Group #38768977 — subgroups

Subgroups — #38768977

Order 25472.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#27163888	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	4	yes	maximal
4	#384×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₉	—	2	1	no	—
7	C₂×C₄⋊Dic₁₉₉	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	C₄×C₁₉₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₉	—	2	2	no	—
15	C₂×C₁₉₉⋊C₈	—	2	2	no	—
16	C₂×C₁₉₉⋊C₈	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₉₉	—	2	4	no	—
19	C₄⋊Dic₁₉₉	—	4	2	no	—
20	C₄⋊Dic₁₉₉	—	4	2	no	—
21	C₂²×C₄×C₁₉₉	—	2	2	no	—
22	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₉₉	—	2	4	no	—
26	C₄⋊C₄×C₁₉₉	—	4	2	no	—
27	C₄⋊C₄×C₁₉₉	—	4	2	no	—
28	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₉₉	Q₈	1	2	yes	—
30	C₂×Dic₁₉₉	—	4	2	no	—
31	C₂×Dic₁₉₉	—	4	2	no	—
32	C₂×Dic₁₉₉	—	4	2	no	—
33	C₁₉₉⋊C₈	—	2	4	no	—
34	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₉	—	4	2	no	—
36	C₂×C₄×C₁₉₉	—	4	2	no	—
37	C₂×C₄×C₁₉₉	—	4	2	no	—
38	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₉₉	D₈	1	2	yes	—
41	C₂×C₄×C₁₉₉	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₉₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₉₉	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉₉	#41	1	2	yes	—
47	C₂×C₄×C₁₉₉	#41	1	2	yes	—
48	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₉₉	#124	1	1	yes	characteristic
50	C₂²×C₁₉₉	#124	1	1	yes	characteristic
51	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₉₉	—	8	2	no	—
56	C₄×C₁₉₉	#106	1	4	yes	—
57	C₄×C₁₉₉	#107	1	4	yes	—
58	C₄×C₁₉₉	#117	1	2	yes	—
59	C₄×C₁₉₉	#125	1	2	yes	—
60	Dic₁₉₉	—	8	2	no	—
61	C₂×C₁₉₉	#382	1	1	yes	characteristic
62	C₂×C₁₉₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₉₉	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₉₉	#486	1	1	yes	characteristic
65	C₂×C₁₉₉	#486	1	1	yes	characteristic
66	C₂×C₁₉₉	#487	1	1	yes	characteristic
67	C₂×C₁₉₉	#487	1	1	yes	characteristic
68	C₁₉₉	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	199	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	199	1	no	—
71	#384	—	199	1	no	—
72	#384	D₁₉₉	1	1	yes	characteristic
73	#340	—	199	4	no	—
74	C₂²×C₈	—	199	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	398	1	no	—
78	C₂×C₄⋊C₄	—	398	1	no	—
79	C₂×C₄⋊C₄	—	199	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
81	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
82	C₄×C₈	—	199	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	398	2	no	—
85	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
88	C₂×C₈	—	398	2	no	—
89	C₂×C₈	—	398	2	no	—
90	C₂×C₈	—	199	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	796	2	no	—
95	C₄⋊C₄	—	796	2	no	—
96	C₄⋊C₄	—	398	4	no	—
97	C₄²	C₄×D₁₉₉	1	2	yes	—
98	C₄²	Dic₃₉₈	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	796	2	no	—
104	C₂×C₄	—	796	2	no	—
105	C₂×C₄	—	796	2	no	—
106	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
108	C₂×C₄	#15691871	1	2	yes	—
109	C₂×C₄	#15691873	1	2	yes	—
110	C₂×C₄	#15691874	1	2	yes	—
111	C₂×C₄	#15691876	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
114	C₂×C₄	#15691885	1	2	yes	—
115	C₂×C₄	#15691887	1	2	yes	—
116	C₈	—	398	4	no	—
117	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
118	C₂²	#20263942	1	1	yes	characteristic
119	C₂²	#20263944	1	1	yes	characteristic
120	C₂²	C₂×#15691871	1	1	yes	characteristic
121	C₂²	C₂×#15691873	1	1	yes	characteristic
122	C₂²	C₂×#15691874	1	1	yes	characteristic
123	C₂²	C₂×#15691876	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1592	2	no	—
126	C₄	#20263886	1	4	yes	—
127	C₄	#20263888	1	4	yes	—
128	C₄	#20263913	1	2	yes	—
129	C₄	#20263946	1	2	yes	—
130	C₂	#27163883	1	1	yes	characteristic
131	C₂	C₂×#20263886	1	1	yes	characteristic
132	C₂	C₂×#20263888	1	1	yes	characteristic
133	C₂	#27164364	1	1	yes	characteristic
134	C₂	#27164366	1	1	yes	characteristic
135	C₂	#27164368	1	1	yes	characteristic
136	C₂	#27164370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₉₉⋊C₈

