Group #49175330 — subgroups

Subgroups — #49175330

Order 37504.

424 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 152 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×#11924078)⋊Q₈	C₂	1	4	yes	maximal
2	#34037078	C₂	1	8	yes	maximal
3	#34037074	C₂	1	2	yes	maximal
4	C₄⋊Q₈×D₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#11924078	C₂²	1	4	yes	—
6	C₂×C₄×#11924078	C₂²	1	4	yes	—
7	(C₂×#11924078)⋊C₄	C₂²	1	8	yes	—
8	(C₂×#11924078)⋊C₄	C₂²	1	4	yes	—
9	(C₂×#11924078)⋊C₄	C₂²	1	8	yes	—
10	C₂×Q₈×D₂₉₃	C₂²	1	2	yes	—
11	#22627624	C₄	1	1	yes	characteristic, complemented
12	#22627623	C₄	1	4	yes	complemented
13	#22627622	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₂₉₃	C₂²	1	4	yes	—
15	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₂₉₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#11924078	C₂³	1	8	yes	—
18	C₄⋊#11924078	—	4	4	no	—
19	C₄×#11924078	—	4	4	no	—
20	C₂×Dic₅₈₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₅₈₆	C₂×C₄	1	4	yes	—
22	Q₈×D₂₉₃	—	4	2	no	—
23	Q₈×D₂₉₃	—	4	2	no	—
24	C₂×C₄×D₂₉₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×D₂₉₃	C₂³	1	4	yes	—
26	C₂×C₄×D₂₉₃	C₂³	1	2	yes	—
27	C₄⋊Dic₂₉₃	C₂×C₄	1	4	yes	—
28	#17989781	C₂×C₄	1	8	yes	—
29	C₄×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₉₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₂₉₃	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₂₉₃	C₂×C₄	1	4	yes	—
33	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924078	—	2	8	no	—
35	C₂×#11924078	—	2	16	no	—
36	C₂²×D₂₉₃	C₂⁴	1	1	yes	characteristic
37	Dic₅₈₆	—	4	2	no	—
38	Dic₅₈₆	—	4	4	no	—
39	C₂×Dic₂₉₃	C₂²×C₄	1	2	yes	—
40	C₂×Dic₂₉₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₂₉₃	C₂²×C₄	1	4	yes	—
42	C₄×D₂₉₃	—	2	2	no	—
43	C₄×D₂₉₃	—	2	4	no	—
44	C₄×D₂₉₃	—	2	4	no	—
45	C₄×D₂₉₃	—	2	4	no	—
46	Q₈×C₂₉₃	—	4	2	no	—
47	C₂×C₄×C₂₉₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₂₉₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₂₉₃	H(5,𝔽₂)	1	2	yes	—
52	C₂×D₂₉₃	2₋¹⁺⁴	1	4	yes	—
53	#11924078	—	4	8	no	—
54	C₄×C₂₉₃	—	2	1	no	—
55	C₄×C₂₉₃	—	2	4	no	—
56	C₄×C₂₉₃	—	2	2	no	—
57	Dic₂₉₃	—	2	4	no	—
58	Dic₂₉₃	—	2	1	no	—
59	Dic₂₉₃	—	2	2	no	—
60	C₂×C₂₉₃	#518	1	1	yes	characteristic
61	C₂×C₂₉₃	#519	1	2	yes	—
62	D₂₉₃	#563	1	4	yes	—
63	C₂₉₃	#2381	1	1	yes	characteristic, complemented, Sylow
64	#2381	—	293	1	no	maximal, Sylow
65	C₂×C₄⋊Q₈	—	293	1	no	—
66	#391	—	293	4	no	—
67	#387	—	293	8	no	—
68	#383	—	293	2	no	—
69	C₂²×Q₈	—	293	2	no	—
70	C₄⋊Q₈	—	293	2	no	—
71	C₄⋊Q₈	—	293	4	no	—
72	C₄⋊Q₈	—	293	1	no	—
73	C₄⋊Q₈	#11924078	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	293	4	no	—
75	C₂×C₄⋊C₄	—	293	4	no	—
76	C₂×C₄²	—	293	1	no	—
77	C₂×C₄²	—	293	4	no	—
78	#95	—	293	8	no	—
79	#95	—	293	4	no	—
80	#95	—	293	8	no	—
81	C₂×Q₈	—	293	4	no	—
82	C₂×Q₈	—	293	2	no	—
83	C₂×Q₈	—	1172	2	no	—
84	C₂×Q₈	—	1172	2	no	—
85	C₂×Q₈	C₂×#11924078	1	2	yes	—
86	C₂²×C₄	—	293	2	no	—
87	C₂²×C₄	—	293	1	no	—
88	C₂²×C₄	—	293	4	no	—
89	C₂²×C₄	—	293	8	no	—
90	C₄⋊C₄	—	293	8	no	—
91	C₄⋊C₄	—	293	4	no	—
92	C₄⋊C₄	—	1172	4	no	—
93	C₄⋊C₄	C₂×#11924078	1	4	yes	—
94	C₄²	—	293	1	no	—
95	C₄²	—	293	2	no	—
96	C₄²	—	1172	4	no	—
97	C₄²	C₂×#11924078	1	1	yes	characteristic
98	C₂³	—	293	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	1172	2	no	—
101	Q₈	—	1172	4	no	—
102	C₂×C₄	—	293	2	no	—
103	C₂×C₄	—	293	1	no	—
104	C₂×C₄	—	293	4	no	—
105	C₂×C₄	—	586	4	no	—
106	C₂×C₄	—	586	4	no	—
107	C₂×C₄	—	586	2	no	—
108	C₂×C₄	—	586	4	no	—
109	C₂×C₄	—	586	8	no	—
110	C₂×C₄	—	586	16	no	—
111	C₂×C₄	C₂²×#11924078	1	2	yes	—
112	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
113	C₂×C₄	C₂²×#11924078	1	4	yes	—
114	C₂²	—	293	2	no	—
115	C₂²	—	293	4	no	—
116	C₂²	C₂³×#11924078	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	586	1	no	—
121	C₄	—	586	4	no	—
122	C₄	—	586	2	no	—
123	C₄	—	1172	8	no	—
124	C₂	—	293	4	no	—
125	C₂	#34037196	1	1	yes	characteristic
126	C₂	#34037197	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₂×#11924078)⋊Q₈

