Group #38768751 — subgroups

Subgroups — #38768751

Order 25472.

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

normal subgroups only

This group has 228 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×#20263890	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₉₉	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#27163783	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691870	C₂²	1	2	yes	—
7	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	#20263890	C₄	1	8	yes	complemented
10	#20263865	—	2	2	no	—
11	C₈×Dic₁₉₉	—	2	2	no	—
12	C₈×Dic₁₉₉	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₉₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₉	Q₈	1	2	yes	—
16	#15691870	—	2	2	no	—
17	#15691870	C₂×C₄	1	4	yes	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₉₉	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₉₉	D₄	1	1	yes	characteristic
23	C₄×Dic₁₉₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₉₉	—	2	2	no	—
26	M₁₆×C₁₉₉	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₉₉	—	2	2	no	—
28	C₂×C₈×C₁₉₉	—	2	2	no	—
29	C₂×C₈×C₁₉₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₉₉	—	2	4	no	—
31	C₂×Dic₁₉₉	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₁₉₉	C₄⋊C₄	1	2	yes	—
33	C₁₉₉⋊C₈	—	2	4	no	—
34	C₁₉₉⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	#41	1	2	yes	—
40	C₈×C₁₉₉	—	2	2	no	—
41	C₈×C₁₉₉	—	2	4	no	—
42	C₂²×C₁₉₉	—	2	2	no	—
43	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₉₉	C₄×Q₈	1	2	yes	—
45	C₄×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₉₉	#117	1	1	yes	characteristic
47	C₄×C₁₉₉	C₄×D₄	1	2	yes	—
48	Dic₁₉₉	—	2	2	no	—
49	Dic₁₉₉	#108	1	4	yes	complemented
50	C₂×C₁₉₉	—	2	2	no	—
51	C₂×C₁₉₉	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₉₉	C₂×#108	1	2	yes	—
53	C₁₉₉	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	199	1	no	maximal, Sylow
55	C₂×#108	—	199	2	no	—
56	C₄×M₁₆	—	199	2	no	—
57	C₂×C₄×C₈	—	199	1	no	—
58	#341	—	199	1	no	—
59	#341	D₁₉₉	1	1	yes	characteristic
60	C₂×M₁₆	—	199	2	no	—
61	C₂×M₁₆	C₂×D₁₉₉	1	2	yes	—
62	C₂²×C₈	—	199	1	no	—
63	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
64	C₂×C₄²	—	199	1	no	—
65	#108	—	199	8	no	—
66	#97	—	398	2	no	—
67	C₄×C₈	—	398	2	no	—
68	C₄×C₈	—	199	4	no	—
69	C₂²×C₄	—	199	2	no	—
70	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	398	2	no	—
73	M₁₆	—	199	4	no	—
74	M₁₆	C₄×D₁₉₉	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	398	2	no	—
78	C₂×C₈	—	199	2	no	—
79	C₂×C₈	—	199	4	no	—
80	C₂×C₈	C₄×D₁₉₉	1	2	yes	—
81	C₄²	—	199	2	no	—
82	C₄²	—	199	1	no	—
83	C₄²	—	199	1	no	—
84	C₂³	Q₈×D₁₉₉	1	1	yes	characteristic
85	C₂×C₄	—	398	4	no	—
86	C₂×C₄	—	199	2	no	—
87	C₂×C₄	—	199	2	no	—
88	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₁₉₉	1	2	yes	—
90	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
91	C₂×C₄	#15691885	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	398	4	no	—
95	C₈	—	199	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₁₉₉	1	1	yes	characteristic
98	C₂²	#20263920	1	2	yes	—
99	C₄	—	398	2	no	—
100	C₄	—	199	4	no	—
101	C₄	C₄²×D₁₉₉	1	1	yes	characteristic
102	C₄	#20263911	1	1	yes	characteristic
103	C₄	(C₂²⋊D₁₉₉)⋊C₄	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#27163869	1	1	yes	characteristic
106	C₂	#108×D₁₉₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263890

