Group #38770414 — subgroups

Subgroups — #38770414

Order 25472.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#27163916	C₂	1	1	yes	maximal, characteristic
2	#27163888	C₂	1	1	yes	maximal, characteristic
3	#27163781	C₂	1	2	yes	maximal
4	#27163781	C₂	1	2	yes	maximal
5	#336×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₉₉	—	2	1	no	—
7	C₂×C₄⋊Dic₁₉₉	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₉⋊C₄	—	2	1	no	—
12	C₁₉₉⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₉	—	2	1	no	—
16	C₂²×Dic₁₉₉	—	2	1	no	—
17	C₂²×Dic₁₉₉	—	2	2	no	—
18	C₄⋊Dic₁₉₉	—	2	1	no	—
19	C₄⋊Dic₁₉₉	—	2	2	no	—
20	C₄⋊Dic₁₉₉	—	2	4	no	—
21	C₄⋊Dic₁₉₉	—	4	2	no	—
22	C₄⋊Dic₁₉₉	—	4	2	no	—
23	C₄⋊Dic₁₉₉	Q₈	1	2	yes	—
24	Dic₁₉₉⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₉₉	—	2	2	no	—
29	C₂×C₈×C₁₉₉	—	2	2	no	—
30	C₂×C₈×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	—	2	2	no	—
32	C₂×Dic₁₉₉	—	2	2	no	—
33	C₂×Dic₁₉₉	—	4	1	no	—
34	C₂×Dic₁₉₉	—	4	1	no	—
35	C₂×Dic₁₉₉	—	4	1	no	—
36	C₂×Dic₁₉₉	—	4	2	no	—
37	C₂×Dic₁₉₉	—	4	1	no	—
38	C₂×Dic₁₉₉	—	4	2	no	—
39	C₂×Dic₁₉₉	—	4	2	no	—
40	C₂×Dic₁₉₉	—	4	1	no	—
41	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	—	2	1	no	—
43	C₂×C₄×C₁₉₉	—	2	1	no	—
44	C₂×C₄×C₁₉₉	—	2	1	no	—
45	C₂×C₄×C₁₉₉	—	2	1	no	—
46	C₂×C₄×C₁₉₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₉₉	Q₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₉₉	#41	1	2	yes	—
50	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₉₉	#41	1	2	yes	—
52	C₈×C₁₉₉	—	4	2	no	—
53	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₉₉	#123	1	1	yes	characteristic
55	C₂²×C₁₉₉	#124	1	1	yes	characteristic
56	C₂²×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₉₉	—	2	2	no	—
61	C₄×C₁₉₉	—	2	2	no	—
62	C₄×C₁₉₉	#125	1	2	yes	—
63	C₄×C₁₉₉	#126	1	2	yes	—
64	Dic₁₉₉	—	4	2	no	—
65	Dic₁₉₉	—	8	1	no	—
66	Dic₁₉₉	—	8	2	no	—
67	C₂×C₁₉₉	#400	1	1	yes	characteristic
68	C₂×C₁₉₉	#476	1	1	yes	characteristic
69	C₂×C₁₉₉	#477	1	1	yes	characteristic
70	C₂×C₁₉₉	#481	1	1	yes	characteristic
71	C₂×C₁₉₉	#484	1	1	yes	characteristic
72	C₂×C₁₉₉	#487	1	1	yes	characteristic
73	C₂×C₁₉₉	#490	1	1	yes	characteristic
74	C₁₉₉	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	199	1	no	maximal, Sylow
76	#389	—	199	1	no	—
77	#384	—	199	1	no	—
78	#340	—	199	2	no	—
79	#340	—	199	2	no	—
80	#336	D₁₉₉	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₉₉	1	2	yes	—
82	C₂×C₄⋊C₄	—	398	1	no	—
83	C₂×C₄⋊C₄	—	398	1	no	—
84	C₂×C₄⋊C₄	—	398	1	no	—
85	C₂×C₄⋊C₄	—	199	2	no	—
86	C₂×C₄⋊C₄	—	199	1	no	—
87	C₂×C₄⋊C₄	—	199	1	no	—
88	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
89	#95	—	398	1	no	—
90	C₂²×C₄	—	398	1	no	—
91	C₂²×C₄	—	398	2	no	—
92	C₂²×C₄	—	398	1	no	—
93	C₂²×C₄	D₇₉₆	1	1	yes	characteristic
94	C₂²×C₄	D₇₉₆	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	796	2	no	—
100	C₄⋊C₄	—	796	2	no	—
101	C₄⋊C₄	—	796	2	no	—
102	C₄⋊C₄	—	398	2	no	—
103	C₄⋊C₄	—	398	4	no	—
104	C₄⋊C₄	—	398	1	no	—
105	C₄⋊C₄	—	199	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	796	1	no	—
113	C₂×C₄	—	796	1	no	—
114	C₂×C₄	—	796	1	no	—
115	C₂×C₄	—	796	2	no	—
116	C₂×C₄	—	796	1	no	—
117	C₂×C₄	—	796	2	no	—
118	C₂×C₄	—	796	2	no	—
119	C₂×C₄	—	796	1	no	—
120	C₂×C₄	—	398	2	no	—
121	C₂×C₄	#15691875	1	2	yes	—
122	C₂×C₄	Dic₇₉₆	1	2	yes	—
123	C₂×C₄	Q₈×D₁₉₉	1	1	yes	characteristic
124	C₂×C₄	#15691885	1	2	yes	—
125	C₂×C₄	#15691885	1	1	yes	characteristic
126	C₂×C₄	#15691886	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#20263935	1	1	yes	characteristic
129	C₂²	#20263939	1	1	yes	characteristic
130	C₂²	#20263943	1	1	yes	characteristic
131	C₂²	C₂×#15691875	1	1	yes	characteristic
132	C₂²	C₂×Dic₇₉₆	1	1	yes	characteristic
133	C₂²	#20263983	1	1	yes	characteristic
134	C₂²	#20263983	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	796	2	no	—
138	C₄	—	1592	1	no	—
139	C₄	—	1592	2	no	—
140	C₄	#20263945	1	2	yes	—
141	C₄	#20263948	1	2	yes	—
142	C₂	#27163958	1	1	yes	characteristic
143	C₂	#27164309	1	1	yes	characteristic
144	C₂	#27164314	1	1	yes	characteristic
145	C₂	#27164337	1	1	yes	characteristic
146	C₂	#27164354	1	1	yes	characteristic
147	C₂	#27164369	1	1	yes	characteristic
148	C₂	#27164383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27163916

