Group #37583033 — subgroups

Subgroups — #37583033

Order 24448.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#27133219	C₂	1	2	yes	maximal
2	#27133067	C₂	1	2	yes	maximal, complemented
3	#27133065	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133062	C₂	1	2	yes	maximal, complemented
5	#27133060	C₂	1	2	yes	maximal, complemented
6	#27132700	C₂	1	2	yes	maximal
7	#27132698	C₂	1	2	yes	maximal
8	C₄×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
12	#20259321	C₂²	1	4	yes	—
13	#20259314	C₂²	1	2	yes	—
14	#20259300	—	2	4	no	—
15	#20259287	—	2	4	no	—
16	#20259280	C₂²	1	4	yes	—
17	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
18	#20259256	C₂²	1	2	yes	—
19	#20259255	C₂²	1	4	yes	—
20	#20259254	C₂²	1	2	yes	—
21	#20259253	C₂²	1	4	yes	—
22	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
25	#125×C₁₉₁	C₂²	1	2	yes	—
26	#107×C₁₉₁	C₂²	1	2	yes	—
27	#106×C₁₉₁	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₃₈₂	—	2	4	no	—
30	C₂×C₄×D₁₉₁	D₄	1	2	yes	—
31	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
32	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
34	C₈×D₁₉₁	Q₈	1	4	yes	complemented
35	C₈×D₁₉₁	Q₈	1	4	yes	complemented
36	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
37	Dic₁₉₁⋊C₄	—	2	4	no	—
38	Dic₁₉₁⋊C₄	—	2	4	no	—
39	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
40	C₄×Dic₁₉₁	—	2	4	no	—
41	C₄×Dic₁₉₁	D₄	1	2	yes	—
42	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₉₁	—	2	4	no	—
46	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
47	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
49	Dic₃₈₂	—	4	8	no	—
50	C₂×Dic₁₉₁	—	2	4	no	—
51	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
53	C₄×D₁₉₁	—	2	4	no	—
54	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
55	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
56	C₁₉₁⋊C₈	C₂×Q₈	1	2	yes	—
57	C₁₉₁⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₉₁	—	2	4	no	—
59	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₉₁	C₂×Q₈	1	2	yes	—
62	C₈×C₁₉₁	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₁₉₁	—	2	2	no	—
65	C₂×D₁₉₁	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₉₁	—	2	2	no	—
67	C₄×C₁₉₁	—	4	4	no	—
68	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₉₁	—	4	4	no	—
