Group #38768894 — subgroups

Subgroups — #38768894

Order 25472.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20263962	C₂	1	1	yes	maximal, characteristic
2	#27164054	C₂	1	2	yes	maximal, complemented
3	#27163992	C₂	1	1	yes	maximal, characteristic
4	#27163774	C₂	1	2	yes	maximal, complemented
5	#410×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	#20263962	—	2	2	no	—
7	#20263962	—	2	2	no	—
8	C₂×#15691870	C₄	1	2	yes	complemented
9	C₂×#15691870	C₂²	1	1	yes	characteristic
10	C₂²×C₁₉₉⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
12	#20263914	C₂²	1	2	yes	—
13	#20263888	—	2	2	no	—
14	#20263886	—	2	2	no	—
15	#20263870	—	2	2	no	—
16	C₂×#41×C₁₉₉	C₂²	1	1	yes	characteristic
17	#117×C₁₉₉	C₂²	1	2	yes	—
18	#99×C₁₉₉	—	2	2	no	—
19	#15691870	—	2	4	no	—
20	#15691870	—	2	2	no	—
21	C₂×C₁₉₉⋊C₈	—	2	2	no	—
22	C₂×C₁₉₉⋊C₈	—	2	2	no	—
23	C₂×C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₁₉₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₉₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₉₉⋊C₈	D₄	1	2	yes	—
27	C₂×C₁₉₉⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₁₉₉	—	2	2	no	—
29	C₂²⋊Dic₁₉₉	—	2	2	no	—
30	C₄×Dic₁₉₉	—	2	2	no	—
31	#41×C₁₉₉	—	2	2	no	—
32	#41×C₁₉₉	—	2	2	no	—
33	C₂×Q₈×C₁₉₉	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₁₉₉	D₄	1	2	yes	—
35	C₂×D₄×C₁₉₉	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉₉	—	2	2	no	—
39	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
40	C₄²×C₁₉₉	—	2	2	no	—
41	C₂×Dic₁₉₉	—	4	2	no	—
42	C₁₉₉⋊C₈	—	2	2	no	—
43	C₁₉₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₁₉₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₁₉₉⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₉₉	C₄⋊C₄	1	2	yes	—
47	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₁₉₉	—	2	2	no	—
49	D₄×C₁₉₉	—	2	4	no	—
50	D₄×C₁₉₉	—	2	2	no	—
51	C₂×C₄×C₁₉₉	—	2	2	no	—
52	C₂×C₄×C₁₉₉	—	2	2	no	—
53	C₂×C₄×C₁₉₉	—	4	2	no	—
54	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉₉	#41	1	2	yes	—
59	C₂²×C₁₉₉	—	2	2	no	—
60	C₂²×C₁₉₉	—	4	1	no	—
61	C₂²×C₁₉₉	—	4	2	no	—
62	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₉₉	#121	1	2	yes	—
64	C₄×C₁₉₉	—	2	2	no	—
65	C₄×C₁₉₉	—	8	2	no	—
66	C₄×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₉₉	#117	1	1	yes	characteristic
68	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
69	Dic₁₉₉	—	8	2	no	—
70	C₂×C₁₉₉	—	2	2	no	—
71	C₂×C₁₉₉	—	2	1	no	—
72	C₂×C₁₉₉	—	4	2	no	—
73	C₂×C₁₉₉	#380	1	1	yes	characteristic
74	C₁₉₉	#1812	1	1	yes	characteristic, complemented, Sylow
75	#1812	—	199	1	no	maximal, Sylow
76	C₂×#131	—	199	1	no	—
77	#427	—	199	2	no	—
78	#410	—	199	1	no	—
79	#410	D₁₉₉	1	1	yes	characteristic
80	H(3,ℤ/4)	—	199	2	no	—
81	C₂×#41	C₂×D₁₉₉	1	1	yes	characteristic
82	#131	—	398	2	no	—
83	#131	—	398	2	no	—
84	C₂×M₁₆	—	199	2	no	—
85	C₂×M₁₆	—	199	1	no	—
86	C₂²×C₈	—	199	2	no	—
87	C₂²×C₈	—	199	1	no	—
88	#117	—	199	2	no	—
89	#117	C₂×D₁₉₉	1	2	yes	—
90	#107	—	398	2	no	—
91	#106	—	398	2	no	—
92	#99	—	2	2	no	—
93	#99	—	398	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₃₉₈	1	1	yes	characteristic
97	C₂×D₄	C₂²⋊D₁₉₉	1	2	yes	—
98	C₂×D₄	Dic₃₉₈	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₁₉₉	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
101	M₁₆	—	398	2	no	—
102	M₁₆	—	398	4	no	—
103	C₂×C₈	—	398	2	no	—
104	C₂×C₈	—	398	2	no	—
105	C₂×C₈	—	199	2	no	—
106	C₂×C₈	—	199	1	no	—
107	C₂×C₈	—	199	1	no	—
108	C₂×C₈	—	199	2	no	—
109	C₂×C₈	—	199	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	398	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	398	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	398	2	no	—
116	C₂³	Dic₁₉₉⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	796	2	no	—
125	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
128	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
129	C₂×C₄	#15691885	1	2	yes	—
130	C₈	—	398	2	no	—
131	C₈	—	199	1	no	—
132	C₈	—	199	1	no	—
133	C₈	—	199	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
138	C₂²	#20263927	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	1592	2	no	—
142	C₄	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic
143	C₄	#20263911	1	1	yes	characteristic
144	C₄	#20263932	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#27163876	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263962

