Group #49168191 — subgroups

Subgroups — #49168191

Order 37504.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#34036700	C₂	1	3	yes	maximal, complemented
2	#34036319	C₂	1	8	yes	maximal, complemented
3	#34036103	C₂	1	3	yes	maximal, complemented
4	#585×C₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic, complemented
6	#22627622	—	2	3	no	—
7	#22627611	—	2	3	no	—
8	#22627605	C₂²	1	12	yes	complemented
9	#22627603	—	2	6	no	—
10	#22627583	C₂²	1	3	yes	complemented
11	#22627539	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₂₉₃	—	2	3	no	—
13	H(5,𝔽₂)×C₂₉₃	—	2	3	no	—
14	H(5,𝔽₂)×C₂₉₃	C₂²	1	4	yes	—
15	C₂×#41×C₂₉₃	—	2	6	no	—
16	C₂×#41×C₂₉₃	C₂²	1	3	yes	—
17	C₂×Dic₅₈₆	—	2	3	no	—
18	C₂×Dic₅₈₆	—	2	3	no	—
19	C₂²×Dic₂₉₃	C₂³	1	2	yes	—
20	C₄⋊Dic₂₉₃	—	2	3	no	—
21	#17989781	—	2	12	no	—
22	C₂²⋊Dic₂₉₃	—	2	12	no	—
23	C₂²⋊Dic₂₉₃	C₂³	1	6	yes	—
24	C₄×Dic₂₉₃	—	2	3	no	—
25	#41×C₂₉₃	—	2	4	no	—
26	#41×C₂₉₃	—	2	12	no	—
27	#41×C₂₉₃	—	2	6	no	—
28	#41×C₂₉₃	—	2	12	no	—
29	#41×C₂₉₃	—	2	6	no	—
30	C₂×Q₈×C₂₉₃	—	2	6	no	—
31	C₂×Q₈×C₂₉₃	D₄	1	3	yes	—
32	C₂×D₄×C₂₉₃	—	2	6	no	—
33	C₂×D₄×C₂₉₃	—	2	12	no	—
34	C₂×D₄×C₂₉₃	D₄	1	3	yes	—
35	C₂×D₄×C₂₉₃	C₂³	1	6	yes	—
36	C₂²×C₄×C₂₉₃	—	2	4	no	—
37	C₂²×C₄×C₂₉₃	D₄	1	6	yes	—
38	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
39	Dic₅₈₆	—	4	1	no	—
40	Dic₅₈₆	—	4	3	no	—
41	C₂×Dic₂₉₃	—	2	6	no	—
42	C₂×Dic₂₉₃	—	2	6	no	—
43	C₂³×C₂₉₃	—	2	4	no	—
44	C₂³×C₂₉₃	C₂×D₄	1	6	yes	—
45	C₂³×C₂₉₃	C₂⁴	1	1	yes	characteristic
46	Q₈×C₂₉₃	—	2	4	no	—
47	Q₈×C₂₉₃	—	2	6	no	—
48	D₄×C₂₉₃	—	2	12	no	—
49	D₄×C₂₉₃	—	2	12	no	—
50	D₄×C₂₉₃	—	2	6	no	—
51	C₂×C₄×C₂₉₃	—	2	12	no	—
52	C₂×C₄×C₂₉₃	—	2	6	no	—
53	C₂×C₄×C₂₉₃	—	2	6	no	—
54	C₂×C₄×C₂₉₃	C₂×D₄	1	6	yes	—
55	C₂×C₄×C₂₉₃	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₂₉₃	C₂×D₄	1	3	yes	—
57	C₂²×C₂₉₃	—	2	6	no	—
58	C₂²×C₂₉₃	—	4	1	no	—
59	C₂²×C₂₉₃	—	4	6	no	—
60	C₂²×C₂₉₃	—	8	4	no	—
61	C₂²×C₂₉₃	C₂²×D₄	1	3	yes	—
62	C₄×C₂₉₃	—	2	6	no	—
63	C₄×C₂₉₃	#120	1	1	yes	characteristic
64	C₄×C₂₉₃	#120	1	3	yes	—
65	Dic₂₉₃	—	4	6	no	—
66	Dic₂₉₃	—	4	2	no	—
67	C₂×C₂₉₃	—	2	3	no	—
68	C₂×C₂₉₃	—	4	6	no	—
69	C₂×C₂₉₃	C₂×#120	1	1	yes	characteristic
70	C₂₉₃	#3028	1	1	yes	characteristic, complemented, Sylow
71	#3028	—	293	1	no	maximal, Sylow
72	#585	D₂₉₃	1	1	yes	characteristic
73	#536	—	293	3	no	—
74	#458	—	293	8	no	—
75	#410	—	293	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₂₉₃	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₂₉₃	1	3	yes	—
81	C₂²×Q₈	—	293	1	no	—
82	C₄⋊Q₈	—	586	3	no	—
83	#124	—	586	3	no	—
84	#123	—	293	12	no	—
85	C₂²⋊Q₈	—	586	6	no	—
86	#117	—	293	3	no	—
87	#99	—	293	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	586	3	no	—
95	C₂×Q₈	—	586	3	no	—
96	C₂×Q₈	C₂²⋊D₂₉₃	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₂₉₃	1	3	yes	—
100	C₂×D₄	C₂²×D₂₉₃	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	293	2	no	—
103	C₂²×C₄	C₂²⋊D₂₉₃	1	6	yes	—
104	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
105	C₄⋊C₄	—	586	3	no	—
106	C₄⋊C₄	—	586	12	no	—
107	C₂²⋊C₄	—	293	6	no	—
108	C₂²⋊C₄	—	586	12	no	—
109	C₄²	—	586	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₂₉₃	1	6	yes	—
112	C₂³	C₂³×D₂₉₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	1172	1	no	—
116	Q₈	—	1172	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	586	6	no	—
124	C₂×C₄	—	586	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₂₉₃	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	1172	6	no	—
135	C₄	—	1172	2	no	—
136	C₄	#22627593	1	1	yes	characteristic
137	C₄	#22627593	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#22627593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036700

