Group #37591078 — subgroups

Subgroups — #1461×C₁₉₃ #37591078

Order 24704.

248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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×C₄×Q₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
2	#431×C₁₉₃	C₂	1	2	yes	maximal
3	#324×C₁₉₃	C₂	1	4	yes	maximal
4	C₂²×Q₈×C₁₉₃	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₁₉₃	—	2	2	no	—
6	C₄×Q₈×C₁₉₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₉₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₉₃	C₄	1	2	yes	—
10	C₂×C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₉₃	—	2	2	no	—
12	C₂²⋊C₈×C₁₉₃	C₂²	1	4	yes	—
13	C₄×C₈×C₁₉₃	—	2	2	no	—
14	C₂×Q₈×C₁₉₃	—	2	2	no	—
15	C₂×Q₈×C₁₉₃	—	2	2	no	—
16	C₂×Q₈×C₁₉₃	C₈	1	4	yes	complemented
17	C₂²×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₉₃	—	2	4	no	—
23	C₄⋊C₄×C₁₉₃	—	2	2	no	—
24	C₄⋊C₄×C₁₉₃	—	2	4	no	—
25	C₄²×C₁₉₃	—	2	2	no	—
26	C₄²×C₁₉₃	—	2	2	no	—
27	C₄²×C₁₉₃	D₄	1	2	yes	—
28	C₄²×C₁₉₃	D₄	1	2	yes	—
29	C₂³×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₁₉₃	—	2	8	no	—
31	C₂×C₄×C₁₉₃	—	2	2	no	—
32	C₂×C₄×C₁₉₃	—	2	2	no	—
33	C₂×C₄×C₁₉₃	—	2	2	no	—
34	C₂×C₄×C₁₉₃	—	2	2	no	—
35	C₂×C₄×C₁₉₃	—	4	1	no	—
36	C₂×C₄×C₁₉₃	—	4	2	no	—
37	C₂×C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₃	C₂×C₈	1	4	yes	—
41	C₂×C₄×C₁₉₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₉₃	M₁₆	1	2	yes	—
43	C₂×C₄×C₁₉₃	C₂×D₄	1	2	yes	—
44	C₈×C₁₉₃	—	4	4	no	—
45	C₂²×C₁₉₃	—	2	1	no	—
46	C₂²×C₁₉₃	—	2	1	no	—
47	C₂²×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
49	C₂²×C₁₉₃	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₁₉₃	—	2	4	no	—
51	C₄×C₁₉₃	—	2	2	no	—
52	C₄×C₁₉₃	—	2	1	no	—
53	C₄×C₁₉₃	—	4	2	no	—
54	C₄×C₁₉₃	C₂²⋊C₈	1	4	yes	—
55	C₄×C₁₉₃	#101	1	2	yes	—
56	C₂×C₁₉₃	—	2	1	no	—
57	C₂×C₁₉₃	—	2	1	no	—
58	C₂×C₁₉₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₁₉₃	#410	1	1	yes	characteristic
60	C₂×C₁₉₃	C₂×#101	1	1	yes	characteristic
61	C₁₉₃	#1461	1	1	yes	characteristic, direct product factor, Sylow
62	#1461	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₄×Q₈	C₂×C₁₉₃	1	1	yes	characteristic
64	#431	C₂×C₁₉₃	1	2	yes	—
65	#324	C₂×C₁₉₃	1	4	yes	—
66	C₂²×Q₈	C₄×C₁₉₃	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	C₂×C₄⋊C₄	C₄×C₁₉₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₁₉₃	1	2	yes	—
71	C₂×C₄²	C₄×C₁₉₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₁₉₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	C₂²×C₁₉₃	1	4	yes	—
75	C₄×C₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	C₈×C₁₉₃	1	4	yes	complemented
79	C₂²×C₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄×C₁₉₃	1	2	yes	—
81	C₂²×C₄	C₂×C₄×C₁₉₃	1	2	yes	—
82	C₂²×C₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
83	C₂²×C₄	C₂³×C₁₉₃	1	1	yes	characteristic
84	C₂×C₈	—	2	4	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	D₄×C₁₉₃	1	2	yes	—
90	C₄²	D₄×C₁₉₃	1	2	yes	—
91	C₂³	C₂²×C₄×C₁₉₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊C₄×C₁₉₃	1	2	yes	—
101	C₂×C₄	C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₈×C₁₉₃	1	4	yes	—
103	C₂×C₄	C₂×C₈×C₁₉₃	1	2	yes	—
104	C₂×C₄	M₁₆×C₁₉₃	1	2	yes	—
105	C₂×C₄	C₂×D₄×C₁₉₃	1	2	yes	—
106	C₈	—	4	4	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂×C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
110	C₂²	C₂²×C₈×C₁₉₃	1	1	yes	characteristic
111	C₂²	C₂×M₁₆×C₁₉₃	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	1	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂²⋊C₈×C₁₉₃	1	4	yes	—
117	C₄	#101×C₁₉₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	C₂×C₂²⋊C₈×C₁₉₃	1	1	yes	characteristic
121	C₂	#410×C₁₉₃	1	1	yes	characteristic
122	C₂	C₂×#101×C₁₉₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₉₃

