Group #38768514 — subgroups

Subgroups — #38768514

Order 25472.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27164395	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164394	C₂	1	2	yes	maximal, complemented
3	#27164366	C₂	1	2	yes	maximal, complemented
4	#27163739	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₅₉₂	—	2	2	no	—
7	C₂×#15691871	—	2	4	no	—
8	#20263952	C₂²	1	2	yes	—
9	#20263875	—	2	4	no	—
10	C₄×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₉₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	#127×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂×D₇₉₆	—	2	2	no	—
16	C₂×D₇₉₆	—	2	2	no	—
17	C₂×D₇₉₆	—	2	2	no	—
18	D₁₅₉₂	—	4	4	no	—
19	#15691871	—	4	8	no	—
20	C₂×C₁₉₉⋊C₈	D₄	1	4	yes	—
21	C₂×Q₈×C₁₉₉	—	2	1	no	—
22	C₂×D₄×C₁₉₉	—	2	2	no	—
23	C₂×D₄×C₁₉₉	—	2	1	no	—
24	SD₁₆×C₁₉₉	—	4	4	no	—
25	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₉₉	—	2	2	no	—
27	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₂²×D₁₉₉	—	4	2	no	—
29	D₇₉₆	—	4	4	no	—
30	D₇₉₆	—	4	4	no	—
31	D₇₉₆	—	4	4	no	—
32	C₁₉₉⋊C₈	—	2	4	no	—
33	C₂³×C₁₉₉	—	4	1	no	—
34	Q₈×C₁₉₉	—	4	2	no	—
35	D₄×C₁₉₉	—	4	2	no	—
36	D₄×C₁₉₉	—	4	4	no	—
37	C₂×C₄×C₁₉₉	—	4	1	no	—
38	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₉₉	D₈	1	4	yes	complemented
41	C₂²×C₁₉₉	—	8	1	no	—
42	C₂²×C₁₉₉	—	8	2	no	—
43	C₂²×C₁₉₉	#127	1	1	yes	characteristic
44	C₂×D₁₉₉	—	8	2	no	—
45	C₂×D₁₉₉	—	8	2	no	—
46	C₂×D₁₉₉	—	8	2	no	—
47	C₄×C₁₉₉	—	8	1	no	—
48	C₄×C₁₉₉	C₂×D₈	1	2	yes	—
49	C₄×C₁₉₉	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₁₉₉	—	16	1	no	—
51	C₂×C₁₉₉	#493	1	1	yes	characteristic
52	C₂×C₁₉₉	#496	1	2	yes	—
53	D₁₉₉	—	16	2	no	—
54	C₁₉₉	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	199	1	no	maximal, Sylow
56	#493	—	199	2	no	—
57	#493	—	199	1	no	—
58	#492	D₁₉₉	1	1	yes	characteristic, complemented
59	#486	—	199	2	no	—
60	#322	—	199	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	398	4	no	—
63	C₂×D₈	—	398	2	no	—
64	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
65	#127	—	199	2	no	—
66	#127	C₂×D₁₉₉	1	1	yes	characteristic
67	#102	—	398	4	no	—
68	C₄×C₈	—	199	2	no	—
69	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	398	2	no	—
74	C₂×D₄	—	398	2	no	—
75	C₂×D₄	—	398	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	796	8	no	—
78	D₈	—	796	4	no	—
79	C₂×C₈	—	199	4	no	—
80	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	796	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	796	4	no	—
89	D₄	—	796	4	no	—
90	D₄	—	796	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
94	C₈	—	398	4	no	—
95	C₈	#15691871	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	1592	2	no	—
99	C₂²	—	1592	2	no	—
100	C₂²	—	1592	2	no	—
101	C₂²	#20263951	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#15691871	1	2	yes	—
104	C₄	#20263982	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	3184	2	no	—
107	C₂	#27164394	1	1	yes	characteristic
108	C₂	#27164404	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164395

C₂

yes

maximal, characteristic, complemented

#27164394

C₂

yes

maximal, complemented

#27164366

C₂

yes

maximal, complemented

#27163739

C₂

yes

maximal, characteristic, complemented

#492×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₅₉₂

—

C₂×#15691871

—

#20263952

C₂²

yes

—

#20263875

—

C₄×C₁₉₉⋊C₈

C₂²

yes

—

C₂×SD₁₆×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

#127×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

D₁₅₉₂

—

#15691871

—

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

SD₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

D₈

yes

complemented

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#127

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×D₈

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#493

yes

characteristic

C₂×C₁₉₉

#496

yes

—

D₁₉₉

—

C₁₉₉

#1721

yes

characteristic, complemented, Sylow

#1721

—

199

maximal, Sylow

#493

—

199

—

#493

—

199

—

#492

D₁₉₉

yes

characteristic, complemented

#486

—

199

—

#322

—

199

—

C₂×SD₁₆

—

C₂×D₈

—

398

—

C₂×D₈

—

398

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

#127

—

199

—

#127

C₂×D₁₉₉

yes

characteristic

#102

—

398

—

C₄×C₈

—

199

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

SD₁₆

—

D₈

—

796

—

D₈

—

796

—

C₂×C₈

—

199

—

C₂×C₈

C₂²⋊D₁₉₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

C₂³

—

796

—

Q₈

—

D₄

—

D₄

—

D₄

—

796

—

D₄

—

796

—

D₄

—

796

—

C₂×C₄

—

C₂×C₄

D₄×D₁₉₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₈

—

398

—

C₈

#15691871

yes

complemented

C₂²

—

C₂²

—

C₂²

—

1592

—

C₂²

—

1592

—

100

C₂²

—

1592

—

101

C₂²

#20263951

yes

characteristic

102

C₄

—

103

C₄

C₂×#15691871

yes

—

104

C₄

#20263982

yes

—

105

C₂

—

106

C₂

—

3184

—

107

C₂

#27164394

yes

characteristic

108

C₂

#27164404

yes

—

	subgroup	G/H
center	C₂²	#20263951
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#492×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#127
hypercenter	#492	D₁₉₉
hypocenter	C₁₉₉	#1721
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20263951

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#492×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27164395	C₂	semidirect
2. #27164394	C₂	semidirect
3. #27164366	C₂	semidirect
4. #27163739	C₂	semidirect
5. #492×C₁₉₉	C₂	semidirect
40. C₈×C₁₉₉	D₈	semidirect
54. C₁₉₉	#1721	semidirect
58. #492	D₁₉₉	semidirect
95. C₈	#15691871	semidirect

H ≅ G/N

type

1. #27164395

C₂

semidirect

2. #27164394

C₂

semidirect

3. #27164366

C₂

semidirect

4. #27163739

C₂

semidirect

5. #492×C₁₉₉

C₂

semidirect

40. C₈×C₁₉₉

D₈

semidirect

54. C₁₉₉

#1721

semidirect

58. #492

D₁₉₉

semidirect

95. C₈

#15691871

semidirect

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

subgroup	index
#27164395	2
#27164394	2
#27164366	2
#27163739	2
#492×C₁₉₉	2
#1721	199

subgroup

index

#492×C₁₉₉

199

Subgroups — #38768514

Conjugacy classes of subgroups

Complements

Maximal subgroups