Group #49174916 — subgroups

Subgroups — #49174916

Order 37504.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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×#22627687	C₂	1	4	yes	maximal
2	#34037088	C₂	1	1	yes	maximal, characteristic
3	#34037073	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₂₉₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#11924078	C₂²	1	2	yes	—
7	#22627687	—	2	8	no	—
8	#22627679	—	2	2	no	—
9	(C₂×#11924078)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₂₉₃	C₂²	1	2	yes	—
11	#22627624	C₄	1	1	yes	characteristic, complemented
12	#22627623	—	2	2	no	—
13	#22627622	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₂₉₃	—	2	2	no	—
15	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₂₉₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#11924078	—	2	2	no	—
18	D₂₉₃⋊C₈	—	2	2	no	—
19	D₂₉₃⋊C₈	D₄	1	4	yes	—
20	C₂×#14556880	—	2	2	no	—
21	C₄⋊#11924078	—	2	4	no	—
22	C₂×Dic₅₈₆	—	2	2	no	—
23	C₂×Dic₅₈₆	C₂×C₄	1	2	yes	—
24	Q₈×D₂₉₃	—	2	4	no	—
25	Q₈×D₂₉₃	—	2	4	no	—
26	C₂×C₄×D₂₉₃	—	2	2	no	—
27	C₂×C₄×D₂₉₃	D₄	1	2	yes	—
28	C₂×C₄×D₂₉₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₂₉₃	—	2	2	no	—
30	#17989781	—	2	4	no	—
31	C₄×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₂₉₃	D₄	1	2	yes	—
33	C₂×Q₈×C₂₉₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₂₉₃	—	2	2	no	—
35	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924078	—	4	2	no	—
37	C₂×#11924078	—	4	4	no	—
38	#14556880	—	2	4	no	—
39	C₂²×D₂₉₃	C₂×D₄	1	1	yes	characteristic
40	Dic₅₈₆	—	2	4	no	—
41	Dic₅₈₆	—	4	4	no	—
42	C₂×Dic₂₉₃	—	2	2	no	—
43	C₂×Dic₂₉₃	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₂₉₃	—	2	4	no	—
46	C₄×D₂₉₃	—	4	2	no	—
47	C₄×D₂₉₃	—	4	2	no	—
48	C₄×D₂₉₃	C₂×D₄	1	2	yes	—
49	C₄×D₂₉₃	#41	1	2	yes	—
50	Q₈×C₂₉₃	—	2	4	no	—
51	C₂×C₄×C₂₉₃	—	2	2	no	—
52	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₂₉₃	#124	1	1	yes	characteristic
56	C₂×D₂₉₃	#127	1	1	yes	characteristic
57	C₂×D₂₉₃	#137	1	4	yes	—
58	#11924078	—	8	2	no	—
59	C₄×C₂₉₃	—	2	2	no	—
60	C₄×C₂₉₃	—	4	2	no	—
61	C₄×C₂₉₃	C₄×D₄	1	2	yes	—
62	Dic₂₉₃	—	2	2	no	—
63	Dic₂₉₃	—	4	2	no	—
64	Dic₂₉₃	#121	1	2	yes	—
65	C₂×C₂₉₃	#390	1	1	yes	characteristic
66	C₂×C₂₉₃	#419	1	2	yes	—
67	D₂₉₃	#491	1	2	yes	—
68	D₂₉₃	#497	1	2	yes	—
69	C₂₉₃	#1960	1	1	yes	characteristic, complemented, Sylow
70	#1960	—	293	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	293	1	no	—
72	C₂×#103	—	293	4	no	—
73	C₂×#97	—	293	1	no	—
74	#382	—	293	1	no	—
75	C₂²×Q₈	—	293	2	no	—
76	C₂²×C₈	—	293	2	no	—
77	C₄⋊Q₈	—	293	2	no	—
78	C₄⋊Q₈	—	293	1	no	—
79	C₄⋊Q₈	—	586	2	no	—
80	C₄⋊Q₈	#11924078	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	293	2	no	—
82	C₂×C₄⋊C₄	—	586	2	no	—
83	C₂×C₄²	—	293	1	no	—
84	#103	—	586	8	no	—
85	#97	—	586	2	no	—
86	#95	—	586	2	no	—
87	C₂×Q₈	—	293	2	no	—
88	C₂×Q₈	—	586	2	no	—
89	C₂×Q₈	—	586	4	no	—
90	C₂×Q₈	—	586	4	no	—
91	C₂×Q₈	C₂×#11924078	1	2	yes	—
92	C₂²×C₄	—	293	1	no	—
93	C₂²×C₄	—	293	2	no	—
94	C₂²×C₄	—	586	2	no	—
95	C₂²×C₄	—	586	2	no	—
96	C₂×C₈	—	293	4	no	—
97	C₂×C₈	—	586	2	no	—
98	C₂×C₈	—	586	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	586	4	no	—
101	C₄⋊C₄	—	586	4	no	—
102	C₄⋊C₄	—	586	2	no	—
103	C₄²	—	293	2	no	—
104	C₄²	—	293	1	no	—
105	C₄²	C₂×#11924078	1	1	yes	characteristic
106	C₂³	—	293	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	586	4	no	—
109	Q₈	—	1172	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	293	1	no	—
112	C₂×C₄	—	293	2	no	—
113	C₂×C₄	—	293	2	no	—
114	C₂×C₄	—	293	2	no	—
115	C₂×C₄	—	586	4	no	—
116	C₂×C₄	—	586	2	no	—
117	C₂×C₄	—	1172	2	no	—
118	C₂×C₄	—	1172	2	no	—
119	C₂×C₄	—	1172	2	no	—
120	C₂×C₄	—	1172	4	no	—
121	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
122	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
123	C₈	—	586	4	no	—
124	C₂²	—	293	1	no	—
125	C₂²	—	293	1	no	—
126	C₂²	—	293	4	no	—
127	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	293	2	no	—
131	C₄	—	2344	2	no	—
132	C₄	—	586	2	no	—
133	C₄	—	1172	2	no	—
134	C₄	D₄×#11924078	1	2	yes	—
135	C₂	—	293	2	no	—
136	C₂	—	293	2	no	—
137	C₂	#34037081	1	1	yes	characteristic
138	C₂	#34037122	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627687