C₂

yes

maximal

#34037078

C₂

yes

maximal

#34037074

C₂

yes

maximal

C₄⋊Q₈×D₂₉₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#11924078

C₂²

yes

—

C₂×C₄×#11924078

C₂²

yes

—

(C₂×#11924078)⋊C₄

C₂²

yes

—

(C₂×#11924078)⋊C₄

C₂²

yes

—

(C₂×#11924078)⋊C₄

C₂²

yes

—

C₂×Q₈×D₂₉₃

C₂²

yes

—

#22627624

C₄

yes

characteristic, complemented

#22627623

C₄

yes

complemented

#22627622

C₄

yes

complemented

C₄⋊C₄×D₂₉₃

C₂²

yes

—

C₄²×D₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₄

yes

characteristic, complemented

C₂²×#11924078

C₂³

yes

—

C₄⋊#11924078

—

C₄×#11924078

—

C₂×Dic₅₈₆

C₂×C₄

yes

—

C₂×Dic₅₈₆

C₂×C₄

yes

—

Q₈×D₂₉₃

—

Q₈×D₂₉₃

—

C₂×C₄×D₂₉₃

C₂³

yes

characteristic

C₂×C₄×D₂₉₃

C₂³

yes

—

C₂×C₄×D₂₉₃

C₂³

yes

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

#17989781

C₂×C₄

yes

—

C₄×Dic₂₉₃

C₂×C₄

yes

characteristic

C₄×Dic₂₉₃

C₂×C₄

yes

—

C₂×Q₈×C₂₉₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₉₃

C₂×C₄

yes

—

C₄²×C₂₉₃

C₂×C₄

yes

characteristic

C₂×#11924078

—

C₂×#11924078

—

C₂²×D₂₉₃

C₂⁴

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

C₂²×C₄

yes

—

C₂×Dic₂₉₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₉₃

C₂²×C₄

yes

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂²×C₄

yes

—

C₂²×C₂₉₃

C₂³×C₄

yes

characteristic

C₂×D₂₉₃

H(5,𝔽₂)

