Group #38768707 — subgroups

Subgroups — #38768707

Order 25472.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊(C₁₉₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#20263866	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27163773	C₂	1	2	yes	maximal
5	#376×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊(C₁₉₉⋊C₈)	C₄	1	8	yes	complemented
9	#20263866	—	2	2	no	—
10	C₄×C₁₉₉⋊C₈	—	2	2	no	—
11	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	#95×C₁₉₉	C₄	1	4	yes	—
15	C₂×C₁₉₉⋊C₈	—	2	2	no	—
16	C₂×C₁₉₉⋊C₈	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₄²×C₁₉₉	—	2	2	no	—
26	C₄²×C₁₉₉	C₂×C₄	1	4	yes	—
27	C₄²×C₁₉₉	D₄	1	2	yes	—
28	C₄²×C₁₉₉	Q₈	1	2	yes	—
29	C₁₉₉⋊C₈	—	2	4	no	—
30	C₁₉₉⋊C₈	—	2	4	no	—
31	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₉₉	—	2	4	no	—
33	C₂×C₄×C₁₉₉	—	2	4	no	—
34	C₂×C₄×C₁₉₉	—	2	2	no	—
35	C₂×C₄×C₁₉₉	—	2	4	no	—
36	C₂×C₄×C₁₉₉	C₄²	1	4	yes	—
37	C₂×C₄×C₁₉₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₉₉	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₉₉	#41	1	2	yes	—
42	C₂²×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₉₉	#117	1	1	yes	characteristic
45	C₂²×C₁₉₉	#131	1	4	yes	—
46	C₄×C₁₉₉	—	2	4	no	—
47	C₄×C₁₉₉	—	2	4	no	—
48	C₄×C₁₉₉	—	4	2	no	—
49	C₄×C₁₉₉	C₄×D₄	1	2	yes	—
50	C₄×C₁₉₉	C₄×Q₈	1	2	yes	—
51	C₂×C₁₉₉	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₉₉	#405	1	2	yes	—
53	C₂×C₁₉₉	#424	1	2	yes	—
54	C₂×C₁₉₉	#433	1	2	yes	—
55	C₁₉₉	#1768	1	1	yes	characteristic, complemented, Sylow
56	#1768	—	199	1	no	maximal, Sylow
57	C₂×C₄⋊C₈	—	199	2	no	—
58	C₂×#97	—	199	1	no	—
59	C₂×C₄×C₈	—	199	1	no	—
60	#376	D₁₉₉	1	1	yes	characteristic
61	#336	—	199	2	no	—
62	C₂²×C₈	—	199	2	no	—
63	C₂²×C₈	—	199	2	no	—
64	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
67	C₄⋊C₈	—	199	8	no	—
68	#97	—	398	2	no	—
69	C₄×C₈	—	398	2	no	—
70	#95	Dic₁₉₉	1	4	yes	—
71	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
73	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
75	C₂×C₈	—	398	2	no	—
76	C₂×C₈	—	398	2	no	—
77	C₂×C₈	—	398	2	no	—
78	C₂×C₈	—	398	2	no	—
79	C₂×C₈	—	199	4	no	—
80	C₂×C₈	—	199	4	no	—
81	C₄²	—	2	2	no	—
82	C₄²	C₄×D₁₉₉	1	4	yes	—
83	C₄²	D₇₉₆	1	2	yes	—
84	C₄²	Dic₃₉₈	1	2	yes	—
85	C₂³	C₂²×Dic₁₉₉	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₄×Dic₁₉₉	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₁₉₉	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₁₉₉	1	2	yes	—
93	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
95	C₂×C₄	#15691886	1	2	yes	—
96	C₈	—	398	4	no	—
97	C₈	—	398	4	no	—
98	C₂²	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
100	C₂²	#20263914	1	1	yes	characteristic
101	C₂²	#20263962	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₇₉₆	1	2	yes	—
106	C₄	C₄×Dic₃₉₈	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₁₉₉	1	1	yes	characteristic
108	C₂	#27163974	1	2	yes	—
109	C₂	#27164045	1	2	yes	—
110	C₂	#27164079	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₉⋊C₈)

C₂

yes

maximal

C₂×#20263866

C₂

yes

maximal, characteristic

C₂×C₄×C₁₉₉⋊C₈

C₂

yes

maximal, characteristic

#27163773

C₂

yes

maximal

#376×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₄⋊(C₁₉₉⋊C₈)

C₄

yes

complemented

#20263866

—

C₄×C₁₉₉⋊C₈

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

#95×C₁₉₉

C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

Q₈

yes

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄²

yes

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

C₂×C₄²

yes

characteristic

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₄×D₄

yes

—

C₄×C₁₉₉

C₄×Q₈

yes

—

C₂×C₁₉₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₉

#405

yes

—

C₂×C₁₉₉

#424

yes

—

C₂×C₁₉₉

#433

yes

—

C₁₉₉

#1768

yes

characteristic, complemented, Sylow

#1768

—

199

maximal, Sylow

C₂×C₄⋊C₈

—

199

—

C₂×#97

—

199

—

C₂×C₄×C₈

—

199

—

#376

D₁₉₉

yes

characteristic

#336

—

199

—

C₂²×C₈

—

199

—

C₂²×C₈

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

199

—

#97

—

398

—

C₄×C₈

—

398

—

#95

Dic₁₉₉

yes

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₂×C₈

—

199

—

C₄²

—

C₄²

C₄×D₁₉₉

yes

—

C₄²

D₇₉₆

yes

—

C₄²

Dic₃₉₈

yes

—

C₂³

C₂²×Dic₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₉

yes

—

C₂×C₄

C₄⋊Dic₁₉₉

yes

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

—

C₂×C₄

C₂×D₇₉₆

yes

characteristic

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

C₂×C₄

#15691886

yes

—

C₈

—

398

—

C₈

—

398

—

C₂²

C₂×C₄×Dic₁₉₉

yes

characteristic

C₂²

C₂×C₄⋊Dic₁₉₉

yes

characteristic

100

C₂²

#20263914

yes

characteristic

101

C₂²

#20263962

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₇₉₆

yes

—

106

C₄

C₄×Dic₃₉₈

yes

—

107

C₂

C₄×C₄⋊Dic₁₉₉

yes

characteristic

108

C₂

#27163974

yes

—

109

C₂

#27164045

yes

—

110

C₂

#27164079

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₉
commutator	C₂²×C₁₉₉	C₂×C₄²
Fitting	#376×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#376	D₁₉₉
hypocenter	C₁₉₉	#1768
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₉

commutator

C₂²×C₁₉₉

C₂×C₄²

Fitting

#376×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊(C₁₉₉⋊C₈)	C₄	semidirect
55. C₁₉₉	#1768	semidirect

H ≅ G/N

type

8. C₄⋊(C₁₉₉⋊C₈)

C₄

semidirect

55. C₁₉₉

#1768

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₉₉⋊C₈)	2
C₂×#20263866	2
C₂×C₄×C₁₉₉⋊C₈	2
#27163773	2
#376×C₁₉₉	2
#1768	199

subgroup

index

C₂×C₄⋊(C₁₉₉⋊C₈)

C₂×#20263866

C₂×C₄×C₁₉₉⋊C₈

#27163773

#376×C₁₉₉

#1768

199

Subgroups — #38768707

Conjugacy classes of subgroups

Complements

Maximal subgroups