Group #37591845 — subgroups

Subgroups — #2228×C₁₉₃ #37591845

Order 24704.

136 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 64 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	#499×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
3	#367×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#366×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₁₆×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#357×C₁₉₃	C₂	1	1	yes	maximal, characteristic
7	#357×C₁₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₈×C₁₉₃	C₂²	1	1	yes	characteristic
9	#125×C₁₉₃	—	2	1	no	—
10	C₄×D₄×C₁₉₃	—	2	1	no	—
11	C₂×C₁₆×C₁₉₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₉₃	C₂²	1	1	yes	characteristic
13	#107×C₁₉₃	C₂²	1	1	yes	characteristic
14	#107×C₁₉₃	C₂²	1	1	yes	characteristic
15	#107×C₁₉₃	C₂²	1	1	yes	characteristic
16	#106×C₁₉₃	—	2	1	no	—
17	#102×C₁₉₃	—	2	1	no	—
18	C₄×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₁₉₃	—	2	1	no	—
20	C₂²×C₄×C₁₉₃	—	4	1	no	—
21	D₈×C₁₉₃	—	2	1	no	—
22	D₈×C₁₉₃	Q₈	1	2	yes	—
23	C₂×C₈×C₁₉₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₉₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₉₃	—	2	1	no	—
26	C₄⋊C₄×C₁₉₃	—	2	1	no	—
27	C₄⋊C₄×C₁₉₃	—	2	1	no	—
28	C₄⋊C₄×C₁₉₃	—	4	1	no	—
29	C₄⋊C₄×C₁₉₃	—	4	1	no	—
30	C₂²⋊C₄×C₁₉₃	—	4	1	no	—
31	C₄²×C₁₉₃	D₄	1	1	yes	characteristic
32	C₁₆×C₁₉₃	—	2	1	no	—
33	C₁₆×C₁₉₃	—	2	1	no	—
34	C₂³×C₁₉₃	—	4	1	no	—
35	D₄×C₁₉₃	—	2	2	no	—
36	D₄×C₁₉₃	—	4	1	no	—
37	C₂×C₄×C₁₉₃	—	4	1	no	—
38	C₂×C₄×C₁₉₃	—	4	1	no	—
39	C₂×C₄×C₁₉₃	—	4	1	no	—
40	C₂×C₄×C₁₉₃	—	8	2	no	—
41	C₂×C₄×C₁₉₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₃	D₈	1	1	yes	characteristic
43	C₂×C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₃	—	2	1	no	—
45	C₈×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₉₃	#41	1	1	yes	characteristic
47	C₂²×C₁₉₃	—	4	2	no	—
48	C₂²×C₁₉₃	—	8	1	no	—
49	C₂²×C₁₉₃	—	8	1	no	—
50	C₂²×C₁₉₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₁₉₃	—	2	1	no	—
52	C₄×C₁₉₃	—	2	1	no	—
53	C₄×C₁₉₃	—	8	1	no	—
54	C₄×C₁₉₃	—	8	1	no	—
55	C₄×C₁₉₃	—	8	1	no	—
56	C₄×C₁₉₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₉₃	#137	1	1	yes	characteristic
58	C₂×C₁₉₃	—	8	2	no	—
59	C₂×C₁₉₃	#474	1	1	yes	characteristic
60	C₂×C₁₉₃	#508	1	1	yes	characteristic
61	C₂×C₁₉₃	#509	1	1	yes	characteristic
62	C₁₉₃	#2228	1	1	yes	characteristic, direct product factor, Sylow
63	#2228	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	#499	C₂×C₁₉₃	1	1	yes	characteristic, complemented
65	C₄×D₈	C₂×C₁₉₃	1	1	yes	characteristic
66	#367	C₂×C₁₉₃	1	1	yes	characteristic, complemented
67	#366	C₂×C₁₉₃	1	1	yes	characteristic, complemented
68	C₄⋊C₁₆	C₂×C₁₉₃	1	1	yes	characteristic, complemented
69	#357	C₂×C₁₉₃	1	1	yes	characteristic
70	#357	C₂×C₁₉₃	1	1	yes	characteristic
71	C₂×D₈	C₂²×C₁₉₃	1	1	yes	characteristic
72	#125	—	2	1	no	—
73	C₄×D₄	—	2	1	no	—
74	C₂×C₁₆	C₂²×C₁₉₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×C₁₉₃	1	1	yes	characteristic
76	#107	C₂²×C₁₉₃	1	1	yes	characteristic
77	#107	C₂²×C₁₉₃	1	1	yes	characteristic
78	#107	C₂²×C₁₉₃	1	1	yes	characteristic
79	#106	—	2	1	no	—
80	#102	—	2	1	no	—
81	C₄×C₈	C₂²×C₁₉₃	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Q₈×C₁₉₃	1	2	yes	—
86	C₂×C₈	D₄×C₁₉₃	1	1	yes	characteristic
87	C₂×C₈	C₂³×C₁₉₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	4	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	D₄×C₁₉₃	1	1	yes	characteristic
95	C₁₆	—	2	1	no	—
96	C₁₆	—	2	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	D₈×C₁₉₃	1	1	yes	characteristic
105	C₂×C₄	D₈×C₁₉₃	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄×C₁₉₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	C₂×Q₈×C₁₉₃	1	1	yes	characteristic
109	C₈	#41×C₁₉₃	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	C₂×D₈×C₁₉₃	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	C₂²⋊Q₈×C₁₉₃	1	1	yes	characteristic
120	C₄	#137×C₁₉₃	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#474×C₁₉₃	1	1	yes	characteristic
123	C₂	#508×C₁₉₃	1	1	yes	characteristic
124	C₂	#509×C₁₉₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#499×C₁₉₃

