Group #38769819 — subgroups

Subgroups — #38769819

Order 25472.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20263947	C₂	1	1	yes	maximal, characteristic
2	#27163888	C₂	1	1	yes	maximal, characteristic
3	#27163781	C₂	1	4	yes	maximal
4	C₂×#97×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#20263947	—	2	2	no	—
6	#20263947	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
10	C₂×Dic₁₉₉⋊C₄	—	2	2	no	—
11	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
13	#97×C₁₉₉	—	2	2	no	—
14	C₂²×Dic₁₉₉	—	2	2	no	—
15	C₂²×Dic₁₉₉	—	2	2	no	—
16	C₄⋊Dic₁₉₉	—	2	2	no	—
17	C₄⋊Dic₁₉₉	—	2	4	no	—
18	C₄⋊Dic₁₉₉	—	4	4	no	—
19	C₄⋊Dic₁₉₉	C₂×C₄	1	4	yes	—
20	Dic₁₉₉⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₉₉	—	2	2	no	—
24	C₂×C₈×C₁₉₉	—	2	2	no	—
25	C₂×C₈×C₁₉₉	Q₈	1	4	yes	—
26	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₉₉	D₄	1	2	yes	—
28	C₂×Dic₁₉₉	—	2	4	no	—
29	C₂×Dic₁₉₉	—	4	2	no	—
30	C₂×Dic₁₉₉	—	4	2	no	—
31	C₂×Dic₁₉₉	—	4	2	no	—
32	C₂×Dic₁₉₉	—	4	4	no	—
33	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₉	—	2	2	no	—
35	C₂×C₄×C₁₉₉	—	2	2	no	—
36	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₉₉	#41	1	2	yes	—
41	C₈×C₁₉₉	—	2	4	no	—
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	C₄×Q₈	1	2	yes	—
49	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
50	Dic₁₉₉	—	4	4	no	—
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	—	199	1	no	—
60	C₂×#97	D₁₉₉	1	1	yes	characteristic
61	#384	—	199	1	no	—
62	#340	—	199	4	no	—
63	C₂²×C₈	C₂×D₁₉₉	1	2	yes	—
64	#125	—	398	2	no	—
65	#125	—	199	4	no	—
66	C₂×C₄⋊C₄	—	398	2	no	—
67	C₂×C₄⋊C₄	—	398	2	no	—
68	C₂×C₄⋊C₄	—	199	2	no	—
69	C₂×C₄⋊C₄	—	199	2	no	—
70	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	398	2	no	—
73	C₂²×C₄	—	398	2	no	—
74	C₂²×C₄	D₇₉₆	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₃₉₈	1	4	yes	—
79	C₄⋊C₄	—	796	4	no	—
80	C₄⋊C₄	—	398	4	no	—
81	C₄⋊C₄	—	398	8	no	—
82	C₄⋊C₄	—	398	2	no	—
83	C₄⋊C₄	—	199	4	no	—
84	C₄²	C₄×D₁₉₉	1	2	yes	—
85	C₄²	C₂²⋊D₁₉₉	1	2	yes	—
86	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	796	2	no	—
90	C₂×C₄	—	796	2	no	—
91	C₂×C₄	—	796	2	no	—
92	C₂×C₄	—	796	4	no	—
93	C₂×C₄	—	398	4	no	—
94	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₃₉₈	1	2	yes	—
98	C₂×C₄	#15691886	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
101	C₂²	#20263943	1	1	yes	characteristic
102	C₂²	#20263955	1	1	yes	characteristic
103	C₂²	#20263983	1	2	yes	—
104	C₂²	#20263988	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	796	4	no	—
107	C₄	—	1592	2	no	—
108	C₄	C₄×Dic₃₉₈	1	2	yes	—
109	C₄	#20263934	1	2	yes	—
110	C₂	#27163924	1	1	yes	characteristic
111	C₂	#27164020	1	2	yes	—
112	C₂	#27164383	1	1	yes	characteristic
113	C₂	#27164387	1	1	yes	characteristic
114	C₂	#27164427	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263947

C₂

yes

maximal, characteristic

#27163888

C₂

yes

maximal, characteristic

#27163781

C₂

yes

maximal

C₂×#97×C₁₉₉

C₂

yes

maximal, characteristic

#20263947

—

#20263947

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₂×Dic₁₉₉⋊C₄

—

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

#97×C₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

C₂×C₄

yes

—

Dic₁₉₉⋊C₄

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

Q₈

yes

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

D₄

yes

—

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

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

—

C₈×C₁₉₉

—

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×Q₈

yes

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

Dic₁₉₉

—

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

—

199

—

C₂×#97

D₁₉₉

yes

characteristic

#384

—

199

—

#340

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

—

#125

—

398

—

#125

—

199

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#97

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

D₇₉₆

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₉₈

yes

—

C₄⋊C₄

—

796

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

199

—

C₄²

C₄×D₁₉₉

yes

—

C₄²

C₂²⋊D₁₉₉

yes

—

C₂³

C₂×D₇₉₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

—

398

—

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

C₂×Dic₃₉₈

yes

—

C₂×C₄

#15691886

yes

—

C₈

—

100

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

101

C₂²

#20263943

yes

characteristic

102

C₂²

#20263955

yes

characteristic

103

C₂²

#20263983

yes

—

104

C₂²

#20263988

yes

—

105

C₄

—

106

C₄

—

796

—

107

C₄

—

1592

—

108

C₄

C₄×Dic₃₉₈

yes

—

109

C₄

#20263934

yes

—

110

C₂

#27163924

yes

characteristic

111

C₂

#27164020

yes

—

112

C₂

#27164383

yes

characteristic

113

C₂

#27164387

yes

characteristic

114

C₂

#27164427

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₉₆
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×#97×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×#97	D₁₉₉
hypocenter	C₁₉₉	#1992
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₇₉₆

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×#97×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂×#97

D₁₉₉

hypocenter

C₁₉₉

#1992

LCS

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #20263947	C₄	semidirect
57. C₁₉₉	#1992	semidirect

H ≅ G/N

type

6. #20263947

C₄

semidirect

57. C₁₉₉

#1992

semidirect

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

subgroup	index
C₂×#20263947	2
#27163888	2
#27163781	2
C₂×#97×C₁₉₉	2
#1992	199

subgroup

index

C₂×#97×C₁₉₉

199

Subgroups — #38769819

Conjugacy classes of subgroups

Complements

Maximal subgroups