Group #49158292 — subgroups

Subgroups — #49158292

Order 37504.

146 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 55 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	#34036527	C₂	1	1	yes	maximal, characteristic
2	#34036425	C₂	1	1	yes	maximal, characteristic
3	#34036340	C₂	1	1	yes	maximal, characteristic
4	#34036259	C₂	1	1	yes	maximal, characteristic
5	#34035877	C₂	1	1	yes	maximal, characteristic
6	#34035876	C₂	1	1	yes	maximal, characteristic
7	#335×C₂₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₁₇₂	—	2	1	no	—
9	#22627624	C₂²	1	1	yes	characteristic
10	#22627624	C₂²	1	1	yes	characteristic
11	C₄×Dic₅₈₆	C₂²	1	1	yes	characteristic
12	#22627558	—	2	1	no	—
13	#22627556	—	2	1	no	—
14	#22627556	—	2	1	no	—
15	C₄⋊#14556873	C₂²	1	1	yes	characteristic
16	#22627553	—	2	1	no	—
17	#22627549	—	2	1	no	—
18	#22627549	—	2	1	no	—
19	#22627534	—	2	1	no	—
20	C₄⋊C₈×C₂₉₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₉₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₈₆	—	2	1	no	—
24	C₂×Dic₅₈₆	—	2	1	no	—
25	C₂×Dic₅₈₆	D₄	1	1	yes	characteristic
26	Dic₁₁₇₂	—	4	2	no	—
27	C₂×#14556873	—	2	1	no	—
28	C₄⋊Dic₂₉₃	—	2	1	no	—
29	C₄⋊Dic₂₉₃	—	2	1	no	—
30	C₄⋊Dic₂₉₃	—	2	1	no	—
31	C₄⋊Dic₂₉₃	—	2	1	no	—
32	C₄⋊Dic₂₉₃	D₄	1	1	yes	characteristic
33	#17989781	—	2	1	no	—
34	C₄×Dic₂₉₃	—	2	1	no	—
35	C₂×C₈×C₂₉₃	—	2	1	no	—
36	C₂×C₈×C₂₉₃	—	2	1	no	—
37	C₂×C₈×C₂₉₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₂₉₃	D₄	1	1	yes	characteristic
39	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
40	Dic₅₈₆	—	2	1	no	—
41	Dic₅₈₆	—	2	1	no	—
42	Dic₅₈₆	—	4	1	no	—
43	Dic₅₈₆	—	4	1	no	—
44	Dic₅₈₆	—	8	1	no	—
45	C₂×Dic₂₉₃	—	2	1	no	—
46	C₂×Dic₂₉₃	—	2	1	no	—
47	C₂×Dic₂₉₃	—	4	1	no	—
48	C₂×Dic₂₉₃	—	4	1	no	—
49	#14556873	—	4	1	no	—
50	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
53	C₈×C₂₉₃	—	2	1	no	—
54	C₈×C₂₉₃	—	4	1	no	—
55	C₈×C₂₉₃	—	4	1	no	—
56	C₈×C₂₉₃	Q₁₆	1	2	yes	complemented
57	C₂²×C₂₉₃	#121	1	1	yes	characteristic
58	C₄×C₂₉₃	—	2	1	no	—
59	C₄×C₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₂₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₂₉₃	#137	1	1	yes	characteristic
62	C₄×C₂₉₃	#137	1	1	yes	characteristic
63	Dic₂₉₃	—	4	1	no	—
64	Dic₂₉₃	—	4	1	no	—
65	Dic₂₉₃	—	8	1	no	—
66	Dic₂₉₃	—	8	1	no	—
67	C₂×C₂₉₃	#462	1	1	yes	characteristic
68	C₂×C₂₉₃	#468	1	1	yes	characteristic
69	C₂×C₂₉₃	#471	1	1	yes	characteristic
70	C₂₉₃	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	293	1	no	maximal, Sylow
72	#498	—	293	1	no	—
73	#477	—	293	1	no	—
74	#462	—	293	1	no	—
75	#446	—	293	1	no	—
76	#335	D₂₉₃	1	1	yes	characteristic
77	#333	—	293	1	no	—
78	#332	—	293	1	no	—
79	C₂×Q₁₆	—	586	1	no	—
80	C₄⋊Q₈	—	293	1	no	—
81	C₄⋊Q₈	—	293	1	no	—
82	C₄×Q₈	—	293	1	no	—
83	#107	—	586	1	no	—
84	#106	—	586	1	no	—
85	#106	—	586	1	no	—
86	C₄⋊C₈	—	293	1	no	—
87	C₄⋊C₈	—	586	1	no	—
88	C₄⋊C₈	C₂×D₂₉₃	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₂₉₃	1	1	yes	characteristic
90	#103	—	586	1	no	—
91	#103	—	586	1	no	—
92	#97	—	586	1	no	—
93	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
94	C₂×Q₈	—	293	1	no	—
95	C₂×Q₈	—	586	1	no	—
96	C₂×Q₈	—	586	1	no	—
97	Q₁₆	—	1172	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	586	1	no	—
101	C₂×C₈	D₁₁₇₂	1	1	yes	characteristic
102	C₂×C₈	D₁₁₇₂	1	1	yes	characteristic
103	C₄⋊C₄	—	293	1	no	—
104	C₄⋊C₄	—	586	1	no	—
105	C₄⋊C₄	—	586	1	no	—
106	C₄⋊C₄	—	586	1	no	—
107	C₄⋊C₄	—	586	1	no	—
108	C₄⋊C₄	—	586	1	no	—
109	C₄²	—	586	1	no	—
110	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
111	Q₈	—	2344	1	no	—
112	Q₈	—	586	1	no	—
113	Q₈	—	586	1	no	—
114	Q₈	—	1172	1	no	—
115	Q₈	—	1172	1	no	—
116	C₂×C₄	—	586	1	no	—
117	C₂×C₄	—	586	1	no	—
118	C₂×C₄	—	1172	1	no	—
119	C₂×C₄	—	1172	1	no	—
120	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
121	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
122	C₂×C₄	#17989803	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	1172	1	no	—
127	C₈	Dic₁₁₇₂	1	2	yes	complemented
128	C₂²	#22627599	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2344	1	no	—
131	C₄	—	2344	1	no	—
132	C₄	—	1172	1	no	—
133	C₄	—	1172	1	no	—
134	C₄	C₂×Dic₁₁₇₂	1	1	yes	characteristic
135	C₄	#22627648	1	1	yes	characteristic
136	C₄	#22627656	1	1	yes	characteristic
137	C₄	#22627657	1	1	yes	characteristic
138	C₂	#34036340	1	1	yes	characteristic
139	C₂	#34036376	1	1	yes	characteristic
140	C₂	#34036393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036527