C₂

yes

maximal, characteristic

#27164054

C₂

yes

maximal, complemented

#27163992

C₂

yes

maximal, characteristic

#27163774

C₂

yes

maximal, complemented

#410×C₁₉₉

C₂

yes

maximal, characteristic

#20263962

—

#20263962

—

C₂×#15691870

C₄

yes

complemented

C₂×#15691870

C₂²

yes

characteristic

C₂²×C₁₉₉⋊C₈

C₄

yes

complemented

C₂²×C₁₉₉⋊C₈

C₂²

yes

characteristic

#20263914

C₂²

yes

—

#20263888

—

#20263886

—

#20263870

—

C₂×#41×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

—

#99×C₁₉₉

—

#15691870

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

—

#41×C₁₉₉

—

#41×C₁₉₉

—

C₂×Q₈×C₁₉₉

Q₈

yes

characteristic

C₂×D₄×C₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

Q₈

yes

characteristic

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₉₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₉₉⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉₉

C₄⋊C₄

yes

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₉

#121

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₉

#117

yes

characteristic

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#380

yes

characteristic

C₁₉₉

#1812

yes

characteristic, complemented, Sylow

#1812

—

199

maximal, Sylow

C₂×#131

—

199

—

#427

—

199

—

#410

—

199

—

#410

D₁₉₉

yes

characteristic

H(3,ℤ/4)

—

199

—

C₂×#41

C₂×D₁₉₉

yes

characteristic

#131

—

398

—

#131

—

398

—

C₂×M₁₆

—

199

—

C₂×M₁₆

—

199

—

C₂²×C₈

—

199

—

C₂²×C₈

—

199

—

#117

—

199

—

#117

C₂×D₁₉₉

yes

—

#107

—

398

—

#106

—

398

—

#99

—

#99

—

398

—

#41

—

#41

—

C₂×Q₈

Dic₃₉₈

yes

characteristic

C₂×D₄

C₂²⋊D₁₉₉

yes

—

C₂×D₄

Dic₃₉₈

yes

characteristic

C₂²×C₄

C₄×D₁₉₉

yes

—

100

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

101

M₁₆

—

398

—

102

M₁₆

—

398

—

103

C₂×C₈

—

398

—

104

C₂×C₈

—

398

—

105

C₂×C₈

—

199

—

106

C₂×C₈

—

199

—

107

C₂×C₈

—

199

—

108

C₂×C₈

—

199

—

109

C₂×C₈

—

199

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

398

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

398

—

114

C₄²

—

115

C₄²

—

398

—

116

C₂³

Dic₁₉₉⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

796

—

125

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

127

C₂×C₄

D₄×D₁₉₉

yes

—

128

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

129

C₂×C₄

#15691885

yes

—

130

C₈

—

398

—

131

C₈

—

199

—

132

C₈

—

199

—

133

C₈

—

199

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₁₉₉⋊C₄

yes

characteristic

138

C₂²

#20263927

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

1592

—

142

C₄

C₂²⋊C₄×D₁₉₉

yes

characteristic

143

C₄

#20263911

yes

characteristic

144

C₄

#20263932

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#27163876

yes

characteristic

	subgroup	G/H
center	C₄	#20263911
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#410×C₁₉₉	C₂
socle	C₂×C₁₉₉	#380
prime radical	C₂×#41×C₁₉₉	C₂²
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#410	D₁₉₉
hypocenter	C₁₉₉	#1812
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#20263911

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#410×C₁₉₉

C₂

socle

C₂×C₁₉₉

#380

prime radical

C₂×#41×C₁₉₉

C₂²

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #27164054	C₂	semidirect
4. #27163774	C₂	semidirect
8. C₂×#15691870	C₄	semidirect
10. C₂²×C₁₉₉⋊C₈	C₄	semidirect
74. C₁₉₉	#1812	semidirect

H ≅ G/N

type

2. #27164054

C₂

semidirect

4. #27163774

C₂

semidirect

8. C₂×#15691870

C₄

semidirect

10. C₂²×C₁₉₉⋊C₈

C₄

semidirect

74. C₁₉₉

#1812

semidirect

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

subgroup	index
C₂×#20263962	2
#27164054	2
#27163992	2
#27163774	2
#410×C₁₉₉	2
#1812	199

subgroup

index

#410×C₁₉₉

199

Subgroups — #38768894

Conjugacy classes of subgroups

Complements

Maximal subgroups