Group #38764342 — subgroups

Subgroups — #1270×C₁₉₉ #38764342

Order 25472.

202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 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	#390×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#336×C₁₉₉	C₂	1	2	yes	maximal, complemented
3	C₂²×D₄×C₁₉₉	C₄	1	1	yes	characteristic
4	C₂²×C₈×C₁₉₉	—	2	2	no	—
5	C₂×C₄⋊C₄×C₁₉₉	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
7	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₂⁴×C₁₉₉	—	2	1	no	—
9	C₂×D₄×C₁₉₉	—	2	1	no	—
10	C₂×D₄×C₁₉₉	—	4	1	no	—
11	C₂×D₄×C₁₉₉	—	4	1	no	—
12	C₂×D₄×C₁₉₉	C₈	1	2	yes	complemented
13	C₂²×C₄×C₁₉₉	—	2	1	no	—
14	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
16	C₂×C₈×C₁₉₉	—	2	4	no	—
17	C₂×C₈×C₁₉₉	—	4	2	no	—
18	C₂×C₈×C₁₉₉	—	4	2	no	—
19	C₄⋊C₄×C₁₉₉	—	4	1	no	—
20	C₂²⋊C₄×C₁₉₉	—	4	2	no	—
21	C₂²⋊C₄×C₁₉₉	—	4	2	no	—
22	C₄²×C₁₉₉	—	4	1	no	—
23	C₂³×C₁₉₉	—	2	2	no	—
24	C₂³×C₁₉₉	—	4	1	no	—
25	C₂³×C₁₉₉	—	4	1	no	—
26	C₂³×C₁₉₉	—	4	2	no	—
27	C₂³×C₁₉₉	—	4	2	no	—
28	C₂³×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
29	D₄×C₁₉₉	—	4	2	no	—
30	D₄×C₁₉₉	—	4	2	no	—
31	C₂×C₄×C₁₉₉	—	2	2	no	—
32	C₂×C₄×C₁₉₉	—	2	1	no	—
33	C₂×C₄×C₁₉₉	—	2	1	no	—
34	C₂×C₄×C₁₉₉	—	2	2	no	—
35	C₂×C₄×C₁₉₉	—	4	1	no	—
36	C₂×C₄×C₁₉₉	—	4	2	no	—
37	C₂×C₄×C₁₉₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	D₈	1	2	yes	—
40	C₂×C₄×C₁₉₉	SD₁₆	1	2	yes	—
41	C₈×C₁₉₉	—	4	4	no	—
42	C₂²×C₁₉₉	—	4	2	no	—
43	C₂²×C₁₉₉	—	4	4	no	—
44	C₂²×C₁₉₉	—	8	2	no	—
45	C₂²×C₁₉₉	—	8	1	no	—
46	C₂²×C₁₉₉	—	8	1	no	—
47	C₂²×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂²×C₁₉₉	#99	1	1	yes	characteristic
49	C₂²×C₁₉₉	#100	1	1	yes	characteristic
50	C₂²×C₁₉₉	#102	1	2	yes	—
51	C₂²×C₁₉₉	#104	1	2	yes	—
52	C₄×C₁₉₉	—	2	1	no	—
53	C₄×C₁₉₉	—	2	1	no	—
54	C₄×C₁₉₉	—	2	2	no	—
55	C₄×C₁₉₉	—	2	2	no	—
56	C₄×C₁₉₉	—	8	1	no	—
57	C₂×C₁₉₉	—	8	2	no	—
58	C₂×C₁₉₉	#323	1	1	yes	characteristic
59	C₂×C₁₉₉	D₄⋊C₈	1	2	yes	—
60	C₂×C₁₉₉	#327	1	2	yes	—
61	C₂×C₁₉₉	#329	1	1	yes	characteristic
62	C₂×C₁₉₉	#331	1	1	yes	characteristic
63	C₁₉₉	#1270	1	1	yes	characteristic, direct product factor, Sylow
64	#1270	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	#390	C₂×C₁₉₉	1	1	yes	characteristic
66	#336	C₂×C₁₉₉	1	2	yes	complemented
67	C₂²×D₄	C₄×C₁₉₉	1	1	yes	characteristic
68	C₂²×C₈	—	2	2	no	—
69	C₂×C₄⋊C₄	C₄×C₁₉₉	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
72	C₂⁴	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	C₈×C₁₉₉	1	2	yes	complemented
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₂×C₄×C₁₉₉	1	1	yes	characteristic
79	C₂²×C₄	D₄×C₁₉₉	1	2	yes	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	2	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₄²	—	4	1	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	1	no	—
89	C₂³	—	4	1	no	—
90	C₂³	—	4	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	C₂²⋊C₄×C₁₉₉	1	1	yes	characteristic
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×C₈×C₁₉₉	1	1	yes	characteristic
102	C₂×C₄	M₁₆×C₁₉₉	1	1	yes	characteristic
103	C₂×C₄	D₈×C₁₉₉	1	2	yes	—
104	C₂×C₄	SD₁₆×C₁₉₉	1	2	yes	—
105	C₈	—	4	4	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	8	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	C₂²⋊C₈×C₁₉₉	1	1	yes	characteristic
112	C₂²	#99×C₁₉₉	1	1	yes	characteristic
113	C₂²	#100×C₁₉₉	1	1	yes	characteristic
114	C₂²	#102×C₁₉₉	1	2	yes	—
115	C₂²	#104×C₁₉₉	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	1	no	—
121	C₂	—	8	2	no	—
122	C₂	#323×C₁₉₉	1	1	yes	characteristic
123	C₂	D₄⋊C₈×C₁₉₉	1	2	yes	—
124	C₂	#327×C₁₉₉	1	2	yes	—
125	C₂	#329×C₁₉₉	1	1	yes	characteristic
126	C₂	#331×C₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#390×C₁₉₉

