Group #38768985 — subgroups

Subgroups — #38768985

Order 25472.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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	#27163884	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	C₂²	1	2	yes	—
7	#20263866	C₄	1	4	yes	complemented
8	C₁₉₉⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₉₉	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₉₉	—	2	2	no	—
13	C₂×C₁₉₉⋊C₈	—	2	2	no	—
14	C₂×C₁₉₉⋊C₈	—	2	2	no	—
15	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₉₉	—	2	4	no	—
17	C₂²×C₄×C₁₉₉	—	2	2	no	—
18	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
19	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₉₉	—	2	4	no	—
21	C₄⋊C₄×C₁₉₉	—	4	4	no	—
22	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₉₉	Q₈	1	2	yes	—
24	C₂×Dic₁₉₉	—	4	2	no	—
25	C₂×Dic₁₉₉	—	4	2	no	—
26	C₂×Dic₁₉₉	—	4	2	no	—
27	C₁₉₉⋊C₈	—	2	4	no	—
28	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	—	2	2	no	—
30	C₂×C₄×C₁₉₉	—	2	2	no	—
31	C₂×C₄×C₁₉₉	—	4	2	no	—
32	C₂×C₄×C₁₉₉	—	4	4	no	—
33	C₂×C₄×C₁₉₉	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	#41	1	2	yes	—
37	C₂×C₄×C₁₉₉	#41	1	2	yes	—
38	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₉₉	#124	1	1	yes	characteristic
40	C₂²×C₁₉₉	#124	1	1	yes	characteristic
41	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₉₉	#137	1	2	yes	—
43	C₄×C₁₉₉	—	2	4	no	—
44	C₄×C₁₉₉	—	8	2	no	—
45	C₄×C₁₉₉	#117	1	2	yes	—
46	C₄×C₁₉₉	#125	1	2	yes	—
47	Dic₁₉₉	—	8	2	no	—
48	C₂×C₁₉₉	#382	1	1	yes	characteristic
49	C₂×C₁₉₉	#428	1	2	yes	—
50	C₂×C₁₉₉	#489	1	2	yes	—
51	C₂×C₁₉₉	#490	1	1	yes	characteristic
52	C₂×C₁₉₉	#491	1	1	yes	characteristic
53	C₁₉₉	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	199	1	no	maximal, Sylow
55	C₂×#97	—	199	1	no	—
56	#384	D₁₉₉	1	1	yes	characteristic
57	#382	—	199	1	no	—
58	#340	—	199	4	no	—
59	C₂²×C₈	—	199	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	199	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
63	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
64	#97	—	199	4	no	—
65	#95	—	398	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	398	2	no	—
68	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
70	C₂×C₈	—	398	2	no	—
71	C₂×C₈	—	398	2	no	—
72	C₂×C₈	—	199	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	398	4	no	—
76	C₄²	C₄×D₁₉₉	1	2	yes	—
77	C₄²	Dic₃₉₈	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	796	2	no	—
84	C₂×C₄	—	796	2	no	—
85	C₂×C₄	—	796	2	no	—
86	C₂×C₄	Dic₁₉₉⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
89	C₂×C₄	#15691885	1	2	yes	—
90	C₂×C₄	#15691887	1	2	yes	—
91	C₈	—	398	4	no	—
92	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
93	C₂²	#20263942	1	1	yes	characteristic
94	C₂²	#20263944	1	1	yes	characteristic
95	C₂²	#20263981	1	2	yes	—
96	C₂²	#20263986	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1592	2	no	—
100	C₄	#20263913	1	2	yes	—
101	C₄	#20263946	1	2	yes	—
102	C₂	#27163883	1	1	yes	characteristic
103	C₂	#27164058	1	2	yes	—
104	C₂	#27164380	1	2	yes	—
105	C₂	#27164382	1	1	yes	characteristic
106	C₂	#27164386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263866

C₂

yes

maximal, characteristic

#27163884

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

#384×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263866

C₄

yes

complemented

C₁₉₉⋊#95

—

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

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

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

—

C₂×C₄×C₁₉₉

C₄⋊C₄

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

yes

—

C₂²×C₁₉₉

#137

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#117

yes

—

C₄×C₁₉₉

#125

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

#382

yes

characteristic

C₂×C₁₉₉

#428

yes

—

C₂×C₁₉₉

#489

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1836

—

199

maximal, Sylow

C₂×#97

—

199

—

#384

D₁₉₉

yes

characteristic

#382

—

199

—

#340

—

199

—

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#97

—

199

—

#95

—

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

—

398

—

C₄²

C₄×D₁₉₉

yes

—

C₄²

Dic₃₉₈

yes

—

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

C₂×C₄

#15691885

yes

—

C₂×C₄

#15691887

yes

—

C₈

—

398

—

C₂²

C₂×Dic₁₉₉⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#20263986

yes

—

C₄

—

C₄

—

C₄

—

1592

—

100

C₄

#20263913

yes

—

101

C₄

#20263946

yes

—

102

C₂

#27163883

yes

characteristic

103

C₂

#27164058

yes

—

104

C₂

#27164380

yes

—

105

C₂

#27164382

yes

characteristic

106

C₂

#27164386

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	#1836
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
7. #20263866	C₄	semidirect
53. C₁₉₉	#1836	semidirect

H ≅ G/N

type

7. #20263866

C₄

semidirect

53. C₁₉₉

#1836

semidirect

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

subgroup	index
C₂×#20263866	2
#27163884	2
#27163780	2
#384×C₁₉₉	2
#1836	199

subgroup

index

#384×C₁₉₉

199

Subgroups — #38768985

Conjugacy classes of subgroups

Complements

Maximal subgroups