Group #49159937 — subgroups

Subgroups — #49159937

Order 37504.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#22627616	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627549	C₂	1	2	yes	maximal
3	#34036035	C₂	1	1	yes	maximal, characteristic
4	#34035892	C₂	1	2	yes	maximal
5	#336×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic
7	#22627616	—	2	4	no	—
8	C₂×C₄⋊Dic₂₉₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
10	C₂×#17989781	—	2	2	no	—
11	#22627549	—	2	4	no	—
12	#22627531	—	2	2	no	—
13	C₂²×C₈×C₂₉₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₈₆	—	2	2	no	—
16	C₂×Dic₅₈₆	—	2	1	no	—
17	C₂×Dic₅₈₆	—	2	2	no	—
18	C₂×Dic₅₈₆	Q₈	1	2	yes	—
19	C₂²×Dic₂₉₃	—	2	1	no	—
20	C₂²×Dic₂₉₃	—	2	2	no	—
21	C₂²×Dic₂₉₃	—	2	1	no	—
22	C₄⋊Dic₂₉₃	—	2	2	no	—
23	C₄⋊Dic₂₉₃	—	2	2	no	—
24	C₄⋊Dic₂₉₃	D₄	1	4	yes	—
25	#17989781	—	2	8	no	—
26	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
27	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₉₃	—	2	2	no	—
29	C₂×C₈×C₂₉₃	—	2	2	no	—
30	C₂×C₈×C₂₉₃	—	2	2	no	—
31	C₄²×C₂₉₃	—	2	2	no	—
32	Dic₅₈₆	—	2	4	no	—
33	Dic₅₈₆	—	4	2	no	—
34	C₂×Dic₂₉₃	—	2	4	no	—
35	C₂×Dic₂₉₃	—	2	2	no	—
36	C₂×Dic₂₉₃	—	4	1	no	—
37	C₂×Dic₂₉₃	—	4	1	no	—
38	C₂×Dic₂₉₃	—	4	2	no	—
39	C₂×Dic₂₉₃	—	4	2	no	—
40	C₂×Dic₂₉₃	—	4	1	no	—
41	C₂×Dic₂₉₃	—	4	2	no	—
42	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉₃	—	2	2	no	—
44	C₂×C₄×C₂₉₃	—	2	2	no	—
45	C₂×C₄×C₂₉₃	—	2	2	no	—
46	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₉₃	#41	1	2	yes	—
50	C₈×C₂₉₃	—	4	2	no	—
51	C₂²×C₂₉₃	#120	1	1	yes	characteristic
52	C₂²×C₂₉₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₂₉₃	#124	1	1	yes	characteristic
54	C₂²×C₂₉₃	#135	1	2	yes	—
55	C₂²×C₂₉₃	#137	1	2	yes	—
56	C₄×C₂₉₃	—	4	2	no	—
57	C₄×C₂₉₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₉₃	#124	1	2	yes	—
59	Dic₂₉₃	—	4	2	no	—
60	Dic₂₉₃	—	4	4	no	—
61	Dic₂₉₃	—	8	1	no	—
62	C₂×C₂₉₃	#393	1	1	yes	characteristic
63	C₂×C₂₉₃	#452	1	2	yes	—
64	C₂×C₂₉₃	#479	1	2	yes	—
65	C₂×C₂₉₃	#488	1	1	yes	characteristic
66	C₂×C₂₉₃	#491	1	1	yes	characteristic
67	C₂₉₃	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	293	1	no	maximal, Sylow
69	C₂×#125	—	293	1	no	—
70	C₂×#103	—	293	2	no	—
71	#391	—	293	1	no	—
72	#340	—	293	2	no	—
73	#336	D₂₉₃	1	1	yes	characteristic
74	C₂²×Q₈	—	293	1	no	—
75	C₂²×C₈	C₂×D₂₉₃	1	2	yes	—
76	#125	—	586	4	no	—
77	C₂×C₄⋊C₄	—	293	2	no	—
78	C₂×C₄⋊C₄	—	293	1	no	—
79	C₂×C₄⋊C₄	—	586	2	no	—
80	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
81	#103	—	586	4	no	—
82	#95	—	586	2	no	—
83	C₂×Q₈	—	293	2	no	—
84	C₂×Q₈	—	586	1	no	—
85	C₂×Q₈	—	586	2	no	—
86	C₂×Q₈	—	586	2	no	—
87	C₂²×C₄	—	586	1	no	—
88	C₂²×C₄	—	586	2	no	—
89	C₂²×C₄	—	586	1	no	—
90	C₂²×C₄	D₁₁₇₂	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	293	4	no	—
96	C₄⋊C₄	—	586	8	no	—
97	C₄⋊C₄	—	586	2	no	—
98	C₄⋊C₄	—	586	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₁₁₇₂	1	1	yes	characteristic
101	Q₈	—	586	4	no	—
102	Q₈	—	1172	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	586	4	no	—
107	C₂×C₄	—	586	2	no	—
108	C₂×C₄	—	1172	1	no	—
109	C₂×C₄	—	1172	2	no	—
110	C₂×C₄	—	1172	1	no	—
111	C₂×C₄	—	1172	1	no	—
112	C₂×C₄	—	1172	2	no	—
113	C₂×C₄	—	1172	2	no	—
114	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
115	C₂×C₄	Q₈×D₂₉₃	1	1	yes	characteristic
116	C₂×C₄	#17989801	1	1	yes	characteristic
117	C₂×C₄	#17989802	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#22627594	1	1	yes	characteristic
120	C₂²	#22627604	1	1	yes	characteristic
121	C₂²	#22627612	1	1	yes	characteristic
122	C₂²	#22627646	1	2	yes	—
123	C₂²	#22627657	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	2344	1	no	—
126	C₄	—	1172	4	no	—
127	C₄	—	1172	2	no	—
128	C₄	#22627602	1	2	yes	—
129	C₄	#22627610	1	2	yes	—
130	C₂	#34036042	1	1	yes	characteristic
131	C₂	#34036293	1	2	yes	—
132	C₂	#34036437	1	2	yes	—
133	C₂	#34036484	1	1	yes	characteristic
134	C₂	#34036498	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627616