C₂

yes

maximal

#34037088

C₂

yes

maximal, characteristic

#34037073

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₂₉₃

C₂

yes

maximal, characteristic

C₂×D₂₉₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#11924078

C₂²

yes

—

#22627687

—

#22627679

—

(C₂×#11924078)⋊C₄

—

C₂×Q₈×D₂₉₃

C₂²

yes

—

#22627624

C₄

yes

characteristic, complemented

#22627623

—

#22627622

C₄

yes

complemented

C₄⋊C₄×D₂₉₃

—

C₄²×D₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₄

yes

characteristic, complemented

C₂²×#11924078

—

D₂₉₃⋊C₈

—

D₂₉₃⋊C₈

D₄

yes

—

C₂×#14556880

—

C₄⋊#11924078

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

C₂×C₄

yes

—

Q₈×D₂₉₃

—

Q₈×D₂₉₃

—

C₂×C₄×D₂₉₃

—

C₂×C₄×D₂₉₃

D₄

yes

—

C₂×C₄×D₂₉₃

C₂³

yes

characteristic

C₄⋊Dic₂₉₃

—

#17989781

—

C₄×Dic₂₉₃

C₂×C₄

yes

characteristic

C₄×Dic₂₉₃

D₄

yes

—

C₂×Q₈×C₂₉₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂×C₄

yes

—

—

—

C₂²×D₂₉₃

C₂×D₄

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

—

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

C₂×D₄

yes

—

C₄×D₂₉₃

#41

yes

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

C₂×D₂₉₃

yes

C₂×D₂₉₃

yes

C₂×D₂₉₃

yes

—

#11924078

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄×D₄

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

#121

yes

—

C₂×C₂₉₃

#390

yes

characteristic

C₂×C₂₉₃

#419

yes

—

D₂₉₃

#491

yes

—

D₂₉₃

#497

yes

—

C₂₉₃

#1960

yes

characteristic, complemented, Sylow

#1960

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

C₂×#103

—

293

—

C₂×#97

—

293

—

#382

—

293

—

C₂²×Q₈

—

293

—

C₂²×C₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

#11924078

yes

characteristic, complemented

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

—

293

—

#103

—

586

—

#97

—

586

—

#95

—

586

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

C₂×#11924078

yes

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₄⋊C₄

—

100

C₄⋊C₄

—

586

—

101

C₄⋊C₄

—

586

—

102

C₄⋊C₄

—

586

—

103

C₄²

—

293

—

104

C₄²

—

293

—

105

C₄²

C₂×#11924078

yes

characteristic

106

C₂³

—

293

—

107

Q₈

—

108

Q₈

—

586

—

109

Q₈

—

1172

—

110

C₂×C₄

—

111

C₂×C₄

—

293

—

112

C₂×C₄

—

293

—

113

C₂×C₄

—

293

—

114

C₂×C₄

—

293

—

115

C₂×C₄

—

586

—

116

C₂×C₄

—

586

—

117

C₂×C₄

—

1172

—

118

C₂×C₄

—

1172

—

119

C₂×C₄

—

1172

—

120

C₂×C₄

—

1172

—

121

C₂×C₄

C₂²⋊#11924078

yes

—

122

C₂×C₄

C₂²×#11924078

yes

characteristic

123

C₈

—

586

—

124

C₂²

—

293

—

125

C₂²

—

293

—

126

C₂²

—

293

—

127

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

293

—

131

C₄

—

2344

—

132

C₄

—

586

—

133

C₄

—

1172

—

134

C₄

D₄×#11924078

yes

—

135

C₂

—

293

—

136

C₂

—

293

—

137

C₂

#34037081

yes

characteristic

138

C₂

#34037122

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#11924078
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#11924078
Fitting	C₄⋊Q₈×C₂₉₃	C₄
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉₃	C₂×D₄
cosocle	C₂×C₄×D₂₉₃	C₂³
hypercenter	C₄⋊Q₈	#11924078
hypocenter	C₂₉₃	#1960
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#11924078

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#11924078

Fitting

C₄⋊Q₈×C₂₉₃

C₄

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉₃

C₂×D₄

cosocle

C₂×C₄×D₂₉₃

C₂³

C₄⋊Q₈

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #22627624	C₄	semidirect
13. #22627622	C₄	semidirect
16. C₄⋊Q₈×C₂₉₃	C₄	semidirect
69. C₂₉₃	#1960	semidirect
80. C₄⋊Q₈	#11924078	semidirect

H ≅ G/N

type

11. #22627624

C₄

semidirect

13. #22627622

C₄

semidirect

16. C₄⋊Q₈×C₂₉₃

C₄

semidirect

69. C₂₉₃

#1960

semidirect

80. C₄⋊Q₈

#11924078

semidirect

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

subgroup	index
C₂×#22627687	2
#34037088	2
#34037073	2
C₄⋊Q₈×D₂₉₃	2
#1960	293

subgroup

index

C₄⋊Q₈×D₂₉₃

293

Subgroups — #49174916

Conjugacy classes of subgroups

Complements

Maximal subgroups