Group #49171463 — subgroups

Subgroups — #49171463

Order 37504.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#34036858	C₂	1	1	yes	maximal, characteristic
2	#34036481	C₂	1	3	yes	maximal
3	#34036421	C₂	1	3	yes	maximal
4	#34036339	C₂	1	3	yes	maximal
5	Q₈×#14556873	C₂	1	1	yes	maximal, characteristic
6	C₄×#17989792	C₂	1	3	yes	maximal
7	Q₈²×C₂₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#17989792	C₂²	1	3	yes	—
9	#22627624	C₂²	1	3	yes	—
10	#22627615	—	2	3	no	—
11	Q₈×Dic₂₉₃	—	2	1	no	—
12	C₄×Dic₅₈₆	C₂²	1	3	yes	—
13	#22627557	C₂²	1	3	yes	—
14	C₄⋊#14556873	C₂²	1	3	yes	—
15	#22627548	C₂²	1	1	yes	characteristic
16	#22627547	C₂²	1	3	yes	—
17	#22627547	C₂²	1	6	yes	—
18	C₄×#14556873	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₂₉₃	—	2	3	no	—
20	C₄⋊Q₈×C₂₉₃	C₂²	1	3	yes	—
21	C₄×Q₈×C₂₉₃	—	2	1	no	—
22	C₄×Q₈×C₂₉₃	C₂²	1	3	yes	—
23	C₄×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₅₈₆	C₂³	1	3	yes	—
25	#17989792	—	2	6	no	—
26	C₂×#14556873	C₂³	1	3	yes	—
27	C₂×#14556873	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₂₉₃	—	2	3	no	—
29	C₄⋊Dic₂₉₃	—	2	3	no	—
30	C₄⋊Dic₂₉₃	C₂³	1	1	yes	characteristic
31	#17989781	—	2	3	no	—
32	C₄×Dic₂₉₃	—	2	3	no	—
33	C₂×Q₈×C₂₉₃	—	2	3	no	—
34	C₂×Q₈×C₂₉₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₂₉₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₉₃	—	2	3	no	—
37	C₄⋊C₄×C₂₉₃	—	2	3	no	—
38	C₄⋊C₄×C₂₉₃	D₄	1	3	yes	—
39	C₄⋊C₄×C₂₉₃	C₂³	1	3	yes	—
40	C₄²×C₂₉₃	—	2	3	no	—
41	C₄²×C₂₉₃	C₂³	1	3	yes	—
42	Dic₅₈₆	—	2	3	no	—
43	Dic₅₈₆	—	2	3	no	—
44	C₂×Dic₂₉₃	—	2	1	no	—
45	C₂×Dic₂₉₃	—	2	3	no	—
46	#14556873	—	2	3	no	—
47	#14556873	—	2	1	no	—
48	Q₈×C₂₉₃	—	2	1	no	—
49	Q₈×C₂₉₃	—	8	3	no	—
50	Q₈×C₂₉₃	Q₁₆	1	4	yes	complemented
51	Q₈×C₂₉₃	#41	1	2	yes	—
52	C₂×C₄×C₂₉₃	—	2	1	no	—
53	C₂×C₄×C₂₉₃	—	2	3	no	—
54	C₂×C₄×C₂₉₃	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₂₉₃	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
57	C₂²×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₂₉₃	—	2	2	no	—
59	C₄×C₂₉₃	—	2	3	no	—
60	C₄×C₂₉₃	—	4	3	no	—
61	C₄×C₂₉₃	C₂×Q₁₆	1	6	yes	—
62	C₄×C₂₉₃	C₂×#41	1	1	yes	characteristic
63	C₄×C₂₉₃	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₂₉₃	—	4	3	no	—