C₂

yes

maximal, characteristic

#431×C₁₉₃

C₂

yes

maximal

#324×C₁₉₃

C₂

yes

maximal

C₂²×Q₈×C₁₉₃

C₄

yes

characteristic

C₄×Q₈×C₁₉₃

—

C₄×Q₈×C₁₉₃

—

C₂×C₄⋊C₄×C₁₉₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉₃

C₂²

yes

—

C₂×C₄²×C₁₉₃

C₄

yes

—

C₂×C₄²×C₁₉₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₃

—

C₂²⋊C₈×C₁₉₃

C₂²

yes

—

C₄×C₈×C₁₉₃

—

C₂×Q₈×C₁₉₃

—

C₂×Q₈×C₁₉₃

—

C₂×Q₈×C₁₉₃

C₈

yes

complemented

C₂²×C₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₃

C₂×C₄

yes

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₃

C₂³

yes

characteristic

C₂×C₈×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄²×C₁₉₃

—

C₄²×C₁₉₃

—

C₄²×C₁₉₃

D₄

yes

—

C₄²×C₁₉₃

D₄

yes

—

C₂³×C₁₉₃

C₂²×C₄

yes

characteristic

Q₈×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₃

C₂×C₈

yes

—

C₂×C₄×C₁₉₃

C₂×C₈

yes

—

C₂×C₄×C₁₉₃

M₁₆

yes

—

C₂×C₄×C₁₉₃

C₂×D₄

yes

—

C₈×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₃

C₂²×C₈

yes

characteristic

C₂²×C₁₉₃

C₂×M₁₆

yes

characteristic

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

C₂²⋊C₈

yes

—

C₄×C₁₉₃

#101

yes

—

C₂×C₁₉₃

—

C₂×C₁₉₃

—

C₂×C₁₉₃

C₂×C₂²⋊C₈

yes

C₂×C₁₉₃

yes

C₂×C₁₉₃

yes

C₁₉₃

yes

characteristic, direct product factor, Sylow

#1461

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₉₃

yes

characteristic

#431

C₂×C₁₉₃

yes

—

#324

C₂×C₁₉₃

yes

—

C₂²×Q₈

C₄×C₁₉₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×C₁₉₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₉₃

yes

—

C₂×C₄²

C₄×C₁₉₃

yes

—

C₂×C₄²

C₂²×C₁₉₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂²×C₁₉₃

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₈×C₁₉₃

yes

complemented

C₂²×C₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₉₃

yes

—

C₂²×C₄

C₂×C₄×C₁₉₃

yes

—

C₂²×C₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂²×C₄

C₂³×C₁₉₃

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₄×C₁₉₃

yes

—

C₄²

D₄×C₁₉₃

yes

—

C₂³

C₂²×C₄×C₁₉₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₉₃

yes

characteristic

100

C₂×C₄

C₂²⋊C₄×C₁₉₃

yes

—

101

C₂×C₄

C₂²⋊C₄×C₁₉₃

yes

characteristic

102

C₂×C₄

C₂×C₈×C₁₉₃

yes

—

103

C₂×C₄

C₂×C₈×C₁₉₃

yes

—

104

C₂×C₄

M₁₆×C₁₉₃

yes

—

105

C₂×C₄

C₂×D₄×C₁₉₃

yes

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₂²⋊C₄×C₁₉₃

yes

characteristic

110

C₂²

C₂²×C₈×C₁₉₃

yes

characteristic

111

C₂²

C₂×M₁₆×C₁₉₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊C₈×C₁₉₃

yes

—

117

C₄

#101×C₁₉₃

yes

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊C₈×C₁₉₃

yes

characteristic

121

C₂

#410×C₁₉₃

yes

characteristic

122

C₂

C₂×#101×C₁₉₃

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉₃	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₁₉₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₉₃
socle	C₂²×C₁₉₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₃ ◁ C₂×C₄²×C₁₉₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉₃

C₂²⋊C₄

commutator

C₂²

C₂²×C₈×C₁₉₃

Frattini, cosocle

C₂²×C₄

C₂³×C₁₉₃

socle

C₂²×C₁₉₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₉₃

C₂²×C₄

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉₃ ◁ C₂×C₄²×C₁₉₃ ◁ G

N	H ≅ G/N	type
16. C₂×Q₈×C₁₉₃	C₈	semidirect
61. C₁₉₃	#1461	direct (N × H)
62. #1461	C₁₉₃	direct (N × H)
78. C₂×Q₈	C₈×C₁₉₃	semidirect

H ≅ G/N

type

16. C₂×Q₈×C₁₉₃

C₈

semidirect

61. C₁₉₃

#1461

direct (N × H)

62. #1461

C₁₉₃

direct (N × H)

78. C₂×Q₈

C₈×C₁₉₃

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₉₃	2
#431×C₁₉₃	2
#324×C₁₉₃	2
#1461	193

subgroup

index

C₂×C₄×Q₈×C₁₉₃

#431×C₁₉₃

#324×C₁₉₃

#1461

193

Subgroups — #1461×C₁₉₃ #37591078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1461×C193 #37591078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1461×C₁₉₃ #37591078