yes

—

C₂×D₂₉₃

2₋¹⁺⁴

yes

—

#11924078

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

#518

yes

characteristic

C₂×C₂₉₃

#519

yes

—

D₂₉₃

#563

yes

—

C₂₉₃

#2381

yes

characteristic, complemented, Sylow

#2381

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

#391

—

293

—

#387

—

293

—

#383

—

293

—

C₂²×Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

#11924078

yes

characteristic, complemented

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄²

—

293

—

C₂×C₄²

—

293

—

#95

—

293

—

#95

—

293

—

#95

—

293

—

C₂×Q₈

—

293

—

C₂×Q₈

—

293

—

C₂×Q₈

—

1172

—

C₂×Q₈

—

1172

—

C₂×Q₈

C₂×#11924078

yes

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

1172

—

C₄⋊C₄

C₂×#11924078

yes

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

1172

—

C₄²

C₂×#11924078

yes

characteristic

C₂³

—

293

—

Q₈

—

100

Q₈

—

1172

—

101

Q₈

—

1172

—

102

C₂×C₄

—

293

—

103

C₂×C₄

—

293

—

104

C₂×C₄

—

293

—

105

C₂×C₄

—

586

—

106

C₂×C₄

—

586

—

107

C₂×C₄

—

586

—

108

C₂×C₄

—

586

—

109

C₂×C₄

—

586

—

110

C₂×C₄

—

586

—

111

C₂×C₄

C₂²×#11924078

yes

—

112

C₂×C₄

C₂²×#11924078

yes

characteristic

113

C₂×C₄

C₂²×#11924078

yes

—

114

C₂²

—

293

—

115

C₂²

—

293

—

116

C₂²

C₂³×#11924078

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

586

—

121

C₄

—

586

—

122

C₄

—

586

—

123

C₄

—

1172

—

124

C₂

—

293

—

125

C₂

#34037196

yes

characteristic

126

C₂

#34037197

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#11924078
commutator, socle	C₂²×C₂₉₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₂₉₃	C₄
prime radical, cosocle	C₂²×D₂₉₃	C₂⁴
hypercenter	C₄⋊Q₈	#11924078
hypocenter	C₂₉₃	#2381
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#11924078

commutator, socle

C₂²×C₂₉₃

C₂³×C₄

Fitting

C₄⋊Q₈×C₂₉₃

C₄

prime radical, cosocle

C₂²×D₂₉₃

C₂⁴

C₄⋊Q₈

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #22627624	C₄	semidirect
12. #22627623	C₄	semidirect
13. #22627622	C₄	semidirect
16. C₄⋊Q₈×C₂₉₃	C₄	semidirect
63. C₂₉₃	#2381	semidirect
73. C₄⋊Q₈	#11924078	semidirect

H ≅ G/N

type

11. #22627624

C₄

semidirect

12. #22627623

C₄

semidirect

13. #22627622

C₄

semidirect

16. C₄⋊Q₈×C₂₉₃

C₄

semidirect

63. C₂₉₃

#2381

semidirect

73. C₄⋊Q₈

#11924078

semidirect

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

subgroup	index
(C₂×#11924078)⋊Q₈	2
#34037078	2
#34037074	2
C₄⋊Q₈×D₂₉₃	2
#2381	293

subgroup

index

C₄⋊Q₈×D₂₉₃

293

Subgroups — #49175330

Conjugacy classes of subgroups

Complements

Maximal subgroups