C₂

yes

maximal, characteristic

C₂×#22627549

C₂

yes

maximal

#34036035

C₂

yes

maximal, characteristic

#34035892

C₂

yes

maximal

#336×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

C₂²

yes

characteristic

#22627616

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×#17989781

—

#22627549

—

#22627531

—

C₂²×C₈×C₂₉₃

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

Q₈

yes

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

—

#17989781

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₄²×C₂₉₃

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

C₂×C₄×C₂₉₃

yes

C₂×C₄×C₂₉₃

yes

—

C₈×C₂₉₃

—

C₂²×C₂₉₃

#120

yes

characteristic

C₂²×C₂₉₃

C₂²⋊Q₈

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

yes

—

C₂²×C₂₉₃

#137

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊Q₈

yes

—

C₄×C₂₉₃

#124

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

#393

yes

characteristic

C₂×C₂₉₃

#452

yes

—

C₂×C₂₉₃

#479

yes

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#2026

—

293

maximal, Sylow

C₂×#125

—

293

—

C₂×#103

—

293

—

#391

—

293

—

#340

—

293

—

#336

D₂₉₃

yes

characteristic

C₂²×Q₈

—

293

—

C₂²×C₈

C₂×D₂₉₃

yes

—

#125

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#103

—

586

—

#95

—

586

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

D₁₁₇₂

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄²

—

100

C₂³

C₂×D₁₁₇₂

yes

characteristic

101

Q₈

—

586

—

102

Q₈

—

1172

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

586

—

107

C₂×C₄

—

586

—

108

C₂×C₄

—

1172

—

109

C₂×C₄

—

1172

—

110

C₂×C₄

—

1172

—

111

C₂×C₄

—

1172

—

112

C₂×C₄

—

1172

—

113

C₂×C₄

—

1172

—

114

C₂×C₄

D₄×D₂₉₃

yes

—

115

C₂×C₄

Q₈×D₂₉₃

yes

characteristic

116

C₂×C₄

#17989801

yes

characteristic

117

C₂×C₄

#17989802

yes

—

118

C₈

—

119

C₂²

#22627594

yes

characteristic

120

C₂²

#22627604

yes

characteristic

121

C₂²

#22627612

yes

characteristic

122

C₂²

#22627646

yes

—

123

C₂²

#22627657

yes

—

124

C₄

—

125

C₄

—

2344

—

126

C₄

—

1172

—

127

C₄

—

1172

—

128

C₄

#22627602

yes

—

129

C₄

#22627610

yes

—

130

C₂

#34036042

yes

characteristic

131

C₂

#34036293

yes

—

132

C₂

#34036437

yes

—

133

C₂

#34036484

yes

characteristic

134

C₂

#34036498

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₁₇₂
commutator, cosocle	C₂²×C₄×C₂₉₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	#336×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	#336	D₂₉₃
hypocenter	C₂₉₃	#2026
LCS	G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₁₇₂

commutator, cosocle

C₂²×C₄×C₂₉₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

#336×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

D₂₉₃

C₂₉₃

G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
67. C₂₉₃	#2026	semidirect

H ≅ G/N

type

67. C₂₉₃

#2026

semidirect

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

subgroup	index
C₂×#22627616	2
C₂×#22627549	2
#34036035	2
#34035892	2
#336×C₂₉₃	2
#2026	293

subgroup

index

#336×C₂₉₃

293

Subgroups — #49159937

Conjugacy classes of subgroups

Complements

Maximal subgroups