C₂

yes

maximal, complemented

#34036319

C₂

yes

maximal, complemented

#34036103

C₂

yes

maximal, complemented

#585×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

C₂²

yes

characteristic, complemented

#22627622

—

#22627611

—

#22627605

C₂²

yes

complemented

#22627603

—

#22627583

C₂²

yes

complemented

#22627539

C₂²

yes

complemented

2₋¹⁺⁴×C₂₉₃

—

H(5,𝔽₂)×C₂₉₃

—

H(5,𝔽₂)×C₂₉₃

C₂²

yes

—

C₂×#41×C₂₉₃

—

C₂×#41×C₂₉₃

C₂²

yes

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂²×Dic₂₉₃

C₂³

yes

—

C₄⋊Dic₂₉₃

—

#17989781

—

C₂²⋊Dic₂₉₃

—

C₂²⋊Dic₂₉₃

C₂³

yes

—

C₄×Dic₂₉₃

—

#41×C₂₉₃

—

#41×C₂₉₃

—

#41×C₂₉₃

—

#41×C₂₉₃

—

#41×C₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

D₄

yes

—

C₂×D₄×C₂₉₃

—

C₂×D₄×C₂₉₃

—

C₂×D₄×C₂₉₃

D₄

yes

—

C₂×D₄×C₂₉₃

C₂³

yes

—

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

—

C₂³×C₂₉₃

C₂×D₄

yes

—

C₂³×C₂₉₃

C₂⁴

yes

characteristic

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂²×D₄

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#120

yes

characteristic

C₄×C₂₉₃

#120

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂×#120

yes

characteristic

C₂₉₃

#3028

yes

characteristic, complemented, Sylow

#3028

—

293

maximal, Sylow

#585

D₂₉₃

yes

characteristic

#536

—

293

—

#458

—

293

—

#410

—

293

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₂₉₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₂₉₃

yes

—

C₂²×Q₈

—

293

—

C₄⋊Q₈

—

586

—

#124

—

586

—

#123

—

293

—

C₂²⋊Q₈

—

586

—

#117

—

293

—

#99

—

293

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

C₂²⋊D₂₉₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₉₃

yes

—

100

C₂×D₄

C₂²×D₂₉₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

293

—

103

C₂²×C₄

C₂²⋊D₂₉₃

yes

—

104

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

105

C₄⋊C₄

—

586

—

106

C₄⋊C₄

—

586

—

107

C₂²⋊C₄

—

293

—

108

C₂²⋊C₄

—

586

—

109

C₄²

—

586

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₂₉₃

yes

—

112

C₂³

C₂³×D₂₉₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

1172

—

116

Q₈

—

1172

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

586

—

124

C₂×C₄

—

586

—

125

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

127

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₂₉₃

yes

—

133

C₄

—

134

C₄

—

1172

—

135

C₄

—

1172

—

136

C₄

#22627593

yes

characteristic

137

C₄

#22627593

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#22627593

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#22627593
commutator, cosocle	C₂³×C₂₉₃	C₂⁴
Frattini	C₂³	C₂³×D₂₉₃
Fitting, prime radical	#585×C₂₉₃	C₂
socle	C₂×C₂₉₃	C₂×#120
hypercenter	#585	D₂₉₃
hypocenter	C₂₉₃	#3028
LCS	G ▷ C₂³×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#22627593

commutator, cosocle

C₂³×C₂₉₃

C₂⁴

Frattini

C₂³

C₂³×D₂₉₃

Fitting, prime radical

#585×C₂₉₃

C₂

C₂×C₂₉₃

D₂₉₃

C₂₉₃

G ▷ C₂³×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #34036700	C₂	semidirect
2. #34036319	C₂	semidirect
3. #34036103	C₂	semidirect
5. C₂²×Dic₅₈₆	C₂²	semidirect
8. #22627605	C₂²	semidirect
10. #22627583	C₂²	semidirect
11. #22627539	C₂²	semidirect
70. C₂₉₃	#3028	semidirect

H ≅ G/N

type

1. #34036700

C₂

semidirect

2. #34036319

C₂

semidirect

3. #34036103

C₂

semidirect

5. C₂²×Dic₅₈₆

C₂²

semidirect

8. #22627605

C₂²

semidirect

10. #22627583

C₂²

semidirect

11. #22627539

C₂²

semidirect

70. C₂₉₃

#3028

semidirect

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

subgroup	index
#34036700	2
#34036319	2
#34036103	2
#585×C₂₉₃	2
#3028	293

subgroup

index

#585×C₂₉₃

293

Subgroups — #49168191

Conjugacy classes of subgroups

Complements

Maximal subgroups