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₁₉₃

C₂

yes

maximal, characteristic

#367×C₁₉₃

C₂

yes

maximal, characteristic, complemented

#366×C₁₉₃

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₁₉₃

C₂

yes

maximal, characteristic, complemented

#357×C₁₉₃

C₂

yes

maximal, characteristic

#357×C₁₉₃

C₂

yes

maximal, characteristic

C₂×D₈×C₁₉₃

C₂²

yes

characteristic

#125×C₁₉₃

—

C₄×D₄×C₁₉₃

—

C₂×C₁₆×C₁₉₃

C₂²

yes

characteristic

C₂×C₁₆×C₁₉₃

C₂²

yes

characteristic

#107×C₁₉₃

C₂²

yes

characteristic

#107×C₁₉₃

C₂²

yes

characteristic

#107×C₁₉₃

C₂²

yes

characteristic

#106×C₁₉₃

—

#102×C₁₉₃

—

C₄×C₈×C₁₉₃

C₂²

yes

characteristic

C₂×D₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

D₈×C₁₉₃

—

D₈×C₁₉₃

Q₈

yes

—

C₂×C₈×C₁₉₃

D₄

yes

characteristic

C₂×C₈×C₁₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₂²⋊C₄×C₁₉₃

—

C₄²×C₁₉₃

D₄

yes

characteristic

C₁₆×C₁₉₃

—

C₁₆×C₁₉₃

—

C₂³×C₁₉₃

—

D₄×C₁₉₃

—

D₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

D₈

yes

characteristic

C₂×C₄×C₁₉₃

D₈

yes

characteristic

C₂×C₄×C₁₉₃

C₂×D₄

yes

characteristic

C₈×C₁₉₃

—

C₈×C₁₉₃

C₂×Q₈

yes

characteristic

C₈×C₁₉₃

#41

yes

characteristic

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₂×D₈

yes

characteristic

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₃

#137

yes

characteristic

C₂×C₁₉₃

—

C₂×C₁₉₃

yes

C₂×C₁₉₃

yes

C₂×C₁₉₃

yes

C₁₉₃

yes

characteristic, direct product factor, Sylow

#2228

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

#499

C₂×C₁₉₃

yes

characteristic, complemented

C₄×D₈

C₂×C₁₉₃

yes

characteristic

#367

C₂×C₁₉₃

yes

characteristic, complemented

#366

C₂×C₁₉₃

yes

characteristic, complemented

C₄⋊C₁₆

C₂×C₁₉₃

yes

characteristic, complemented

#357

C₂×C₁₉₃

yes

characteristic

#357

C₂×C₁₉₃

yes

characteristic

C₂×D₈

C₂²×C₁₉₃

yes

characteristic

#125

—

C₄×D₄

—

C₂×C₁₆

C₂²×C₁₉₃

yes

characteristic

C₂×C₁₆

C₂²×C₁₉₃

yes

characteristic

#107

C₂²×C₁₉₃

yes

characteristic

#107

C₂²×C₁₉₃

yes

characteristic

#107

C₂²×C₁₉₃

yes

characteristic

#106

—

#102

—

C₄×C₈