C₂

yes

maximal, characteristic

#27163888

C₂

yes

maximal, characteristic

#27163781

C₂

yes

maximal

#27163781

C₂

yes

maximal

#336×C₁₉₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×Dic₁₉₉⋊C₄

—

C₁₉₉⋊#95

—

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

Q₈

yes

—

Dic₁₉₉⋊C₄

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

Q₁₆

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

C₂²⋊Q₈

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₉

C₂×Q₁₆

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#125

yes

—

C₄×C₁₉₉

#126

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#2100

—

199

maximal, Sylow

#389

—

199

—

#384

—

199

—

#340

—

199

—

#340

—

199

—

#336

D₁₉₉

yes

characteristic

C₂²×C₈

C₂×D₁₉₉

yes

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#95

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

D₇₉₆

yes

characteristic

C₂²×C₄

D₇₉₆

yes

characteristic

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

796

—

100

C₄⋊C₄

—

796

—

101

C₄⋊C₄

—

796

—

102

C₄⋊C₄

—

398

—

103

C₄⋊C₄

—

398

—

104

C₄⋊C₄

—

398

—

105

C₄⋊C₄

—

199

—

106

C₄²

—

107

C₂³

C₂×D₇₉₆

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

796

—

113

C₂×C₄

—

796

—

114

C₂×C₄

—

796

—

115

C₂×C₄

—

796

—

116

C₂×C₄

—

796

—

117

C₂×C₄

—

796

—

118

C₂×C₄

—

796

—

119

C₂×C₄

—

796

—

120

C₂×C₄

—

398

—

121

C₂×C₄

#15691875

yes

—

122

C₂×C₄

Dic₇₉₆

yes

—

123

C₂×C₄

Q₈×D₁₉₉

yes

characteristic

124

C₂×C₄

#15691885

yes

—

125

C₂×C₄

#15691885

yes

characteristic

126

C₂×C₄

#15691886

yes

—

127

C₈

—

128

C₂²

#20263935

yes

characteristic

129

C₂²

#20263939

yes

characteristic

130

C₂²

#20263943

yes

characteristic

131

C₂²

C₂×#15691875

yes

characteristic

132

C₂²

C₂×Dic₇₉₆

yes

characteristic

133

C₂²

#20263983

yes

characteristic

134

C₂²

#20263983

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

796

—

138

C₄

—

1592

—

139

C₄

—

1592

—

140

C₄

#20263945

yes

—

141

C₄

#20263948

yes

—

142

C₂

#27163958

yes

characteristic

143

C₂

#27164309

yes

characteristic

144

C₂

#27164314

yes

characteristic

145

C₂

#27164337

yes

characteristic

146

C₂

#27164354

yes

characteristic

147

C₂

#27164369

yes

characteristic

148

C₂

#27164383

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₉₆
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#336×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
hypercenter	#336	D₁₉₉
hypocenter	C₁₉₉	#2100
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₉₆

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#336×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
74. C₁₉₉	#2100	semidirect

H ≅ G/N

type

74. C₁₉₉

#2100

semidirect

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

subgroup	index
#27163916	2
#27163888	2
#27163781	2
#336×C₁₉₉	2
#2100	199

subgroup

index

#336×C₁₉₉

199

Subgroups — #38770414

Conjugacy classes of subgroups

Complements

Maximal subgroups