Group #37591752 — subgroups

Subgroups — #2135×C₁₉₃ #37591752

Order 24704.

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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	#536×C₁₉₃	C₂	1	1	yes	maximal, characteristic
2	#413×C₁₉₃	C₂	1	2	yes	maximal
3	#356×C₁₉₃	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₁₉₃	C₂²	1	1	yes	characteristic
5	C₂²×Q₈×C₁₉₃	C₄	1	1	yes	characteristic
6	C₂×M₁₆×C₁₉₃	—	2	2	no	—
7	C₄⋊Q₈×C₁₉₃	C₂²	1	2	yes	—
8	#124×C₁₉₃	C₄	1	2	yes	—
9	#123×C₁₉₃	—	2	2	no	—
10	C₂²⋊Q₈×C₁₉₃	—	2	2	no	—
11	#117×C₁₉₃	C₄	1	1	yes	characteristic
12	#101×C₁₉₃	C₂²	1	4	yes	—
13	#100×C₁₉₃	—	2	2	no	—
14	#41×C₁₉₃	—	4	2	no	—
15	C₂×Q₈×C₁₉₃	—	2	1	no	—
16	C₂×Q₈×C₁₉₃	—	4	2	no	—
17	C₂×Q₈×C₁₉₃	C₂×C₄	1	2	yes	—
18	C₂×Q₈×C₁₉₃	C₂³	1	1	yes	characteristic
19	C₂×D₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₁₉₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₉₃	—	2	1	no	—
22	C₂²×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₃	D₄	1	2	yes	—
24	M₁₆×C₁₉₃	—	2	4	no	—
25	M₁₆×C₁₉₃	—	4	2	no	—
26	C₂×C₈×C₁₉₃	—	4	2	no	—
27	C₄⋊C₄×C₁₉₃	—	2	1	no	—
28	C₄⋊C₄×C₁₉₃	—	2	4	no	—
29	C₂²⋊C₄×C₁₉₃	—	2	1	no	—
30	C₂²⋊C₄×C₁₉₃	—	2	4	no	—
31	C₄²×C₁₉₃	C₂×C₄	1	2	yes	—
32	C₂³×C₁₉₃	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₁₉₃	—	4	1	no	—
35	Q₈×C₁₉₃	—	4	2	no	—
36	Q₈×C₁₉₃	—	4	2	no	—
37	D₄×C₁₉₃	—	4	1	no	—
38	D₄×C₁₉₃	—	4	2	no	—
39	C₂×C₄×C₁₉₃	—	2	2	no	—
40	C₂×C₄×C₁₉₃	—	2	2	no	—
41	C₂×C₄×C₁₉₃	—	2	2	no	—
42	C₂×C₄×C₁₉₃	—	2	2	no	—
43	C₂×C₄×C₁₉₃	—	2	2	no	—
44	C₂×C₄×C₁₉₃	—	4	2	no	—
45	C₂×C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₃	C₂×D₄	1	2	yes	—
48	C₈×C₁₉₃	—	4	4	no	—
49	C₂²×C₁₉₃	—	2	1	no	—
50	C₂²×C₁₉₃	—	2	2	no	—
51	C₂²×C₁₉₃	—	4	2	no	—
52	C₂²×C₁₉₃	—	4	1	no	—
53	C₂²×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₁₉₃	—	2	2	no	—
55	C₄×C₁₉₃	—	2	1	no	—
56	C₄×C₁₉₃	—	4	2	no	—
57	C₄×C₁₉₃	—	4	2	no	—
58	C₄×C₁₉₃	#99	1	2	yes	—
59	C₂×C₁₉₃	—	2	1	no	—
60	C₂×C₁₉₃	—	4	1	no	—
61	C₂×C₁₉₃	—	4	2	no	—
62	C₂×C₁₉₃	C₂×#99	1	1	yes	characteristic
63	C₁₉₃	#2135	1	1	yes	characteristic, direct product factor, Sylow
64	#2135	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	#536	C₂×C₁₉₃	1	1	yes	characteristic
66	#413	C₂×C₁₉₃	1	2	yes	—
67	#356	C₂×C₁₉₃	1	4	yes	complemented
68	C₂×#41	C₂²×C₁₉₃	1	1	yes	characteristic
69	C₂²×Q₈	C₄×C₁₉₃	1	1	yes	characteristic
70	C₂×M₁₆	—	2	2	no	—
71	C₄⋊Q₈	C₂²×C₁₉₃	1	2	yes	—
72	#124	C₄×C₁₉₃	1	2	yes	—
73	#123	—	2	2	no	—
74	C₂²⋊Q₈	—	2	2	no	—
75	#117	C₄×C₁₉₃	1	1	yes	characteristic
76	#101	C₂²×C₁₉₃	1	4	yes	—
77	#100	—	2	2	no	—
78	#41	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×C₄×C₁₉₃	1	2	yes	—
82	C₂×Q₈	C₂³×C₁₉₃	1	1	yes	characteristic
83	C₂×D₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
84	C₂×D₄	D₄×C₁₉₃	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
87	C₂²×C₄	D₄×C₁₉₃	1	2	yes	—
88	M₁₆	—	2	4	no	—
89	M₁₆	—	4	2	no	—
90	C₂×C₈	—	4	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₄²	C₂×C₄×C₁₉₃	1	2	yes	—
96	C₂³	C₂²⋊C₄×C₁₉₃	1	2	yes	—
97	C₂³	C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×C₄×C₁₉₃	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄×C₁₉₃	1	2	yes	—
112	C₈	—	4	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	C₂×C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	#99×C₁₉₃	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	4	1	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×#99×C₁₉₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#536×C₁₉₃