C₂

yes

maximal, characteristic

#27163888

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

#384×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₄×C₁₉₉⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

Q₈

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

D₈

yes

—

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

Q₁₆

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

C₂×D₈

yes

characteristic

C₂²×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

#106

yes

—

C₄×C₁₉₉

#107

yes

—

C₄×C₁₉₉

#117

yes

—

C₄×C₁₉₉

#125

yes

—

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

#1834

—

199

maximal, Sylow

C₂×C₄×C₈

—

199

—

#384

—

199

—

#384

D₁₉₉

yes

characteristic

#340

—

199

—

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄×C₈

—

199

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

796

—

C₄⋊C₄

—

796

—

C₄⋊C₄

—

398

—

C₄²

C₄×D₁₉₉

yes

—

C₄²

Dic₃₉₈

yes

—

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

796

—

104

C₂×C₄

—

796

—

105

C₂×C₄

—

796

—

106

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

107

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

108

C₂×C₄

#15691871

yes

—

109

C₂×C₄

#15691873

yes

—

110

C₂×C₄

#15691874

yes

—

111

C₂×C₄

#15691876

yes

—

112

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

113

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

114

C₂×C₄

#15691885

yes

—

115

C₂×C₄

#15691887

yes

—

116

C₈

—

398

—

117

C₂²

C₂×Dic₁₉₉⋊C₄

yes

characteristic

118

C₂²

#20263942

yes

characteristic

119

C₂²

#20263944

yes

characteristic

120

C₂²

C₂×#15691871

yes

characteristic

121

C₂²

C₂×#15691873

yes

characteristic

122

C₂²

C₂×#15691874

yes

characteristic

123

C₂²

C₂×#15691876

yes

characteristic

124

C₄

—

125

C₄

—

1592

—

126

C₄

#20263886

yes

—

127

C₄

#20263888

yes

—

128

C₄

#20263913

yes

—

129

C₄

#20263946

yes

—

130

C₂

#27163883

yes

characteristic

131

C₂

C₂×#20263886

yes

characteristic

132

C₂

C₂×#20263888

yes

characteristic

133

C₂

#27164364

yes

characteristic

134

C₂

#27164366

yes

characteristic

135

C₂

#27164368

yes

characteristic

136

C₂

#27164370

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#384×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#384	D₁₉₉
hypocenter	C₁₉₉	#1834
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#384×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₁₉₉⋊C₈	C₄	semidirect
68. C₁₉₉	#1834	semidirect

H ≅ G/N

type

9. C₄×C₁₉₉⋊C₈

C₄

semidirect

68. C₁₉₉

#1834

semidirect

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

subgroup	index
C₂×C₄×C₁₉₉⋊C₈	2
#27163888	2
#27163780	2
#384×C₁₉₉	2
#1834	199

subgroup

index

C₂×C₄×C₁₉₉⋊C₈

#27163888

#27163780

#384×C₁₉₉

#1834

199

Subgroups — #38768977

Conjugacy classes of subgroups

Complements

Maximal subgroups