Group #38766680 — subgroups

Subgroups — #38766680

Order 25472.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#27164071	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263866	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉₉⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₁₉₉⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₁₉₉⋊C₈)	—	2	1	no	—
7	#20263866	—	2	1	no	—
8	#20263866	—	2	1	no	—
9	C₄×C₁₉₉⋊C₈	C₄	1	2	yes	—
10	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₉₉	—	2	1	no	—
12	C₂²⋊C₈×C₁₉₉	—	2	1	no	—
13	C₄×C₈×C₁₉₉	C₄	1	2	yes	—
14	C₂×C₁₉₉⋊C₈	—	2	2	no	—
15	C₂×C₁₉₉⋊C₈	—	2	2	no	—
16	C₂×C₁₉₉⋊C₈	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₉₉	—	2	2	no	—
22	C₄²×C₁₉₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
25	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
26	C₁₉₉⋊C₈	—	2	4	no	—
27	C₁₉₉⋊C₈	—	4	2	no	—
28	C₂³×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	—	4	1	no	—
30	C₂×C₄×C₁₉₉	—	4	1	no	—
31	C₂×C₄×C₁₉₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₉	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₉₉	M₁₆	1	2	yes	—
39	C₈×C₁₉₉	—	4	2	no	—
40	C₂²×C₁₉₉	—	2	1	no	—
41	C₂²×C₁₉₉	—	2	1	no	—
42	C₂²×C₁₉₉	#95	1	1	yes	characteristic
43	C₂²×C₁₉₉	C₄⋊C₈	1	2	yes	—
44	C₄×C₁₉₉	—	2	1	no	—
45	C₄×C₁₉₉	—	2	1	no	—
46	C₄×C₁₉₉	—	2	1	no	—
47	C₄×C₁₉₉	—	2	1	no	—
48	C₄×C₁₉₉	C₄×C₈	1	1	yes	characteristic
49	C₄×C₁₉₉	#97	1	1	yes	characteristic
50	C₄×C₁₉₉	C₂²⋊C₈	1	2	yes	—
51	C₂×C₁₉₉	—	2	2	no	—
52	C₂×C₁₉₉	#336	1	1	yes	characteristic
53	C₂×C₁₉₉	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₉₉	#338	1	1	yes	characteristic
55	C₁₉₉	#1275	1	1	yes	characteristic, complemented, Sylow
56	#1275	—	199	1	no	maximal, Sylow
57	#431	—	199	1	no	—
58	#431	D₁₉₉	1	1	yes	characteristic
59	C₂×#97	—	199	1	no	—
60	C₂²×C₈	—	199	2	no	—
61	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	398	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	398	1	no	—
66	#97	—	398	1	no	—
67	#97	—	398	1	no	—
68	C₄×C₈	—	199	2	no	—
69	C₄×C₈	Dic₁₉₉	1	2	yes	—
70	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
72	C₂²×C₄	Dic₃₉₈	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	398	2	no	—
75	C₂×C₈	—	398	2	no	—
76	C₂×C₈	—	398	2	no	—
77	C₂×C₈	—	199	4	no	—
78	C₄²	C₄×D₁₉₉	1	1	yes	characteristic
79	C₄²	C₂×Dic₁₉₉	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₉₉	1	1	yes	characteristic
81	C₄²	D₇₉₆	1	1	yes	characteristic
82	C₂³	Dic₁₉₉⋊C₄	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
89	C₂×C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₉₉	1	1	yes	characteristic
91	C₂×C₄	C₈×D₁₉₉	1	2	yes	—
92	C₂×C₄	#15691869	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	796	2	no	—
95	C₈	—	398	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₁₉₉⋊#95	1	1	yes	characteristic
99	C₂²	Dic₁₉₉⋊C₈	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₉₉	1	1	yes	characteristic
105	C₄	#20263865	1	1	yes	characteristic
106	C₄	(C₂×D₁₉₉)⋊C₈	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#27163772	1	1	yes	characteristic
109	C₂	#27163774	1	1	yes	characteristic
110	C₂	C₁₉₉⋊#338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164071