C₂

yes

maximal, characteristic

#413×C₁₉₃

C₂

yes

maximal

#356×C₁₉₃

C₂

yes

maximal, complemented

C₂×#41×C₁₉₃

C₂²

yes

characteristic

C₂²×Q₈×C₁₉₃

C₄

yes

characteristic

C₂×M₁₆×C₁₉₃

—

C₄⋊Q₈×C₁₉₃

C₂²

yes

—

#124×C₁₉₃

C₄

yes

—

#123×C₁₉₃

—

C₂²⋊Q₈×C₁₉₃

—

#117×C₁₉₃

C₄

yes

characteristic

#101×C₁₉₃

C₂²

yes

—

#100×C₁₉₃

—

#41×C₁₉₃

—

C₂×Q₈×C₁₉₃

—

C₂×Q₈×C₁₉₃

—

C₂×Q₈×C₁₉₃

C₂×C₄

yes

—

C₂×Q₈×C₁₉₃

C₂³

yes

characteristic

C₂×D₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₉₃

D₄

yes

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₃

D₄

yes

—

M₁₆×C₁₉₃

—

M₁₆×C₁₉₃

—

C₂×C₈×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₄⋊C₄×C₁₉₃

—

C₂²⋊C₄×C₁₉₃

—

C₂²⋊C₄×C₁₉₃

—

C₄²×C₁₉₃

C₂×C₄

yes

—

C₂³×C₁₉₃

C₂²⋊C₄

yes

—

C₂³×C₁₉₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₉₃

—

Q₈×C₁₉₃

—

Q₈×C₁₉₃

—

D₄×C₁₉₃

—

D₄×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

characteristic

C₂×C₄×C₁₉₃

C₂×D₄

yes

—

C₈×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

#99

yes

—

C₂×C₁₉₃

—

C₂×C₁₉₃

—

C₂×C₁₉₃

—

C₂×C₁₉₃

C₂×#99

yes

characteristic

C₁₉₃

#2135

yes

characteristic, direct product factor, Sylow

#2135

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

#536

C₂×C₁₉₃

yes

characteristic

#413

C₂×C₁₉₃

yes

—

#356

C₂×C₁₉₃

yes

complemented

C₂×#41

C₂²×C₁₉₃

yes

characteristic

C₂²×Q₈

C₄×C₁₉₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₁₉₃

yes

—

#124

C₄×C₁₉₃

yes

—

#123

—

C₂²⋊Q₈

—

#117

C₄×C₁₉₃

yes

characteristic

#101

C₂²×C₁₉₃

yes

—

#100

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₁₉₃

yes

—

C₂×Q₈

C₂³×C₁₉₃

yes

characteristic

C₂×D₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂×D₄

D₄×C₁₉₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂²×C₄

D₄×C₁₉₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₁₉₃

yes

—

C₂³

C₂²⋊C₄×C₁₉₃

yes

—

C₂³

C₂²⋊C₄×C₁₉₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊C₄×C₁₉₃

yes

characteristic

110

C₂×C₄

C₂²×C₄×C₁₉₃

yes

characteristic

111

C₂×C₄

C₂×D₄×C₁₉₃

yes

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₂²⋊C₄×C₁₉₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#99×C₁₉₃

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×#99×C₁₉₃

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉₃	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₁₉₃
Frattini, cosocle	C₂×Q₈	C₂³×C₁₉₃
prime radical	C₂×#41×C₁₉₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉₃ ◁ C₂×C₄×C₁₉₃ ◁ C₂×#41×C₁₉₃ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉₃

C₂×#99

commutator

C₂×C₄

C₂²×C₄×C₁₉₃

Frattini, cosocle

C₂×Q₈

C₂³×C₁₉₃

prime radical

C₂×#41×C₁₉₃

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉₃ ◁ C₂×C₄×C₁₉₃ ◁ C₂×#41×C₁₉₃ ◁ G

N	H ≅ G/N	type
3. #356×C₁₉₃	C₂	semidirect
63. C₁₉₃	#2135	direct (N × H)
64. #2135	C₁₉₃	direct (N × H)
67. #356	C₂×C₁₉₃	semidirect

H ≅ G/N

type

3. #356×C₁₉₃

C₂

semidirect

63. C₁₉₃

#2135

direct (N × H)

64. #2135

C₁₉₃

direct (N × H)

67. #356

C₂×C₁₉₃

semidirect

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

subgroup	index
#536×C₁₉₃	2
#413×C₁₉₃	2
#356×C₁₉₃	2
#2135	193

subgroup

index

#536×C₁₉₃

#413×C₁₉₃

#356×C₁₉₃

#2135

193

Subgroups — #2135×C₁₉₃ #37591752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2135×C193 #37591752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2135×C₁₉₃ #37591752