Group #38769820 — subgroups

Subgroups — #38769820

Order 25472.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20263866	C₂	1	1	yes	maximal, characteristic
2	#27163888	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	4	yes	maximal
4	C₂×#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
7	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
8	#20263866	—	2	2	no	—
9	#125×C₁₉₉	—	2	2	no	—
10	#125×C₁₉₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₉₉	—	2	2	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	Q₈	1	4	yes	—
18	C₄⋊Dic₁₉₉	—	2	4	no	—
19	C₄⋊Dic₁₉₉	—	4	4	no	—
20	C₂²×C₄×C₁₉₉	—	2	2	no	—
21	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₉₉	—	2	8	no	—
24	C₄⋊C₄×C₁₉₉	—	2	2	no	—
25	C₄⋊C₄×C₁₉₉	C₂×C₄	1	4	yes	—
26	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₉₉	D₄	1	2	yes	—
28	C₂×Dic₁₉₉	—	4	2	no	—
29	C₂×Dic₁₉₉	—	4	4	no	—
30	C₁₉₉⋊C₈	—	2	4	no	—
31	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₉₉	—	2	2	no	—
33	C₂×C₄×C₁₉₉	—	2	2	no	—
34	C₂×C₄×C₁₉₉	—	2	4	no	—
35	C₂×C₄×C₁₉₉	—	4	2	no	—
36	C₂×C₄×C₁₉₉	—	4	2	no	—
37	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₉₉	#41	1	2	yes	—
42	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₉₉	#124	1	1	yes	characteristic
44	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₉₉	#137	1	2	yes	—
47	C₄×C₁₉₉	—	2	4	no	—
48	C₄×C₁₉₉	—	4	4	no	—
49	C₄×C₁₉₉	C₄×Q₈	1	2	yes	—
50	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₉₉	—	8	2	no	—
52	C₂×C₁₉₉	#391	1	1	yes	characteristic
53	C₂×C₁₉₉	#417	1	2	yes	—
54	C₂×C₁₉₉	#490	1	1	yes	characteristic
55	C₂×C₁₉₉	#491	1	1	yes	characteristic
56	C₂×C₁₉₉	#501	1	2	yes	—
57	C₁₉₉	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	199	1	no	maximal, Sylow
59	C₂×#125	D₁₉₉	1	1	yes	characteristic
60	C₂×#97	—	199	1	no	—
61	#384	—	199	1	no	—
62	#340	—	199	4	no	—
63	C₂²×C₈	—	199	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₉₉	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	398	2	no	—
68	C₂×C₄⋊C₄	—	199	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
70	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
71	#97	—	398	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	398	2	no	—
74	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
76	C₂×C₈	—	398	2	no	—
77	C₂×C₈	—	398	2	no	—
78	C₂×C₈	—	199	4	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	796	4	no	—
82	C₄⋊C₄	—	398	4	no	—
83	C₄⋊C₄	C₂×Dic₁₉₉	1	4	yes	—
84	C₄²	C₂×Dic₁₉₉	1	2	yes	—
85	C₄²	C₂²⋊D₁₉₉	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	796	2	no	—
93	C₂×C₄	—	796	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₉₉	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₉₉	1	2	yes	—
98	C₂×C₄	#15691887	1	2	yes	—
99	C₈	—	398	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
101	C₂²	#20263944	1	1	yes	characteristic
102	C₂²	#20263953	1	1	yes	characteristic
103	C₂²	#20263980	1	2	yes	—
104	C₂²	#20263985	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	1592	2	no	—
108	C₄	Q₈×Dic₁₉₉	1	2	yes	—
109	C₄	#20263931	1	2	yes	—
110	C₂	#27163925	1	1	yes	characteristic
111	C₂	#27164019	1	2	yes	—
112	C₂	#27164384	1	1	yes	characteristic
113	C₂	#27164388	1	1	yes	characteristic
114	C₂	#27164428	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263866

C₂

yes

maximal, characteristic

#27163888

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

C₂×#125×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263866

—

#125×C₁₉₉

—

#125×C₁₉₉

C₄

yes

complemented

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

Q₈

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

D₄

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×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

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#124

yes

characteristic

C₂²×C₁₉₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₉₉

#137

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₄×Q₈

yes

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

#391

yes

characteristic

C₂×C₁₉₉

#417

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₁₉₉

#1992

yes

characteristic, complemented, Sylow

#1992

—

199

maximal, Sylow

C₂×#125

D₁₉₉

yes

characteristic

C₂×#97

—

199

—

#384

—

199

—

#340

—

199

—

C₂²×C₈

—

199

—

#125

—

#125

Dic₁₉₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#97

—

398

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

796

—

C₄⋊C₄

—

398

—

C₄⋊C₄

C₂×Dic₁₉₉

yes

—

C₄²

C₂×Dic₁₉₉

yes

—

C₄²

C₂²⋊D₁₉₉

yes

—

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

C₂×C₄

C₂²×Dic₁₉₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

Q₈×D₁₉₉

yes

—

C₂×C₄

#15691887

yes

—

C₈

—

398

—

100

C₂²

C₂×C₂²⋊Dic₁₉₉

yes

characteristic

101

C₂²

#20263944

yes

characteristic

102

C₂²

#20263953

yes

characteristic

103

C₂²

#20263980

yes

—

104

C₂²

#20263985

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1592

—

108

C₄

Q₈×Dic₁₉₉

yes

—

109

C₄

#20263931

yes

—

110

C₂

#27163925

yes

characteristic

111

C₂

#27164019

yes

—

112

C₂

#27164384

yes

characteristic

113

C₂

#27164388

yes

characteristic

114

C₂

#27164428

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×#125×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×#125	D₁₉₉
hypocenter	C₁₉₉	#1992
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×#125×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 ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₉₉	C₄	semidirect
57. C₁₉₉	#1992	semidirect
65. #125	Dic₁₉₉	semidirect

H ≅ G/N

type

10. #125×C₁₉₉

C₄

semidirect

57. C₁₉₉

#1992

semidirect

65. #125

Dic₁₉₉

semidirect

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

subgroup	index
C₂×#20263866	2
#27163888	2
#27163780	2
C₂×#125×C₁₉₉	2
#1992	199

subgroup

index

C₂×#125×C₁₉₉

199

Subgroups — #38769820

Conjugacy classes of subgroups

Complements

Maximal subgroups