Group #38767988 — subgroups

Subgroups — #38767988

Order 25472.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27164748	C₂	1	1	yes	maximal, characteristic
2	#27164310	C₂	1	2	yes	maximal
3	#27163767	C₂	1	1	yes	maximal, characteristic
4	#27163750	C₂	1	2	yes	maximal
5	#494×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	#20263954	—	2	2	no	—
7	#20263953	—	2	2	no	—
8	C₄×Dic₃₉₈	C₂²	1	2	yes	—
9	#20263920	—	2	2	no	—
10	#20263888	—	2	2	no	—
11	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	—
12	#20263878	—	2	2	no	—
13	C₂×Q₁₆×C₁₉₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₃₉₈	D₄	1	2	yes	—
19	C₂×C₁₉₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₉₉	D₄	1	2	yes	—
21	Dic₁₉₉⋊C₄	—	2	4	no	—
22	Dic₁₉₉⋊C₄	—	2	2	no	—
23	C₄×Dic₁₉₉	—	2	2	no	—
24	C₄×Dic₁₉₉	—	2	2	no	—
25	C₂×Q₈×C₁₉₉	—	2	1	no	—
26	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
27	Q₁₆×C₁₉₉	—	4	4	no	—
28	C₂×C₈×C₁₉₉	—	2	1	no	—
29	C₄⋊C₄×C₁₉₉	—	2	2	no	—
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
32	Dic₃₉₈	—	2	2	no	—
33	Dic₃₉₈	—	8	2	no	—
34	Dic₃₉₈	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₉₉	—	2	2	no	—
36	C₂×Dic₁₉₉	—	2	2	no	—
37	C₁₉₉⋊C₈	—	4	2	no	—
38	Q₈×C₁₉₉	—	4	2	no	—
39	Q₈×C₁₉₉	—	4	2	no	—
40	C₂×C₄×C₁₉₉	—	2	2	no	—
41	C₂×C₄×C₁₉₉	—	4	1	no	—
42	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₉	—	2	2	no	—
45	C₂²×C₁₉₉	#120	1	1	yes	characteristic
46	C₄×C₁₉₉	—	2	1	no	—
47	C₄×C₁₉₉	—	4	2	no	—
48	C₄×C₁₉₉	—	8	1	no	—
49	C₄×C₁₉₉	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₉₉	#137	1	2	yes	—
51	Dic₁₉₉	—	2	4	no	—
52	Dic₁₉₉	—	4	2	no	—
53	C₂×C₁₉₉	#451	1	2	yes	—
54	C₂×C₁₉₉	#453	1	1	yes	characteristic
55	C₁₉₉	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	199	1	no	maximal, Sylow
57	Q₈²	—	199	1	no	—
58	#494	D₁₉₉	1	1	yes	characteristic
59	#477	—	199	2	no	—
60	#333	—	199	1	no	—
61	Q₈⋊C₈	—	199	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	398	2	no	—
64	C₄⋊Q₈	—	398	2	no	—
65	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
67	C₄×Q₈	—	398	2	no	—
68	C₄×Q₈	—	199	2	no	—
69	#107	—	398	2	no	—
70	C₄⋊C₈	—	199	2	no	—
71	#103	—	398	2	no	—
72	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	398	2	no	—
75	C₂×Q₈	—	199	2	no	—
76	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	398	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	398	4	no	—
83	C₄⋊C₄	—	398	2	no	—
84	C₄⋊C₄	—	199	2	no	—
85	C₄²	—	398	2	no	—
86	C₄²	—	398	2	no	—
87	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	1592	2	no	—
91	Q₈	—	398	2	no	—
92	Q₈	—	199	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	398	2	no	—
96	C₂×C₄	—	398	2	no	—
97	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	796	2	no	—
101	C₂²	#20263923	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	796	2	no	—
106	C₄	—	398	4	no	—
107	C₄	Q₁₆×D₁₉₉	1	2	yes	—
108	C₄	#20263987	1	2	yes	—
109	C₂	#27164172	1	2	yes	—
110	C₂	#27164185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164748

C₂

yes

maximal, characteristic

#27164310

C₂

yes

maximal

#27163767

C₂

yes

maximal, characteristic

#27163750

C₂

yes

maximal

#494×C₁₉₉

C₂

yes

maximal, characteristic

#20263954

—

#20263953

—

C₄×Dic₃₉₈

C₂²

yes

—

#20263920

—

#20263888

—

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

—

#20263878

—

C₂×Q₁₆×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

D₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

D₄

yes

—

Dic₁₉₉⋊C₄

—

Dic₁₉₉⋊C₄

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

Q₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

Q₁₆

yes

complemented

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×Q₁₆

yes

—

C₄×C₁₉₉

#137

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

#451

yes

—

C₂×C₁₉₉

#453

yes

characteristic

C₁₉₉

#1633

yes

characteristic, complemented, Sylow

#1633

—

199

maximal, Sylow

Q₈²

—

199

—

#494

D₁₉₉

yes

characteristic

#477

—

199

—

#333

—

199

—

Q₈⋊C₈

—

199

—

C₂×Q₁₆

—

C₄⋊Q₈

—

398

—

C₄⋊Q₈

—

398

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

C₄×Q₈

—

398

—

C₄×Q₈

—

199

—

#107

—

398

—

C₄⋊C₈

—

199

—

#103

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

199

—

C₄²

—

398

—

C₄²

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

1592

—

Q₈

—

398

—

Q₈

—

199

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

D₄×D₁₉₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₈

—

100

C₈

—

796

—

101

C₂²

#20263923

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

796

—

106

C₄

—

398

—

107

C₄

Q₁₆×D₁₉₉

yes

—

108

C₄

#20263987

yes

—

109

C₂

#27164172

yes

—

110

C₂

#27164185

yes

characteristic

	subgroup	G/H
center	C₂²	#20263923
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#494×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	#120
hypercenter	#494	D₁₉₉
hypocenter	C₁₉₉	#1633
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#20263923

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#494×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₃₉₈	Q₁₆	semidirect
55. C₁₉₉	#1633	semidirect

H ≅ G/N

type

34. Dic₃₉₈

Q₁₆

semidirect

55. C₁₉₉

#1633

semidirect

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

subgroup	index
#27164748	2
#27164310	2
#27163767	2
#27163750	2
#494×C₁₉₉	2
#1633	199

subgroup

index

#494×C₁₉₉

199

Subgroups — #38767988

Conjugacy classes of subgroups

Complements

Maximal subgroups