C₂

yes

maximal

M₁₆×Dic₁₉₉

C₂

yes

maximal

C₂×C₈×Dic₁₉₉

C₂

yes

maximal, characteristic

#27163783

C₂

yes

maximal, characteristic

#341×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

C₂²×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

#20263890

C₄

yes

complemented

#20263865

—

C₈×Dic₁₉₉

—

C₈×Dic₁₉₉

C₄

yes

—

C₂×M₁₆×C₁₉₉

C₂²

yes

—

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

Q₈

yes

—

#15691870

—

#15691870

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₉₉

C₂×C₄

yes

—

C₄×Dic₁₉₉

D₄

yes

characteristic

C₄×Dic₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

M₁₆×C₁₉₉

C₂×C₄

yes

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

C₂×C₄

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

C₄⋊C₄

yes

—

C₂×Dic₁₉₉

C₄⋊C₄

yes

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

C₄²

yes

—

C₂³×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₉

C₄×Q₈

yes

—

C₄×C₁₉₉

C₂×C₄²

yes

characteristic

C₄×C₁₉₉

#117

yes

characteristic

C₄×C₁₉₉

C₄×D₄

yes

—

Dic₁₉₉

—

Dic₁₉₉

#108

yes

complemented

C₂×C₁₉₉

—

C₂×C₁₉₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₉

C₂×#108

yes

—

C₁₉₉

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

199

maximal, Sylow

C₂×#108

—

199

—

C₄×M₁₆

—

199

—

C₂×C₄×C₈

—

199

—

#341

—

199

—

#341

D₁₉₉

yes

characteristic

C₂×M₁₆

—

199

—

C₂×M₁₆

C₂×D₁₉₉

yes

—

C₂²×C₈

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

C₂×C₄²

—

199

—

#108

—

199

—

#97

—

398

—

C₄×C₈

—

398

—

C₄×C₈

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

M₁₆

—

398

—

M₁₆

—

199

—

M₁₆

C₄×D₁₉₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₂×C₈

—

199

—

C₂×C₈

C₄×D₁₉₉

yes

—

C₄²

—

199

—

C₄²

—

199

—

C₄²

—

199

—

C₂³

Q₈×D₁₉₉

yes

characteristic

C₂×C₄

—

398

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₉

yes

—

C₂×C₄

D₄×D₁₉₉

yes

characteristic

C₂×C₄

#15691885

yes

—

C₈

—

C₈

—

C₈

—

398

—

C₈

—

199

—

C₂²

—

C₂²

C₄⋊C₄×D₁₉₉

yes

characteristic

C₂²

#20263920

yes

—

C₄

—

398

—

100

C₄

—

199

—

101

C₄

C₄²×D₁₉₉

yes

characteristic

102

C₄

#20263911

yes

characteristic

103

C₄

(C₂²⋊D₁₉₉)⋊C₄

yes

—

104

C₂

—

105

C₂

#27163869

yes

characteristic

106

C₂

#108×D₁₉₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#341×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#341	D₁₉₉
hypocenter	C₁₉₉	C₄×#108
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₉

commutator

C₄×C₁₉₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#341×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #20263890	C₄	semidirect
49. Dic₁₉₉	#108	semidirect
53. C₁₉₉	C₄×#108	semidirect

H ≅ G/N

type

9. #20263890

C₄

semidirect

49. Dic₁₉₉

#108

semidirect

53. C₁₉₉

C₄×#108

semidirect

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

subgroup	index
C₂×#20263890	2
M₁₆×Dic₁₉₉	2
C₂×C₈×Dic₁₉₉	2
#27163783	2
#341×C₁₉₉	2
C₄×#108	199

subgroup

index

C₂×#20263890

M₁₆×Dic₁₉₉

C₂×C₈×Dic₁₉₉

#27163783

#341×C₁₉₉

C₄×#108

199

Subgroups — #38768751

Conjugacy classes of subgroups

Complements

Maximal subgroups