Group #38768969 — subgroups

Subgroups — #38768969

Order 25472.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	1	yes	maximal, characteristic
2	#27163864	C₂	1	1	yes	maximal, characteristic
3	#27163772	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₉₉	C₂²	1	2	yes	—
7	C₄⋊(C₁₉₉⋊C₈)	—	2	2	no	—
8	C₁₉₉⋊#95	C₄	1	2	yes	—
9	C₁₉₉⋊#95	C₄	1	2	yes	—
10	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₉₉	—	2	2	no	—
13	C₂²×Dic₁₉₉	C₂×C₄	1	4	yes	—
14	C₂×C₁₉₉⋊C₈	—	2	2	no	—
15	C₂×C₁₉₉⋊C₈	—	2	4	no	—
16	C₄×Dic₁₉₉	—	2	2	no	—
17	C₄×Dic₁₉₉	—	2	2	no	—
18	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₉₉	—	2	2	no	—
22	C₂×C₈×C₁₉₉	—	2	4	no	—
23	C₄²×C₁₉₉	D₄	1	2	yes	—
24	C₄²×C₁₉₉	Q₈	1	2	yes	—
25	C₂×Dic₁₉₉	—	2	4	no	—
26	C₂×Dic₁₉₉	—	2	4	no	—
27	C₂×Dic₁₉₉	—	2	4	no	—
28	C₁₉₉⋊C₈	—	4	2	no	—
29	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₉	—	2	2	no	—
31	C₂×C₄×C₁₉₉	—	2	2	no	—
32	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	#41	1	2	yes	—
37	C₂×C₄×C₁₉₉	#41	1	2	yes	—
38	C₈×C₁₉₉	—	4	2	no	—
39	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₉₉	#117	1	2	yes	—
41	C₂²×C₁₉₉	#131	1	2	yes	—
42	C₂²×C₁₉₉	#131	1	2	yes	—
43	C₄×C₁₉₉	—	2	2	no	—
44	C₄×C₁₉₉	—	2	2	no	—
45	C₄×C₁₉₉	#124	1	1	yes	characteristic
46	C₄×C₁₉₉	#124	1	1	yes	characteristic
47	C₄×C₁₉₉	#125	1	2	yes	—
48	Dic₁₉₉	—	4	4	no	—
49	C₂×C₁₉₉	#382	1	1	yes	characteristic
50	C₂×C₁₉₉	#424	1	1	yes	characteristic
51	C₂×C₁₉₉	#424	1	1	yes	characteristic
52	C₂×C₁₉₉	#433	1	4	yes	—
53	C₁₉₉	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	199	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	199	1	no	—
56	C₂×C₄⋊C₈	D₁₉₉	1	1	yes	characteristic
57	#376	—	199	1	no	—
58	#336	—	199	4	no	—
59	C₂²×C₈	—	199	2	no	—
60	C₂²×C₈	C₂×D₁₉₉	1	2	yes	—
61	C₂×C₄²	—	199	2	no	—
62	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	398	2	no	—
65	#95	—	199	2	no	—
66	#95	—	199	2	no	—
67	C₂²×C₄	—	199	4	no	—
68	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	398	2	no	—
74	C₂×C₈	—	398	4	no	—
75	C₄²	—	398	2	no	—
76	C₄²	—	398	2	no	—
77	C₄²	C₂²⋊D₁₉₉	1	2	yes	—
78	C₄²	Dic₃₉₈	1	2	yes	—
79	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	398	4	no	—
83	C₂×C₄	—	398	4	no	—
84	C₂×C₄	—	398	4	no	—
85	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
86	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
89	C₂×C₄	#15691885	1	2	yes	—
90	C₂×C₄	#15691887	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	796	2	no	—
93	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
94	C₂²	#20263913	1	2	yes	—
95	C₂²	#20263961	1	2	yes	—
96	C₂²	#20263963	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	796	4	no	—
100	C₄	#20263942	1	1	yes	characteristic
101	C₄	#20263944	1	1	yes	characteristic
102	C₄	#20263946	1	2	yes	—
103	C₂	#27163883	1	1	yes	characteristic
104	C₂	#27164044	1	1	yes	characteristic
105	C₂	#27164046	1	1	yes	characteristic
106	C₂	#27164078	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₉⋊C₈)

C₂

yes

maximal, characteristic

#27163864

C₂

yes

maximal, characteristic

#27163772

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₉₉

C₂²

yes

—

C₄⋊(C₁₉₉⋊C₈)

—

C₁₉₉⋊#95

C₄

yes

—

C₁₉₉⋊#95

C₄

yes

—

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

—

C₂²×Dic₁₉₉

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

Q₈

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₉

#117

yes

—

C₂²×C₁₉₉

#131

yes

—

C₂²×C₁₉₉

#131

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₁₉₉

#1832

yes

characteristic, complemented, Sylow

#1832

—

199

maximal, Sylow

C₂×C₄⋊C₈

—

199

—

C₂×C₄⋊C₈

D₁₉₉

yes

characteristic

#376

—

199

—

#336

—

199

—

C₂²×C₈

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

—

C₂×C₄²

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

398

—

#95

—

199

—

#95

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₄²

—

398

—

C₄²

—

398

—

C₄²

C₂²⋊D₁₉₉

yes

—

C₄²

Dic₃₉₈

yes

—

C₂³

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

C₂×C₄

Dic₁₉₉⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

C₂×C₄

#15691885

yes

—

C₂×C₄

#15691887

yes

—

C₈

—

C₈

—

796

—

C₂²

C₂×Dic₁₉₉⋊C₄

yes

characteristic

C₂²

#20263913

yes

—

C₂²

#20263961

yes

—

C₂²

#20263963

yes

—

C₄

—

C₄

—

C₄

—

796

—

100

C₄

#20263942

yes

characteristic

101

C₄

#20263944

yes

characteristic

102

C₄

#20263946

yes

—

103

C₂

#27163883

yes

characteristic

104

C₂

#27164044

yes

characteristic

105

C₂

#27164046

yes

characteristic

106

C₂

#27164078

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₉
commutator, socle, prime radical	C₂³×C₁₉₉	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₉₉	C₂
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₉₉
hypocenter	C₁₉₉	#1832
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₉

commutator, socle, prime radical

C₂³×C₁₉₉

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₉₉

C₂

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₉₉

C₁₉₉

G ▷ C₂³×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₁₉₉	#1832	semidirect

H ≅ G/N

type

53. C₁₉₉

#1832

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
#27163864	2
#27163772	2
C₂×C₄⋊C₈×C₁₉₉	2
#1832	199

subgroup

index

C₂×C₄⋊(C₁₉₉⋊C₈)

#27163864

#27163772

C₂×C₄⋊C₈×C₁₉₉

#1832

199

Subgroups — #38768969

Conjugacy classes of subgroups

Complements

Maximal subgroups