C₂²×C₁₉₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

Q₈×C₁₉₃

yes

—

C₂×C₈

D₄×C₁₉₃

yes

characteristic

C₂×C₈

C₂³×C₁₉₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₁₉₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

D₈×C₁₉₃

yes

characteristic

105

C₂×C₄

D₈×C₁₉₃

yes

characteristic

106

C₂×C₄

C₂×D₄×C₁₉₃

yes

characteristic

107

C₈

—

108

C₈

C₂×Q₈×C₁₉₃

yes

characteristic

109

C₈

#41×C₁₉₃

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×D₈×C₁₉₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂²⋊Q₈×C₁₉₃

yes

characteristic

120

C₄

#137×C₁₉₃

yes

characteristic

121

C₂

—

122

C₂

#474×C₁₉₃

yes

characteristic

123

C₂

#508×C₁₉₃

yes

characteristic

124

C₂

#509×C₁₉₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₉₃
prime radical	C₂×D₈×C₁₉₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₃ ◁ C₄²×C₁₉₃ ◁ C₄×C₈×C₁₉₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉₃

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₁₉₃

prime radical

C₂×D₈×C₁₉₃

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₃ ◁ C₄²×C₁₉₃ ◁ C₄×C₈×C₁₉₃ ◁ G

N	H ≅ G/N	type
1. #499×C₁₉₃	C₂	semidirect
3. #367×C₁₉₃	C₂	semidirect
4. #366×C₁₉₃	C₂	semidirect
5. C₄⋊C₁₆×C₁₉₃	C₂	semidirect
62. C₁₉₃	#2228	direct (N × H)
63. #2228	C₁₉₃	direct (N × H)
64. #499	C₂×C₁₉₃	semidirect
66. #367	C₂×C₁₉₃	semidirect
67. #366	C₂×C₁₉₃	semidirect
68. C₄⋊C₁₆	C₂×C₁₉₃	semidirect

H ≅ G/N

type

1. #499×C₁₉₃

C₂

semidirect

3. #367×C₁₉₃

C₂

semidirect

4. #366×C₁₉₃

C₂

semidirect

5. C₄⋊C₁₆×C₁₉₃

C₂

semidirect

62. C₁₉₃

#2228

direct (N × H)

63. #2228

C₁₉₃

direct (N × H)

64. #499

C₂×C₁₉₃

semidirect

66. #367

C₂×C₁₉₃

semidirect

67. #366

C₂×C₁₉₃

semidirect

68. C₄⋊C₁₆

C₂×C₁₉₃

semidirect

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

subgroup	index
#499×C₁₉₃	2
C₄×D₈×C₁₉₃	2
#367×C₁₉₃	2
#366×C₁₉₃	2
C₄⋊C₁₆×C₁₉₃	2
#357×C₁₉₃	2
#2228	193

subgroup

index

#499×C₁₉₃

C₄×D₈×C₁₉₃

#367×C₁₉₃

#366×C₁₉₃

C₄⋊C₁₆×C₁₉₃

#357×C₁₉₃

#2228

193

Subgroups — #2228×C₁₉₃ #37591845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2228×C193 #37591845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2228×C₁₉₃ #37591845