71	Dic₁₉₁	C₄⋊Q₈	1	2	yes	—
72	Dic₁₉₁	#135	1	4	yes	—
73	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₉₁	C₂×#135	1	2	yes	—
75	D₁₉₁	—	2	2	no	—
76	C₁₉₁	#3160	1	1	yes	characteristic, complemented, Sylow
77	#3160	—	191	1	no	maximal, Sylow
78	#533	—	191	2	no	—
79	#500	—	191	2	no	—
80	#499	—	191	2	no	—
81	#499	—	191	1	no	—
82	#499	D₁₉₁	1	1	yes	characteristic, complemented
83	#498	—	191	2	no	—
84	#427	—	191	2	no	—
85	#427	—	191	2	no	—
86	C₂×C₄×C₈	—	191	1	no	—
87	C₂²×C₈	—	191	1	no	—
88	C₂²×C₈	—	191	1	no	—
89	C₄⋊Q₈	—	191	4	no	—
90	#125	—	191	2	no	—
91	#125	C₂×D₁₉₁	1	2	yes	—
92	C₂²⋊Q₈	—	382	4	no	—
93	C₄×Q₈	—	382	4	no	—
94	#117	—	191	4	no	—
95	C₂×C₄²	—	191	1	no	—
96	#107	—	191	2	no	—
97	#107	—	191	4	no	—
98	#107	C₂×D₁₉₁	1	2	yes	—
99	#106	—	191	2	no	—
100	#106	—	191	4	no	—
101	#106	C₂×D₁₉₁	1	2	yes	—
102	C₄×C₈	—	191	1	no	—
103	C₄×C₈	—	191	1	no	—
104	C₄×C₈	—	191	1	no	—
105	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
106	C₂×Q₈	—	382	4	no	—
107	C₂²×C₄	—	191	2	no	—
108	C₂²×C₄	—	191	1	no	—
109	C₂×C₈	—	191	1	no	—
110	C₂×C₈	—	191	1	no	—
111	C₂×C₈	—	191	4	no	—
112	C₂×C₈	—	191	4	no	—
113	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	382	4	no	—
117	C₄⋊C₄	—	382	4	no	—
118	C₄⋊C₄	—	191	4	no	—
119	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
120	C₂²⋊C₄	—	382	4	no	—
121	C₄²	—	382	4	no	—
122	C₄²	—	191	2	no	—
123	C₄²	—	191	1	no	—
124	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
125	C₂³	—	191	1	no	—
126	Q₈	—	764	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	382	4	no	—
129	C₂×C₄	—	382	4	no	—
130	C₂×C₄	—	191	1	no	—
131	C₂×C₄	—	191	2	no	—
132	C₂×C₄	—	191	2	no	—
133	C₂×C₄	—	191	2	no	—
134	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
135	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
136	C₈	—	191	2	no	—
137	C₈	—	191	2	no	—
138	C₈	Q₈×D₁₉₁	1	2	yes	—
139	C₈	Q₈×D₁₉₁	1	2	yes	—
140	C₂²	—	382	2	no	—
141	C₂²	—	191	2	no	—
142	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	191	4	no	—
146	C₄	—	191	2	no	—
147	C₄	—	764	4	no	—
148	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
150	C₂	—	382	2	no	—
151	C₂	C₄⋊Q₈×D₁₉₁	1	1	yes	characteristic
152	C₂	#135×D₁₉₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133219