C₂

yes

maximal, characteristic

#336×C₁₉₉

C₂

yes

maximal, complemented

C₂²×D₄×C₁₉₉

C₄

yes

characteristic

C₂²×C₈×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₉

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂⁴×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

C₈

yes

complemented

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂²⋊C₄

yes

characteristic

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₉

M₁₆

yes

characteristic

C₂×C₄×C₁₉₉

D₈

yes

—

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂²⋊C₈

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

—

C₂²×C₁₉₉

#104

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#323

yes

characteristic

C₂×C₁₉₉

D₄⋊C₈

yes

—

C₂×C₁₉₉

#327

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, direct product factor, Sylow

#1270

C₁₉₉

yes

maximal, characteristic, direct product factor, Sylow

#390

C₂×C₁₉₉

yes

characteristic

#336

C₂×C₁₉₉

yes

complemented

C₂²×D₄

C₄×C₁₉₉

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

C₄×C₁₉₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₁₉₉

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₈×C₁₉₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₉₉

yes

characteristic

C₂²×C₄

D₄×C₁₉₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₁₉₉

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₈×C₁₉₉

yes

characteristic

102

C₂×C₄

M₁₆×C₁₉₉

yes

characteristic

103

C₂×C₄

D₈×C₁₉₉

yes

—

104

C₂×C₄

SD₁₆×C₁₉₉

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊C₈×C₁₉₉

yes

characteristic

112

C₂²

#99×C₁₉₉

yes

characteristic

113

C₂²

#100×C₁₉₉

yes

characteristic

114

C₂²

#102×C₁₉₉

yes

—

115

C₂²

#104×C₁₉₉

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#323×C₁₉₉

yes

characteristic

123

C₂

D₄⋊C₈×C₁₉₉

yes

—

124

C₂

#327×C₁₉₉

yes

—

125

C₂

#329×C₁₉₉

yes

characteristic

126

C₂

#331×C₁₉₉

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉₉	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₈×C₁₉₉
Frattini, cosocle	C₂×C₄²	C₂²×C₁₉₉
prime radical	C₂²×D₄×C₁₉₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₉ ◁ C₂×C₄²×C₁₉₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₉₉

C₂²⋊C₄

commutator

C₂×C₄

C₂×C₈×C₁₉₉

Frattini, cosocle

C₂×C₄²

C₂²×C₁₉₉

prime radical

C₂²×D₄×C₁₉₉

C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₉₉ ◁ C₂×C₄²×C₁₉₉ ◁ G

N	H ≅ G/N	type
2. #336×C₁₉₉	C₂	semidirect
12. C₂×D₄×C₁₉₉	C₈	semidirect
63. C₁₉₉	#1270	direct (N × H)
64. #1270	C₁₉₉	direct (N × H)
66. #336	C₂×C₁₉₉	semidirect
76. C₂×D₄	C₈×C₁₉₉	semidirect

H ≅ G/N

type

2. #336×C₁₉₉

C₂

semidirect

12. C₂×D₄×C₁₉₉

C₈

semidirect

63. C₁₉₉

#1270

direct (N × H)

64. #1270

C₁₉₉

direct (N × H)

66. #336

C₂×C₁₉₉

semidirect

76. C₂×D₄

C₈×C₁₉₉

semidirect

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

subgroup	index
#390×C₁₉₉	2
#336×C₁₉₉	2
#1270	199

subgroup

index

#390×C₁₉₉

#336×C₁₉₉

#1270

199

Subgroups — #1270×C₁₉₉ #38764342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1270×C199 #38764342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1270×C₁₉₉ #38764342