Group #49158409 — subgroups

Subgroups — #49158409

Order 37504.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#34036532	C₂	1	1	yes	maximal, characteristic, complemented
2	#34036525	C₂	1	1	yes	maximal, characteristic, complemented
3	#34036503	C₂	1	1	yes	maximal, characteristic
4	#34036502	C₂	1	1	yes	maximal, characteristic
5	#34036477	C₂	1	1	yes	maximal, characteristic
6	#34035851	C₂	1	1	yes	maximal, characteristic, complemented
7	#487×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#17989791	—	2	2	no	—
9	C₂×#17989789	—	2	2	no	—
10	#22627624	C₂²	1	1	yes	characteristic
11	#22627621	C₂²	1	1	yes	characteristic
12	#22627557	—	2	2	no	—
13	#22627555	—	2	2	no	—
14	#22627544	—	2	2	no	—
15	C₄×#14556873	C₂²	1	1	yes	characteristic
16	C₄×#14556873	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
19	#103×C₂₉₃	—	2	2	no	—
20	C₄×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₈₆	—	2	1	no	—
22	C₂×D₁₁₇₂	—	2	1	no	—
23	C₂×D₁₁₇₂	—	2	1	no	—
24	C₂×D₁₁₇₂	—	2	1	no	—
25	#17989791	—	4	4	no	—
26	#17989789	—	4	4	no	—
27	C₂×#14556873	D₄	1	2	yes	—
28	C₂×#14556873	D₄	1	2	yes	—
29	C₄⋊Dic₂₉₃	—	2	1	no	—
30	C₄⋊Dic₂₉₃	—	2	1	no	—
31	C₂×Q₈×C₂₉₃	—	2	1	no	—
32	C₂×Q₈×C₂₉₃	—	2	1	no	—
33	C₂×C₈×C₂₉₃	D₄	1	2	yes	—
34	C₄⋊C₄×C₂₉₃	—	2	1	no	—
35	C₄⋊C₄×C₂₉₃	—	2	1	no	—
36	C₄⋊C₄×C₂₉₃	—	2	1	no	—
37	C₄⋊C₄×C₂₉₃	—	2	1	no	—
38	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
39	C₂²×D₂₉₃	—	4	1	no	—
40	Dic₅₈₆	—	4	2	no	—
41	D₁₁₇₂	—	4	2	no	—
42	D₁₁₇₂	—	4	2	no	—
43	D₁₁₇₂	—	4	2	no	—
44	C₂×Dic₂₉₃	—	4	1	no	—
45	#14556873	—	2	2	no	—
46	#14556873	SD₁₆	1	4	yes	complemented
47	Q₈×C₂₉₃	—	4	2	no	—
48	Q₈×C₂₉₃	—	4	2	no	—
49	C₂×C₄×C₂₉₃	—	4	1	no	—
50	C₂×C₄×C₂₉₃	—	4	1	no	—
51	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
54	C₈×C₂₉₃	—	2	2	no	—
55	C₂²×C₂₉₃	#127	1	1	yes	characteristic
56	C₂×D₂₉₃	—	8	1	no	—
57	C₂×D₂₉₃	—	8	1	no	—
58	C₂×D₂₉₃	—	8	1	no	—
59	C₄×C₂₉₃	—	8	1	no	—
60	C₄×C₂₉₃	—	8	1	no	—
61	C₄×C₂₉₃	C₂×SD₁₆	1	2	yes	—
62	C₄×C₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₂₉₃	#137	1	2	yes	—
64	Dic₂₉₃	—	8	1	no	—
65	C₂×C₂₉₃	#492	1	1	yes	characteristic
66	C₂×C₂₉₃	#496	1	1	yes	characteristic
67	C₂×C₂₉₃	#497	1	1	yes	characteristic
68	D₂₉₃	—	16	1	no	—
69	C₂₉₃	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	293	1	no	maximal, Sylow
71	#500	—	293	1	no	—
72	#498	—	293	1	no	—
73	#492	—	293	1	no	—
74	#492	—	293	1	no	—
75	#487	D₂₉₃	1	1	yes	characteristic, complemented
76	#486	—	293	1	no	—
77	#322	—	293	1	no	—
78	C₂×SD₁₆	—	586	2	no	—
79	C₂×SD₁₆	—	586	2	no	—
80	C₄⋊Q₈	—	293	1	no	—
81	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
83	#127	—	293	1	no	—
84	#107	—	586	2	no	—
85	#106	—	586	2	no	—
86	#103	—	2	2	no	—
87	#102	—	586	2	no	—
88	C₄×C₈	—	293	1	no	—
89	C₄×C₈	—	293	1	no	—
90	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	586	1	no	—
94	C₂×D₄	—	586	1	no	—
95	C₂×D₄	—	586	1	no	—
96	C₂×D₄	—	586	1	no	—
97	SD₁₆	—	1172	4	no	—
98	SD₁₆	—	1172	4	no	—
99	C₂×C₈	—	293	2	no	—
100	C₂×C₈	—	293	2	no	—
101	C₂×C₈	C₂²⋊D₂₉₃	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	586	1	no	—
107	C₄⋊C₄	—	586	1	no	—
108	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
109	C₂³	—	1172	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	1172	2	no	—
113	D₄	—	1172	2	no	—
114	D₄	—	1172	2	no	—
115	D₄	—	1172	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	1172	1	no	—
119	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₉₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	293	4	no	—
124	C₈	—	586	2	no	—
125	C₂²	—	2344	1	no	—
126	C₂²	—	2344	1	no	—
127	C₂²	—	2344	1	no	—
128	C₂²	#22627620	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	2344	1	no	—
132	C₄	SD₁₆×D₂₉₃	1	2	yes	—
133	C₄	#22627649	1	2	yes	—
134	C₄	#22627656	1	2	yes	—
135	C₂	—	4688	1	no	—
136	C₂	#34036500	1	1	yes	characteristic
137	C₂	#34036519	1	1	yes	characteristic
138	C₂	#34036522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036532