C₂

yes

maximal, characteristic

C₂×#20263866

C₂

yes

maximal, characteristic

#431×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₄

yes

—

C₄⋊(C₁₉₉⋊C₈)

—

C₂²⋊(C₁₉₉⋊C₈)

—

#20263866

—

#20263866

—

C₄×C₁₉₉⋊C₈

C₄

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

—

C₂²⋊C₈×C₁₉₉

—

C₄×C₈×C₁₉₉

C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₈

yes

complemented

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

Q₈

yes

characteristic

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

C₂×C₄

yes

characteristic

C₄²×C₁₉₉

C₂×C₄

yes

characteristic

C₄²×C₁₉₉

D₄

yes

characteristic

C₄²×C₁₉₉

D₄

yes

characteristic

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄²

yes

characteristic

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×C₈

yes

—

C₂×C₄×C₁₉₉

M₁₆

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#95

yes

characteristic

C₂²×C₁₉₉

C₄⋊C₈

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₄×C₈

yes

characteristic

C₄×C₁₉₉

#97

yes

characteristic

C₄×C₁₉₉

C₂²⋊C₈

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1275

—

199

maximal, Sylow

#431

—

199

—

#431

D₁₉₉

yes

characteristic

C₂×#97

—

199

—

C₂²×C₈

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

398

—

C₂²⋊C₈

—

C₂²⋊C₈

—

398

—

#97

—

398

—

#97

—

398

—

C₄×C₈

—

199

—

C₄×C₈

Dic₁₉₉

yes

—

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

Dic₃₉₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₄²

C₄×D₁₉₉

yes

characteristic

C₄²

C₂×Dic₁₉₉

yes

characteristic

C₄²

C₂²⋊D₁₉₉

yes

characteristic

C₄²

D₇₉₆

yes

characteristic

C₂³

Dic₁₉₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₉

yes

characteristic

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₉₉

yes

characteristic

C₂×C₄

Dic₁₉₉⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₉

yes

characteristic

C₂×C₄

C₈×D₁₉₉

yes

—

C₂×C₄

#15691869

yes

—

C₈

—

C₈

—

796

—

C₈

—

398

—

C₂²

—

C₂²

—

C₂²

C₁₉₉⋊#95

yes

characteristic

C₂²

Dic₁₉₉⋊C₈

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₉₉

yes

characteristic

105

C₄

#20263865

yes

characteristic

106

C₄

(C₂×D₁₉₉)⋊C₈

yes

—

107

C₂

—

108

C₂

#27163772

yes

characteristic

109

C₂

#27163774

yes

characteristic

110

C₂

C₁₉₉⋊#338

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₉₉
commutator	C₄×C₁₉₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₉₉
Fitting	#431×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#95
prime radical	C₂³×C₁₉₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉₉	C₂²
hypercenter	#431	D₁₉₉
hypocenter	C₁₉₉	#1275
LCS	G ▷ C₄×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₁₉₉

commutator

C₄×C₁₉₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₉₉

Fitting

#431×C₁₉₉

C₂

socle

C₂²×C₁₉₉

#95

prime radical

C₂³×C₁₉₉

C₄⋊C₄

cosocle

C₂×C₄²×C₁₉₉

C₂²

D₁₉₉

C₁₉₉

G ▷ C₄×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×C₁₉₉⋊C₈	C₈	semidirect
55. C₁₉₉	#1275	semidirect

H ≅ G/N

type

17. C₂×C₁₉₉⋊C₈

C₈

semidirect

55. C₁₉₉

#1275

semidirect

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

subgroup	index
#27164071	2
C₂×#20263866	2
#431×C₁₉₉	2
#1275	199

subgroup

index

#27164071

C₂×#20263866

#431×C₁₉₉

#1275

199

Subgroups — #38766680

Conjugacy classes of subgroups

Complements

Maximal subgroups