C₂

yes

maximal

#27133067

C₂

yes

maximal, complemented

#27133065

C₂

yes

maximal, characteristic, complemented

#27133062

C₂

yes

maximal, complemented

#27133060

C₂

yes

maximal, complemented

#27132700

C₂

yes

maximal

#27132698

C₂

yes

maximal

C₄×C₈×D₁₉₁

C₂

yes

maximal, characteristic

#499×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

#20259321

C₂²

yes

—

#20259314

C₂²

yes

—

#20259300

—

#20259287

—

#20259280

C₂²

yes

—

C₄²×D₁₉₁

C₂²

yes

characteristic

#20259256

C₂²

yes

—

#20259255

C₂²

yes

—

#20259254

C₂²

yes

—

#20259253

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

#125×C₁₉₁

C₂²

yes

—

#107×C₁₉₁

C₂²

yes

—

#106×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₈×D₁₉₁

Q₈

yes

complemented

C₈×D₁₉₁

Q₈

yes

complemented

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

C₂×Q₈

yes

—

C₄×D₁₉₁

C₂×Q₈

yes

—

C₁₉₁⋊C₈

C₂×Q₈

yes

—

C₁₉₁⋊C₈

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

C₂×Q₈

yes

—

C₈×C₁₉₁

C₂×Q₈

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

C₄⋊Q₈

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²×Q₈

yes

characteristic

C₄×C₁₉₁

C₂²×Q₈

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

C₄⋊Q₈

yes

—

Dic₁₉₁

#135

yes

—

C₂×C₁₉₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₁

C₂×#135

yes

—

D₁₉₁

—

C₁₉₁

#3160

yes

characteristic, complemented, Sylow

#3160

—

191

maximal, Sylow

#533

—

191

—

#500

—

191

—

#499

—

191

—

#499

—

191

—

#499

D₁₉₁

yes

characteristic, complemented

#498

—

191

—

#427

—

191

—

#427

—

191

—

C₂×C₄×C₈

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

191

—

#125

—

191

—

#125

C₂×D₁₉₁

yes

—

C₂²⋊Q₈

—

382

—

C₄×Q₈

—

382

—

#117

—

191

—

C₂×C₄²

—

191

—

#107

—

191

—

#107

—

191

—

#107

C₂×D₁₉₁

yes

—

#106

—

191

—

100

#106

—

191

—

101

#106

C₂×D₁₉₁

yes

—

102

C₄×C₈

—

191

—

103

C₄×C₈

—

191

—

104

C₄×C₈

—

191

—

105

C₄×C₈

C₂×D₁₉₁

yes

characteristic

106

C₂×Q₈

—

382

—

107

C₂²×C₄

—

191

—

108

C₂²×C₄

—

191

—

109

C₂×C₈

—

191

—

110

C₂×C₈

—

191

—

111

C₂×C₈

—

191

—

112

C₂×C₈

—

191

—

113

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

114

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

382

—

117

C₄⋊C₄

—

382

—

118

C₄⋊C₄

—

191

—

119

C₄⋊C₄

C₂²×D₁₉₁

yes

—

120

C₂²⋊C₄

—

382

—

121

C₄²

—

382

—

122

C₄²

—

191

—

123

C₄²

—

191

—

124

C₄²

C₂²×D₁₉₁

yes

characteristic

125

C₂³

—

191

—

126

Q₈

—

764

—

127

C₂×C₄

—

128

C₂×C₄

—

382

—

129

C₂×C₄

—

382

—

130

C₂×C₄

—

191

—

131

C₂×C₄

—

191

—

132

C₂×C₄

—

191

—

133

C₂×C₄

—

191

—

134

C₂×C₄

D₄×D₁₉₁

yes

—

135

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

136

C₈

—

191

—

137

C₈

—

191

—

138

C₈

Q₈×D₁₉₁

yes

—

139

C₈

Q₈×D₁₉₁

yes

—

140

C₂²

—

382

—

141

C₂²

—

191

—

142

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

191

—

146

C₄

—

191

—

147

C₄

—

764

—

148

C₄

C₂×Q₈×D₁₉₁

yes

characteristic

149

C₄

C₂×Q₈×D₁₉₁

yes

characteristic

150

C₂

—

382

—

151

C₂

C₄⋊Q₈×D₁₉₁

yes

characteristic

152

C₂

#135×D₁₉₁

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#499×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂²×D₁₉₁	C₂×D₄
hypercenter	#499	D₁₉₁
hypocenter	C₁₉₁	#3160
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#499×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂²×D₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #27133067	C₂	semidirect
3. #27133065	C₂	semidirect
4. #27133062	C₂	semidirect
5. #27133060	C₂	semidirect
9. #499×C₁₉₁	C₂	semidirect
34. C₈×D₁₉₁	Q₈	semidirect
35. C₈×D₁₉₁	Q₈	semidirect
76. C₁₉₁	#3160	semidirect
82. #499	D₁₉₁	semidirect

H ≅ G/N

type

2. #27133067

C₂

semidirect

3. #27133065

C₂

semidirect

4. #27133062

C₂

semidirect

5. #27133060

C₂

semidirect

9. #499×C₁₉₁

C₂

semidirect

34. C₈×D₁₉₁

Q₈

semidirect

35. C₈×D₁₉₁

Q₈

semidirect

76. C₁₉₁

#3160

semidirect

82. #499

D₁₉₁

semidirect

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

subgroup	index
#27133219	2
#27133067	2
#27133065	2
#27133062	2
#27133060	2
#27132700	2
#27132698	2
C₄×C₈×D₁₉₁	2
#499×C₁₉₁	2
#3160	191

subgroup

index

C₄×C₈×D₁₉₁

#499×C₁₉₁

#3160

191

Subgroups — #37583033

Conjugacy classes of subgroups

Complements

Maximal subgroups