Group #38770360 — subgroups

Subgroups — #38770360

Order 25472.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27164061	C₂	1	2	yes	maximal
2	C₂×#20263887	C₂	1	1	yes	maximal, characteristic
3	#27163786	C₂	1	2	yes	maximal
4	#27163783	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691870	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
8	#20263888	—	2	2	no	—
9	#20263887	—	2	2	no	—
10	#20263886	—	2	2	no	—
11	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	#117×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₉	—	2	2	no	—
15	C₂²×Dic₁₉₉	—	2	2	no	—
16	#15691870	—	2	2	no	—
17	#15691870	Q₈	1	4	yes	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₉₉	—	2	2	no	—
20	C₄⋊Dic₁₉₉	—	2	2	no	—
21	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₉₉	—	2	2	no	—
23	C₂×C₈×C₁₉₉	—	2	2	no	—
24	C₂×C₈×C₁₉₉	—	2	1	no	—
25	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
28	C₄²×C₁₉₉	—	2	1	no	—
29	C₂×Dic₁₉₉	—	4	2	no	—
30	C₂×Dic₁₉₉	—	8	2	no	—
31	C₁₉₉⋊C₈	—	2	4	no	—
32	C₂³×C₁₉₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₉₉	—	2	2	no	—
34	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₉₉	#41	1	2	yes	—
38	C₈×C₁₉₉	—	2	2	no	—
39	C₈×C₁₉₉	—	4	1	no	—
40	C₂²×C₁₉₉	—	2	2	no	—
41	C₂²×C₁₉₉	#121	1	1	yes	characteristic
42	C₂²×C₁₉₉	#125	1	2	yes	—
43	C₄×C₁₉₉	—	4	2	no	—
44	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₉₉	#123	1	1	yes	characteristic
46	C₄×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₉₉	—	8	2	no	—
48	C₂×C₁₉₉	—	2	2	no	—
49	C₂×C₁₉₉	#398	1	1	yes	characteristic
50	C₂×C₁₉₉	#471	1	2	yes	—
51	C₁₉₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	199	1	no	maximal, Sylow
53	#428	—	199	2	no	—
54	C₂×#106	—	199	1	no	—
55	#424	D₁₉₉	1	1	yes	characteristic
56	#343	—	199	2	no	—
57	#341	—	199	1	no	—
58	C₂×M₁₆	—	199	2	no	—
59	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
61	#117	C₂×D₁₉₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	199	2	no	—
63	#107	—	398	2	no	—
64	#106	—	398	2	no	—
65	#106	—	398	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	398	2	no	—
68	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	398	2	no	—
71	M₁₆	—	199	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	398	2	no	—
75	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
76	C₄⋊C₄	—	398	2	no	—
77	C₄⋊C₄	—	398	2	no	—
78	C₄⋊C₄	D₇₉₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#15691886	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	796	2	no	—
84	C₂×C₄	—	1592	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₉₉	1	2	yes	—
88	C₂×C₄	#15691885	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	398	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#20263929	1	1	yes	characteristic
94	C₂²	#20263945	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	1592	2	no	—
97	C₄	#20263935	1	2	yes	—
98	C₄	#20263936	1	1	yes	characteristic
99	C₄	#20263953	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#27163948	1	1	yes	characteristic
102	C₂	#27164281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164061

C₂

yes

maximal

C₂×#20263887

C₂

yes

maximal, characteristic

#27163786

C₂

yes

maximal

#27163783

C₂

yes

maximal, characteristic

#424×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263888

—

#20263887

—

#20263886

—

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

—

C₂²×Dic₁₉₉

—

#15691870

—

#15691870

Q₈

yes

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#121

yes

characteristic

C₂²×C₁₉₉

#125

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#123

yes

characteristic

C₄×C₁₉₉

C₄⋊Q₈

yes

characteristic

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#398

yes

characteristic

C₂×C₁₉₉

#471

yes

—

C₁₉₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

199

maximal, Sylow

#428

—

199

—

C₂×#106

—

199

—

#424

D₁₉₉

yes

characteristic

#343

—

199

—

#341

—

199

—

C₂×M₁₆

—

199

—

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

#117

C₂×D₁₉₉

yes

characteristic

C₂×C₄⋊C₄

—

199

—

#107

—

398

—

#106

—

398

—

#106

—

398

—

C₄⋊C₈

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

M₁₆

—

398

—

M₁₆

—

199

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

C₂²⋊D₁₉₉

yes

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

D₇₉₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#15691886

yes

characteristic

C₂×C₄

—

C₂×C₄

—

796

—

C₂×C₄

—

1592

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

C₂×D₇₉₆

yes

characteristic

C₂×C₄

Q₈×D₁₉₉

yes

—

C₂×C₄

#15691885

yes

—

C₈

—

C₈

—

C₈

—

398

—

C₂²

—

C₂²

#20263929

yes

characteristic

C₂²

#20263945

yes

—

C₄

—

C₄

—

1592

—

C₄

#20263935

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#27163948

yes

characteristic

102

C₂

#27164281

yes

—

	subgroup	G/H
center	C₂²	#20263929
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#424×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#121
prime radical	C₂³×C₁₉₉	#41
hypercenter	#424	D₁₉₉
hypocenter	C₁₉₉	#2090
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#20263929

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#424×C₁₉₉

C₂

C₂²×C₁₉₉

C₂³×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
51. C₁₉₉	#2090	semidirect

H ≅ G/N

type

51. C₁₉₉

#2090

semidirect

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

subgroup	index
#27164061	2
C₂×#20263887	2
#27163786	2
#27163783	2
#424×C₁₉₉	2
#2090	199

subgroup

index

#424×C₁₉₉

199

Subgroups — #38770360

Conjugacy classes of subgroups

Complements

Maximal subgroups