₂×#17989802

C₂²

yes

characteristic

C₂²×#14556873

C₂²

yes

—

C₄×Dic₅₈₆

C₄

yes

—

C₄×D₁₁₇₂

C₄

yes

—

C₄×C₂²⋊D₂₉₃

C₄

yes

—

#22627583

C₂²

yes

characteristic

#22627581

C₂²

yes

—

C₄²×D₂₉₃

C₂²

yes

—

#22627553

C₂²

yes

—

#22627536

C₂²

yes

—

#22627535

C₂²

yes

—

#22627535

C₂²

yes

—

C₂×M₁₆×C₂₉₃

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

#97×C₂₉₃

C₂²

yes

—

C₄×C₈×C₂₉₃

C₂²

yes

—

#17989802

—

C₂×Dic₅₈₆

C₂×C₄

yes

characteristic

C₂×D₁₁₇₂

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₂₉₃

C₂×C₄

yes

—

C₂×C₄×D₂₉₃

C₂×C₄

yes

—

C₂×C₄×D₂₉₃

C₂³

yes

—

C₂×#14556873

—

C₂×#14556873

C₂³

yes

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

#17989781

C₂×C₄

yes

—

C₂²⋊Dic₂₉₃

C₂×C₄

yes

—

#17989779

C₂×C₄

yes

—

C₄×Dic₂₉₃

C₂×C₄

yes

—

C₄×Dic₂₉₃

C₂³

yes

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

M₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

C₂³

yes

—

C₄²×C₂₉₃

C₂×C₄

yes

—

C₄²×C₂₉₃

C₂³

yes

characteristic

C₄²×C₂₉₃

C₂³

yes

characteristic

C₂²×D₂₉₃

C₂²×C₄

yes

—

Dic₅₈₆

—

D₁₁₇₂

—

C₂²⋊D₂₉₃

—

C₂×Dic₂₉₃

C₂²×C₄

yes

—

C₂×Dic₂₉₃

C₂²×C₄

yes

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

#14556873

—

C₂³×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

#117

yes

—

C₂²×C₂₉₃

C₂³×C₄

yes

characteristic

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#117

yes

—

C₄×C₂₉₃

C₂×#41

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂×#117

yes

characteristic

C₂×C₂₉₃

#568

yes

—

D₂₉₃

—

C₂₉₃

#2923

yes

characteristic, complemented, Sylow

#2923

—

293

maximal, Sylow

C₄×#41

—

293

—

#433

—

293

—

#432

—

293

—

#424

—

293

—

#408

—

293

—

C₄×M₁₆

D₂₉₃

yes

characteristic, complemented

C₂×#97

—

293

—

C₂×#41

—

293

—

C₂×M₁₆

C₂×D₂₉₃

yes

—

C₂²×C₈

—

293

—

C₄×Q₈

—

293

—

C₄×D₄

—

293

—

C₄×D₄

—

293

—

#117

—

293

—

#117

—

293

—

C₂×C₄²

—

293

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

C₄⋊C₈

—

293

—

C₂²⋊C₈

—

293

—

#97

—

293

—

#97

—

293

—

100

#97

C₂×D₂₉₃

yes

—

101

C₄×C₈

C₂×D₂₉₃

yes

—

102

#41

—

586

—

103

C₂×Q₈

—

293

—

104

C₂×D₄

—

293

—

105

C₂×D₄

—

293

—

106

C₂²×C₄

—

293

—

107

C₂²×C₄

—

293

—

108

C₂²×C₄

C₄×D₂₉₃

yes

—

109

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

293

—

112

C₂×C₈

—

586

—

113

C₂×C₈

C₂²×D₂₉₃

yes

—

114

C₄⋊C₄

—

293

—

115

C₄⋊C₄

—

293

—

116

C₂²⋊C₄

—

293

—

117

C₂²⋊C₄

—

293

—

118

C₄²

—

293

—

119

C₄²

—

293

—

120

C₄²

C₄×D₂₉₃

yes

—

121

C₄²

C₂²×D₂₉₃

yes

characteristic

122

C₄²

C₂²×D₂₉₃

yes

characteristic

123

C₂³

—

293

—

124

C₂³

C₂×C₄×D₂₉₃

yes

characteristic

125

Q₈

—

586

—

126

D₄

—

586

—

127

D₄

—

586

—

128

C₂×C₄

—

129

C₂×C₄

—

293

—

130

C₂×C₄

—

293

—

131

C₂×C₄

—

586

—

132

C₂×C₄

—

1172

—

133

C₂×C₄

C₂×C₄×D₂₉₃

yes

—

134

C₂×C₄

C₂×C₄×D₂₉₃

yes

—

135

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

136

C₂×C₄

#17989802

yes

—

137

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

138

C₈

—

139

C₈

—

586

—

140

C₂²

—

141

C₂²

—

586

—

142

C₂²

—

586

—

143

C₂²

#22627581

yes

—

144

C₂²

C₂²×C₄×D₂₉₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

586

—

148

C₄

—

586

—

149

C₄

#22627581

yes

—

150

C₄

C₂×#17989802

yes

—

151

C₂

—

152

C₂

—

1172

—

153

C₂

C₂×#22627581

yes

characteristic

154

C₂

#34036895

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉₃
commutator, socle	C₂²×C₂₉₃	C₂³×C₄
Fitting	C₄×M₁₆×C₂₉₃	C₂
prime radical	C₂×#17989802	C₂²
cosocle	C₂×C₄×C₂₉₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₉₃
hypocenter	C₂₉₃	#2923
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₉₃

commutator, socle

C₂²×C₂₉₃

C₂³×C₄

Fitting

C₄×M₁₆×C₂₉₃

C₂

prime radical

C₂×#17989802

C₂²

cosocle

C₂×C₄×C₂₉₃

C₂⁴

hypercenter

C₄×M₁₆

D₂₉₃

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #34036188	C₂	semidirect
3. #34036184	C₂	semidirect
4. #34036157	C₂	semidirect
6. C₂×#22627535	C₂	semidirect
7. C₄×M₁₆×C₂₉₃	C₂	semidirect
77. C₂₉₃	#2923	semidirect
84. C₄×M₁₆	D₂₉₃	semidirect

H ≅ G/N

type

2. #34036188

C₂

semidirect

3. #34036184

C₂

semidirect

4. #34036157

C₂

semidirect

6. C₂×#22627535

C₂

semidirect

7. C₄×M₁₆×C₂₉₃

C₂

semidirect

77. C₂₉₃

#2923

semidirect

84. C₄×M₁₆

D₂₉₃

semidirect

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

subgroup	index
C₄×#17989802	2
#34036188	2
#34036184	2
#34036157	2
#34036097	2
C₂×#22627535	2
C₄×M₁₆×C₂₉₃	2
#2923	293

subgroup

index

C₄×M₁₆×C₂₉₃

293

Subgroups — #49167437

Conjugacy classes of subgroups

Complements

Maximal subgroups