Group #38768951 — subgroups

Subgroups — #38768951

Order 25472.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#27164002	C₂	1	1	yes	maximal, characteristic
3	#27163777	C₂	1	2	yes	maximal, complemented
4	#27163774	C₂	1	2	yes	maximal, complemented
5	#410×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691870	C₂²	1	2	yes	—
7	Q₈×Dic₁₉₉	—	2	1	no	—
8	D₄×Dic₁₉₉	—	2	2	no	—
9	D₄×Dic₁₉₉	—	2	1	no	—
10	#20263914	C₄	1	2	yes	complemented
11	#20263914	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₉₉	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
14	#20263874	—	2	1	no	—
15	#20263872	—	2	1	no	—
16	C₂×#41×C₁₉₉	C₂²	1	1	yes	characteristic
17	#117×C₁₉₉	C₂²	1	2	yes	—
18	#99×C₁₉₉	—	2	2	no	—
19	C₂²×Dic₁₉₉	—	2	2	no	—
20	C₂²×Dic₁₉₉	—	2	1	no	—
21	#15691870	—	2	2	no	—
22	#15691870	—	2	2	no	—
23	C₂×C₁₉₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₉	—	2	2	no	—
25	C₄⋊Dic₁₉₉	—	2	1	no	—
26	C₂²⋊Dic₁₉₉	—	2	2	no	—
27	C₂²⋊Dic₁₉₉	—	2	1	no	—
28	C₄×Dic₁₉₉	—	2	2	no	—
29	C₄×Dic₁₉₉	—	2	1	no	—
30	C₄×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₉₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₉₉	—	2	2	no	—
34	#41×C₁₉₉	—	2	2	no	—
35	C₂×Q₈×C₁₉₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₉₉	D₄	1	2	yes	—
37	C₂×D₄×C₁₉₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉₉	—	2	2	no	—
41	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
42	C₄²×C₁₉₉	—	2	2	no	—
43	C₂×Dic₁₉₉	—	2	1	no	—
44	C₂×Dic₁₉₉	—	2	2	no	—
45	C₂×Dic₁₉₉	—	2	1	no	—
46	C₂×Dic₁₉₉	—	4	2	no	—
47	C₂×Dic₁₉₉	—	4	4	no	—
48	C₁₉₉⋊C₈	—	4	2	no	—
49	C₂³×C₁₉₉	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₉₉	—	2	2	no	—
52	D₄×C₁₉₉	—	2	4	no	—
53	D₄×C₁₉₉	—	2	2	no	—
54	C₂×C₄×C₁₉₉	—	2	2	no	—
55	C₂×C₄×C₁₉₉	—	2	2	no	—
56	C₂×C₄×C₁₉₉	—	4	2	no	—
57	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₉	#41	1	2	yes	—
61	C₂×C₄×C₁₉₉	#41	1	2	yes	—
62	C₂²×C₁₉₉	—	2	2	no	—
63	C₂²×C₁₉₉	—	4	1	no	—
64	C₂²×C₁₉₉	—	4	2	no	—
65	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉₉	#123	1	2	yes	—
67	C₄×C₁₉₉	—	2	2	no	—
68	C₄×C₁₉₉	—	8	2	no	—
69	C₄×C₁₉₉	#117	1	1	yes	characteristic
70	C₄×C₁₉₉	#117	1	1	yes	characteristic
71	C₄×C₁₉₉	#123	1	2	yes	—
72	Dic₁₉₉	—	2	2	no	—
73	Dic₁₉₉	—	4	2	no	—
74	Dic₁₉₉	—	4	1	no	—
75	C₂×C₁₉₉	—	2	1	no	—
76	C₂×C₁₉₉	—	2	2	no	—
77	C₂×C₁₉₉	—	4	2	no	—
78	C₂×C₁₉₉	#381	1	1	yes	characteristic
79	C₁₉₉	#1827	1	1	yes	characteristic, complemented, Sylow
80	#1827	—	199	1	no	maximal, Sylow
81	C₄×#41	—	199	1	no	—
82	#413	—	199	1	no	—
83	#410	D₁₉₉	1	1	yes	characteristic
84	#339	—	199	2	no	—
85	H(3,ℤ/4)	—	199	2	no	—
86	C₂×#41	C₂×D₁₉₉	1	1	yes	characteristic
87	C₂×M₁₆	—	199	2	no	—
88	C₄×Q₈	—	398	1	no	—
89	C₄×D₄	—	398	2	no	—
90	C₄×D₄	—	398	1	no	—
91	#117	—	199	2	no	—
92	#117	—	199	1	no	—
93	#117	C₂×D₁₉₉	1	2	yes	—
94	C₂×C₄²	—	199	2	no	—
95	C₂×C₄²	—	199	1	no	—
96	#101	—	398	1	no	—
97	#100	—	398	1	no	—
98	#99	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	D₇₉₆	1	1	yes	characteristic
102	C₂×D₄	C₂²⋊D₁₉₉	1	2	yes	—
103	C₂×D₄	D₇₉₆	1	1	yes	characteristic
104	C₂²×C₄	—	398	2	no	—
105	C₂²×C₄	—	398	1	no	—
106	C₂²×C₄	C₄×D₁₉₉	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
108	M₁₆	—	398	2	no	—
109	M₁₆	—	398	2	no	—
110	C₂×C₈	—	398	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	398	2	no	—
113	C₄⋊C₄	—	398	1	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	398	2	no	—
116	C₂²⋊C₄	—	398	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	398	2	no	—
119	C₄²	—	398	1	no	—
120	C₄²	—	199	1	no	—
121	C₄²	—	199	2	no	—
122	C₄²	—	199	1	no	—
123	C₂³	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	796	2	no	—
132	C₂×C₄	—	796	4	no	—
133	C₂×C₄	—	398	2	no	—
134	C₂×C₄	—	398	1	no	—
135	C₂×C₄	—	398	1	no	—
136	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
139	C₂×C₄	#15691885	1	2	yes	—
140	C₂×C₄	#15691885	1	2	yes	—
141	C₈	—	796	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
146	C₂²	#20263936	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	796	1	no	—
150	C₄	—	796	2	no	—
151	C₄	—	398	2	no	—
152	C₄	#20263911	1	1	yes	characteristic
153	C₄	#20263911	1	1	yes	characteristic
154	C₄	#20263939	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#27163879	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₁₉₉