65	Dic₂₉₃	—	4	1	no	—
66	C₂×C₂₉₃	#548	1	1	yes	characteristic
67	C₂×C₂₉₃	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₂₉₃	#577	1	1	yes	characteristic
69	C₂₉₃	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	293	1	no	maximal, Sylow
71	Q₈²	D₂₉₃	1	1	yes	characteristic
72	#557	—	293	1	no	—
73	#487	—	293	3	no	—
74	#477	—	293	3	no	—
75	#462	—	293	3	no	—
76	C₈×Q₈	—	293	1	no	—
77	C₄×Q₁₆	—	293	3	no	—
78	C₂×Q₁₆	—	293	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	293	3	no	—
81	C₄⋊Q₈	C₂×D₂₉₃	1	3	yes	—
82	#125	—	586	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	293	3	no	—
85	C₄×Q₈	—	586	1	no	—
86	C₄×Q₈	C₂×D₂₉₃	1	3	yes	—
87	C₄×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
88	#107	—	293	3	no	—
89	C₄⋊C₈	—	293	3	no	—
90	#103	—	293	1	no	—
91	#103	—	293	3	no	—
92	#103	—	293	6	no	—
93	C₄×C₈	—	293	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	293	3	no	—
96	C₂×Q₈	C₂²⋊D₂₉₃	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₂₉₃	1	1	yes	characteristic
98	Q₁₆	—	586	6	no	—
99	C₂×C₈	—	293	1	no	—
100	C₂×C₈	—	293	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	293	1	no	—
104	C₄⋊C₄	—	586	3	no	—
105	C₄⋊C₄	—	586	3	no	—
106	C₄⋊C₄	—	586	3	no	—
107	C₄⋊C₄	C₂²⋊D₂₉₃	1	3	yes	—
108	C₄⋊C₄	C₂²×D₂₉₃	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	586	3	no	—
111	C₄²	C₂²×D₂₉₃	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	586	3	no	—
115	Q₈	—	586	3	no	—
116	Q₈	#17989792	1	4	yes	complemented
117	Q₈	#17989803	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	586	1	no	—
121	C₂×C₄	—	586	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	3	yes	—
124	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
125	C₈	—	586	3	no	—
126	C₈	—	586	1	no	—
127	C₂²	C₂²×C₂²⋊D₂₉₃	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	1172	3	no	—
132	C₄	—	1172	1	no	—
133	C₄	C₂×#17989792	1	6	yes	—
134	C₄	C₂×#17989803	1	1	yes	characteristic
135	C₄	#22627672	1	1	yes	characteristic
136	C₂	#34036790	1	1	yes	characteristic
137	C₂	C₂²×#17989792	1	1	yes	characteristic
138	C₂	#34036940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036858