C₂

yes

maximal, characteristic, complemented

#34036525

C₂

yes

maximal, characteristic, complemented

#34036503

C₂

yes

maximal, characteristic

#34036502

C₂

yes

maximal, characteristic

#34036477

C₂

yes

maximal, characteristic

#34035851

C₂

yes

maximal, characteristic, complemented

#487×C₂₉₃

C₂

yes

maximal, characteristic, complemented

C₂×#17989791

—

C₂×#17989789

—

#22627624

C₂²

yes

characteristic

#22627621

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#14556873

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

#103×C₂₉₃

—

C₄×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

#17989791

—

#17989789

—

C₂×#14556873

D₄

yes

—

C₂×#14556873

D₄

yes

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×C₈×C₂₉₃

D₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

C₂²×D₂₉₃

—

Dic₅₈₆

—

D₁₁₇₂

—

D₁₁₇₂

—

D₁₁₇₂

—

C₂×Dic₂₉₃

—

#14556873

—

#14556873

SD₁₆

yes

complemented

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

#127

yes

characteristic

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×SD₁₆

yes

—

C₄×C₂₉₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₉₃

#137

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

D₂₉₃

—

C₂₉₃

#1714

yes

characteristic, complemented, Sylow

#1714

—

293

maximal, Sylow

#500

—

293

—

#498

—

293

—

#492

—

293

—

#492

—

293

—

#487

D₂₉₃

yes

characteristic, complemented

#486

—

293

—

#322

—

293

—

C₂×SD₁₆

—

586

—

C₂×SD₁₆

—

586

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

#127

—

293

—

#107

—

586

—

#106

—

586

—

#103

—

#102

—

586

—

C₄×C₈

—

293

—

C₄×C₈

—

293

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

586

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

SD₁₆

—

1172

—

SD₁₆

—

1172

—

C₂×C₈

—

293

—

100

C₂×C₈

—

293

—

101

C₂×C₈

C₂²⋊D₂₉₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

586

—

107

C₄⋊C₄

—

586

—

108

C₄²

C₂²×D₂₉₃

yes

characteristic

109

C₂³

—

1172

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

1172

—

113

D₄

—

1172

—

114

D₄

—

1172

—

115

D₄

—

1172

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

1172

—

119

C₂×C₄

D₄×D₂₉₃

yes

characteristic

120

C₂×C₄

D₄×D₂₉₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

122

C₈

—

123

C₈

—

293

—

124

C₈

—

586

—

125

C₂²

—

2344

—

126

C₂²

—

2344

—

127

C₂²

—

2344

—

128

C₂²

#22627620

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

2344

—

132

C₄

SD₁₆×D₂₉₃

yes

—

133

C₄

#22627649

yes

—

134

C₄

#22627656

yes

—

135

C₂

—

4688

—

136

C₂

#34036500

yes

characteristic

137

C₂

#34036519

yes

characteristic

138

C₂

#34036522

yes

characteristic

	subgroup	G/H
center	C₂²	#22627620
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#487×C₂₉₃	C₂
socle	C₂²×C₂₉₃	#127
prime radical	#22627621	C₂²
hypercenter	#487	D₂₉₃
hypocenter	C₂₉₃	#1714
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#22627620

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#487×C₂₉₃

C₂

C₂²×C₂₉₃

C₂²

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #34036532	C₂	semidirect
2. #34036525	C₂	semidirect
6. #34035851	C₂	semidirect
7. #487×C₂₉₃	C₂	semidirect
46. #14556873	SD₁₆	semidirect
69. C₂₉₃	#1714	semidirect
75. #487	D₂₉₃	semidirect

H ≅ G/N

type

1. #34036532

C₂

semidirect

2. #34036525

C₂

semidirect

6. #34035851

C₂

semidirect

7. #487×C₂₉₃

C₂

semidirect

46. #14556873

SD₁₆

semidirect

69. C₂₉₃

#1714

semidirect

75. #487

D₂₉₃

semidirect

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

subgroup	index
#34036532	2
#34036525	2
#34036503	2
#34036502	2
#34036477	2
#34035851	2
#487×C₂₉₃	2
#1714	293

subgroup

index

#487×C₂₉₃

293

Subgroups — #49158409

Conjugacy classes of subgroups

Complements

Maximal subgroups