Group #38768476 — subgroups

Subgroups — #38768476

Order 25472.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27164416	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164390	C₂	1	2	yes	maximal, complemented
3	#27164364	C₂	1	2	yes	maximal, complemented
4	#27163739	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691874	—	2	4	no	—
7	#20263955	C₂²	1	2	yes	—
8	#20263887	—	2	2	no	—
9	#20263876	—	2	4	no	—
10	C₄×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	C₂×D₈×C₁₉₉	—	2	2	no	—
12	#127×C₁₉₉	C₂²	1	1	yes	characteristic
13	#127×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₈	—	2	2	no	—
16	#15691874	—	4	8	no	—
17	C₂×C₁₉₉⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₁₉₉	—	2	2	no	—
19	C₄⋊Dic₁₉₉	—	2	2	no	—
20	C₂×D₄×C₁₉₉	—	2	2	no	—
21	C₂×D₄×C₁₉₉	—	2	1	no	—
22	C₂×D₄×C₁₉₉	—	2	1	no	—
23	C₂×D₄×C₁₉₉	—	2	2	no	—
24	D₈×C₁₉₉	—	4	4	no	—
25	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
26	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
27	Dic₃₉₈	—	4	4	no	—
28	C₂×Dic₁₉₉	—	4	2	no	—
29	C₁₉₉⋊C₈	—	2	4	no	—
30	C₂³×C₁₉₉	—	4	1	no	—
31	C₂³×C₁₉₉	—	4	1	no	—
32	D₄×C₁₉₉	—	4	2	no	—
33	D₄×C₁₉₉	—	4	2	no	—
34	D₄×C₁₉₉	—	4	4	no	—
35	D₄×C₁₉₉	—	4	4	no	—
36	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₉₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₉₉	—	8	1	no	—
40	C₂²×C₁₉₉	—	8	1	no	—
41	C₂²×C₁₉₉	—	8	2	no	—
42	C₂²×C₁₉₉	—	8	2	no	—
43	C₂²×C₁₉₉	#127	1	1	yes	characteristic
44	C₄×C₁₉₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₉₉	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₉₉	—	8	2	no	—
47	C₂×C₁₉₉	—	16	1	no	—
48	C₂×C₁₉₉	—	16	1	no	—
49	C₂×C₁₉₉	#492	1	1	yes	characteristic
50	C₂×C₁₉₉	#496	1	2	yes	—
51	C₁₉₉	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	199	1	no	maximal, Sylow
53	#498	—	199	1	no	—
54	#493	D₁₉₉	1	1	yes	characteristic
55	#492	—	199	2	no	—
56	#486	—	199	2	no	—
57	#322	—	199	1	no	—
58	C₂×SD₁₆	—	398	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	199	2	no	—
61	#127	C₂×D₁₉₉	1	1	yes	characteristic
62	#127	C₂×D₁₉₉	1	1	yes	characteristic
63	#106	—	398	2	no	—
64	#102	—	398	4	no	—
65	C₄×C₈	—	199	2	no	—
66	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
67	C₂×Q₈	—	398	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	796	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	199	4	no	—
75	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
76	C₄⋊C₄	—	398	2	no	—
77	C₄⋊C₄	—	398	2	no	—
78	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	796	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	796	2	no	—
87	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
89	C₈	—	398	4	no	—
90	C₈	#15691874	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#20263951	1	1	yes	characteristic
96	C₄	—	1592	2	no	—
97	C₄	C₂×#15691874	1	2	yes	—
98	C₄	#20263979	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#27164390	1	1	yes	characteristic
102	C₂	#27164406	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164416

C₂

yes

maximal, characteristic, complemented

#27164390

C₂

yes

maximal, complemented

#27164364

C₂

yes

maximal, complemented

#27163739

C₂

yes

maximal, characteristic, complemented

#493×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691874

—

#20263955

C₂²

yes

—

#20263887

—

#20263876

—

C₄×C₁₉₉⋊C₈

C₂²

yes

—

C₂×D₈×C₁₉₉

—

#127×C₁₉₉

C₂²

yes

characteristic

#127×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

—

#15691874

—

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

D₈×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

SD₁₆

yes

complemented

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#127

yes

characteristic

C₄×C₁₉₉

C₂×SD₁₆

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#492

yes

characteristic

C₂×C₁₉₉

#496

yes

—

C₁₉₉

#1715

yes

characteristic, complemented, Sylow

#1715

—

199

maximal, Sylow

#498

—

199

—

#493

D₁₉₉

yes

characteristic

#492

—

199

—

#486

—

199

—

#322

—

199

—

C₂×SD₁₆

—

398

—

C₂×D₈

—

C₄⋊Q₈

—

199

—

#127

C₂×D₁₉₉

yes

characteristic

#127

C₂×D₁₉₉

yes

characteristic

#106

—

398

—

#102

—

398

—

C₄×C₈

—

199

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

398

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

796

—

D₈

—

C₂×C₈

—

199

—

C₂×C₈

C₂²⋊D₁₉₉

yes

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

796

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

796

—

C₂×C₄

D₄×D₁₉₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₈

—

398

—

C₈

#15691874

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#20263951

yes

characteristic

C₄

—

1592

—

C₄

C₂×#15691874

yes

—

C₄

#20263979

yes

—

C₂

—

100

C₂

—

101

C₂

#27164390

yes

characteristic

102

C₂

#27164406

yes

—

	subgroup	G/H
center	C₂²	#20263951
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting, prime radical	#493×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#127
hypercenter	#493	D₁₉₉
hypocenter	C₁₉₉	#1715
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#20263951

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting, prime radical

#493×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #27164416	C₂	semidirect
2. #27164390	C₂	semidirect
3. #27164364	C₂	semidirect
4. #27163739	C₂	semidirect
38. C₈×C₁₉₉	SD₁₆	semidirect
51. C₁₉₉	#1715	semidirect
90. C₈	#15691874	semidirect

H ≅ G/N

type

1. #27164416

C₂

semidirect

2. #27164390

C₂

semidirect

3. #27164364

C₂

semidirect

4. #27163739

C₂

semidirect

38. C₈×C₁₉₉

SD₁₆

semidirect

51. C₁₉₉

#1715

semidirect

90. C₈

#15691874

semidirect

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

subgroup	index
#27164416	2
#27164390	2
#27164364	2
#27163739	2
#493×C₁₉₉	2
#1715	199

subgroup

index

#493×C₁₉₉

199

Subgroups — #38768476

Conjugacy classes of subgroups

Complements

Maximal subgroups