C₂

yes

maximal, characteristic

#34036481

C₂

yes

maximal

#34036421

C₂

yes

maximal

#34036339

C₂

yes

maximal

Q₈×#14556873

C₂

yes

maximal, characteristic

C₄×#17989792

C₂

yes

maximal

Q₈²×C₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989792

C₂²

yes

—

#22627624

C₂²

yes

—

#22627615

—

Q₈×Dic₂₉₃

—

C₄×Dic₅₈₆

C₂²

yes

—

#22627557

C₂²

yes

—

C₄⋊#14556873

C₂²

yes

—

#22627548

C₂²

yes

characteristic

#22627547

C₂²

yes

—

#22627547

C₂²

yes

—

C₄×#14556873

C₂²

yes

—

C₄⋊Q₈×C₂₉₃

—

C₄⋊Q₈×C₂₉₃

C₂²

yes

—

C₄×Q₈×C₂₉₃

—

C₄×Q₈×C₂₉₃

C₂²

yes

—

C₄×Q₈×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

C₂³

yes

—

#17989792

—

C₂×#14556873

C₂³

yes

—

C₂×#14556873

C₂³

yes

characteristic

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂³

yes

characteristic

#17989781

—

C₄×Dic₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

D₄

yes

characteristic

C₂×Q₈×C₂₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

D₄

yes

—

C₄⋊C₄×C₂₉₃

C₂³

yes

—

C₄²×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

#14556873

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

Q₁₆

yes

complemented

Q₈×C₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₂²×C₂₉₃

C₂²×D₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×Q₁₆

yes

—

C₄×C₂₉₃

C₂×#41

yes

characteristic

C₄×C₂₉₃

2₋¹⁺⁴

yes

characteristic

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

#548

yes

characteristic

C₂×C₂₉₃

C₂²×Q₁₆

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#3363

—

293

maximal, Sylow

Q₈²

D₂₉₃

yes

characteristic

#557

—

293

—

#487

—

293

—

#477

—

293

—

#462

—

293

—

C₈×Q₈

—

293

—

C₄×Q₁₆

—

293

—

C₂×Q₁₆

—

293

—

C₄⋊Q₈

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

C₂×D₂₉₃

yes

—

#125

—

586

—

C₄×Q₈

—

C₄×Q₈

—

293

—

C₄×Q₈

—

586

—

C₄×Q₈

C₂×D₂₉₃

yes

—

C₄×Q₈

C₂×D₂₉₃

yes

characteristic

#107

—

293

—

C₄⋊C₈

—

293

—

#103

—

293

—

#103

—

293

—

#103

—

293

—

C₄×C₈

—

293

—

C₂×Q₈

—

C₂×Q₈

—

293

—

C₂×Q₈

C₂²⋊D₂₉₃

yes

characteristic

C₂×Q₈

C₂²×D₂₉₃

yes

characteristic

Q₁₆

—

586

—

C₂×C₈

—

293

—

100

C₂×C₈

—

293

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

293

—

104

C₄⋊C₄

—

586

—

105

C₄⋊C₄

—

586

—

106

C₄⋊C₄

—

586

—

107

C₄⋊C₄

C₂²⋊D₂₉₃

yes

—

108

C₄⋊C₄

C₂²×D₂₉₃

yes

—

109

C₄²

—

110

C₄²

—

586

—

111

C₄²

C₂²×D₂₉₃

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

586

—

115

Q₈

—

586

—

116

Q₈

#17989792

yes

complemented

117

Q₈

#17989803

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

586

—

121

C₂×C₄

—

586

—

122

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

124

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

125

C₈

—

586

—

126

C₈

—

586

—

127

C₂²

C₂²×C₂²⋊D₂₉₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

1172

—

132

C₄

—

1172

—

133

C₄

C₂×#17989792

yes

—

134

C₄

C₂×#17989803

yes

characteristic

135

C₄

#22627672

yes

characteristic

136

C₂

#34036790

yes

characteristic

137

C₂

C₂²×#17989792

yes

characteristic

138

C₂

#34036940

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₉₃
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	Q₈²×C₂₉₃	C₂
socle, prime radical	C₂²×C₂₉₃	C₂²×D₄
hypercenter	Q₈²	D₂₉₃
hypocenter	C₂₉₃	#3363
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₉₃

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

Q₈²×C₂₉₃

C₂

socle, prime radical

C₂²×C₂₉₃

C₂²×D₄

hypercenter

Q₈²

D₂₉₃

hypocenter

C₂₉₃

#3363

LCS

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
50. Q₈×C₂₉₃	Q₁₆	semidirect
69. C₂₉₃	#3363	semidirect
116. Q₈	#17989792	semidirect

H ≅ G/N

type

50. Q₈×C₂₉₃

Q₁₆

semidirect

69. C₂₉₃

#3363

semidirect

116. Q₈

#17989792

semidirect

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

subgroup	index
#34036858	2
#34036481	2
#34036421	2
#34036339	2
Q₈×#14556873	2
C₄×#17989792	2
Q₈²×C₂₉₃	2
#3363	293

subgroup

index

Q₈²×C₂₉₃

293

Subgroups — #49171463

Conjugacy classes of subgroups

Complements

Maximal subgroups