C₂

yes

maximal, characteristic

#34036425

C₂

yes

maximal, characteristic

#34036340

C₂

yes

maximal, characteristic

#34036259

C₂

yes

maximal, characteristic

#34035877

C₂

yes

maximal, characteristic

#34035876

C₂

yes

maximal, characteristic

#335×C₂₉₃

C₂

yes

maximal, characteristic

C₂×Dic₁₁₇₂

—

#22627624

C₂²

yes

characteristic

#22627624

C₂²

yes

characteristic

C₄×Dic₅₈₆

C₂²

yes

—

—

—

C₂²

yes

—

—

—

—

C₄⋊C₈×C₂₉₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₉₃

C₂²

yes

characteristic

C₄×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

D₄

yes

characteristic

Dic₁₁₇₂

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

characteristic

#17989781

—

C₄×Dic₂₉₃

—

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

D₄

yes

characteristic

C₂×C₈×C₂₉₃

D₄

yes

characteristic

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

characteristic

C₈×C₂₉₃

—

C₈×C₂₉₃

—

C₈×C₂₉₃

—

C₈×C₂₉₃

Q₁₆

yes

complemented

C₂²×C₂₉₃

#121

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×Q₁₆

yes

C₄×C₂₉₃

yes

C₄×C₂₉₃

yes

C₄×C₂₉₃

yes

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#1692

—

293

maximal, Sylow

#498

—

293

—

#477

—

293

—

#462

—

293

—

#446

—

293

—

#335

D₂₉₃

yes

characteristic

#333

—

293

—

#332

—

293

—

C₂×Q₁₆

—

586

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄×Q₈

—

293

—

#107

—

586

—

#106

—

586

—

#106

—

586

—

C₄⋊C₈

—

293

—

C₄⋊C₈

—

586

—

C₄⋊C₈

C₂×D₂₉₃

yes

characteristic

C₄⋊C₈

C₂×D₂₉₃

yes

characteristic

#103

—

586

—

#103

—

586

—

#97

—

586

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

Q₁₆

—

1172

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

586

—

101

C₂×C₈

D₁₁₇₂

yes

characteristic

102

C₂×C₈

D₁₁₇₂

yes

characteristic

103

C₄⋊C₄

—

293

—

104

C₄⋊C₄

—

586

—

105

C₄⋊C₄

—

586

—

106

C₄⋊C₄

—

586

—

107

C₄⋊C₄

—

586

—

108

C₄⋊C₄

—

586

—

109

C₄²

—

586

—

110

C₄²

C₂²×D₂₉₃

yes

characteristic

111

Q₈

—

2344

—

112

Q₈

—

586

—

113

Q₈

—

586

—

114

Q₈

—

1172

—

115

Q₈

—

1172

—

116

C₂×C₄

—

586

—

117

C₂×C₄

—

586

—

118

C₂×C₄

—

1172

—

119

C₂×C₄

—

1172

—

120

C₂×C₄

D₄×D₂₉₃

yes

characteristic

121

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

122

C₂×C₄

#17989803

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

1172

—

127

C₈

Dic₁₁₇₂

yes

complemented

128

C₂²

#22627599

yes

characteristic

129

C₄

—

130

C₄

—

2344

—

131

C₄

—

2344

—

132

C₄

—

1172

—

133

C₄

—

1172

—

134

C₄

C₂×Dic₁₁₇₂

yes

characteristic

135

C₄

#22627648

yes

characteristic

136

C₄

#22627656

yes

characteristic

137

C₄

#22627657

yes

characteristic

138

C₂

#34036340

yes

characteristic

139

C₂

#34036376

yes

characteristic

140

C₂

#34036393

yes

characteristic

	subgroup	G/H
center	C₂²	#22627599
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#335×C₂₉₃	C₂
socle, prime radical	C₂²×C₂₉₃	#121
hypercenter	#335	D₂₉₃
hypocenter	C₂₉₃	#1692
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#22627599

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#335×C₂₉₃

C₂

C₂²×C₂₉₃

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
56. C₈×C₂₉₃	Q₁₆	semidirect
70. C₂₉₃	#1692	semidirect
127. C₈	Dic₁₁₇₂	semidirect

H ≅ G/N

type

56. C₈×C₂₉₃

Q₁₆

semidirect

70. C₂₉₃

#1692

semidirect

127. C₈

Dic₁₁₇₂

semidirect

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

subgroup	index
#34036527	2
#34036425	2
#34036340	2
#34036259	2
#34035877	2
#34035876	2
#335×C₂₉₃	2
#1692	293

subgroup

index

#335×C₂₉₃

293

Subgroups — #49158292

Conjugacy classes of subgroups

Complements

Maximal subgroups