C₂

yes

maximal, characteristic

#27164002

C₂

yes

maximal, characteristic

#27163777

C₂

yes

maximal, complemented

#27163774

C₂

yes

maximal, complemented

#410×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

Q₈×Dic₁₉₉

—

D₄×Dic₁₉₉

—

D₄×Dic₁₉₉

—

#20263914

C₄

yes

complemented

#20263914

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₉

C₄

yes

complemented

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

#20263874

—

#20263872

—

C₂×#41×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

—

#99×C₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

#15691870

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

C₂×C₄

yes

characteristic

C₄×Dic₁₉₉

C₂×C₄

yes

—

C₄×Dic₁₉₉

C₂×C₄

yes

characteristic

#41×C₁₉₉

—

#41×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

characteristic

C₂×D₄×C₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂²⋊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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#123

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#381

yes

characteristic

C₁₉₉

#1827

yes

characteristic, complemented, Sylow

#1827

—

199

maximal, Sylow

C₄×#41

—

199

—

#413

—

199

—

#410

D₁₉₉

yes

characteristic

#339

—

199

—

H(3,ℤ/4)

—

199

—

C₂×#41

C₂×D₁₉₉

yes

characteristic

C₂×M₁₆

—

199

—

C₄×Q₈

—

398

—

C₄×D₄

—

398

—

C₄×D₄

—

398

—

#117

—

199

—

#117

—

199

—

#117

C₂×D₁₉₉

yes

—

C₂×C₄²

—

199

—

C₂×C₄²

—

199

—

#101

—

398

—

#100

—

398

—

#99

—

#41

—

100

#41

—

101

C₂×Q₈

D₇₉₆

yes

characteristic

102

C₂×D₄

C₂²⋊D₁₉₉

yes

—

103

C₂×D₄

D₇₉₆

yes

characteristic

104

C₂²×C₄

—

398

—

105

C₂²×C₄

—

398

—

106

C₂²×C₄

C₄×D₁₉₉

yes

—

107

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

108

M₁₆

—

398

—

109

M₁₆

—

398

—

110

C₂×C₈

—

398

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

398

—

113

C₄⋊C₄

—

398

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

398

—

116

C₂²⋊C₄

—

398

—

117

C₄²

—

118

C₄²

—

398

—

119

C₄²

—

398

—

120

C₄²

—

199

—

121

C₄²

—

199

—

122

C₄²

—

199

—

123

C₂³

(C₂×D₁₉₉)⋊C₄

yes

—

124

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

796

—

132

C₂×C₄

—

796

—

133

C₂×C₄

—

398

—

134

C₂×C₄

—

398

—

135

C₂×C₄

—

398

—

136

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

137

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

138

C₂×C₄

C₂×D₇₉₆

yes

characteristic

139

C₂×C₄

#15691885

yes

—

140

C₂×C₄

#15691885

yes

—

141

C₈

—

796

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

146

C₂²

#20263936

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

796

—

150

C₄

—

796

—

151

C₄

—

398

—

152

C₄

#20263911

yes

characteristic

153

C₄

#20263911

yes

characteristic

154

C₄

#20263939

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#27163879

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	#381
prime radical	C₂×#41×C₁₉₉	C₂²
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#410	D₁₉₉
hypocenter	C₁₉₉	#1827
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

#381

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
3. #27163777	C₂	semidirect
4. #27163774	C₂	semidirect
10. #20263914	C₄	semidirect
12. C₂×C₄×Dic₁₉₉	C₄	semidirect
79. C₁₉₉	#1827	semidirect

H ≅ G/N

type

3. #27163777

C₂

semidirect

4. #27163774

C₂

semidirect

10. #20263914

C₄

semidirect

12. C₂×C₄×Dic₁₉₉

C₄

semidirect

79. C₁₉₉

#1827

semidirect

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

subgroup	index
#41×Dic₁₉₉	2
#27164002	2
#27163777	2
#27163774	2
#410×C₁₉₉	2
#1827	199

subgroup

index

#41×Dic₁₉₉

#410×C₁₉₉

199

Subgroups — #38768951

Conjugacy classes of subgroups

Complements

Maximal subgroups