Group #38768516 — subgroups

Subgroups — #38768516

Order 25472.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27164395	C₂	1	2	yes	maximal, complemented
3	#27164391	C₂	1	1	yes	maximal, characteristic, complemented
4	#27164365	C₂	1	2	yes	maximal, complemented
5	#322×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691875	—	2	2	no	—
7	C₂×D₁₅₉₂	—	2	4	no	—
8	C₂×D₁₅₉₂	—	2	2	no	—
9	#20263955	C₂²	1	1	yes	characteristic
10	#20263952	C₂²	1	2	yes	—
11	#20263952	C₂²	1	1	yes	characteristic
12	#20263877	—	2	4	no	—
13	C₄×C₈×C₁₉₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₈	—	2	1	no	—
16	C₂×D₇₉₆	—	2	2	no	—
17	C₂×D₇₉₆	—	2	2	no	—
18	C₂×D₇₉₆	—	2	2	no	—
19	C₂×D₇₉₆	—	2	1	no	—
20	C₂×D₇₉₆	—	2	2	no	—
21	#15691875	—	4	4	no	—
22	D₁₅₉₂	—	4	8	no	—
23	D₁₅₉₂	—	4	4	no	—
24	C₄⋊Dic₁₉₉	—	2	2	no	—
25	C₂×C₈×C₁₉₉	D₄	1	4	yes	—
26	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
27	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₂²×D₁₉₉	—	4	1	no	—
29	C₂²×D₁₉₉	—	4	2	no	—
30	Dic₃₉₈	—	4	2	no	—
31	D₇₉₆	—	4	4	no	—
32	D₇₉₆	—	4	2	no	—
33	D₇₉₆	—	4	4	no	—
34	D₇₉₆	—	4	4	no	—
35	D₇₉₆	—	4	4	no	—
36	C₂×Dic₁₉₉	—	4	1	no	—
37	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
39	C₈×C₁₉₉	—	2	4	no	—
40	C₈×C₁₉₉	D₈	1	4	yes	complemented
41	C₂²×C₁₉₉	#127	1	1	yes	characteristic
42	C₂×D₁₉₉	—	8	2	no	—
43	C₂×D₁₉₉	—	8	1	no	—
44	C₂×D₁₉₉	—	8	2	no	—
45	C₂×D₁₉₉	—	8	2	no	—
46	C₂×D₁₉₉	—	8	2	no	—
47	C₄×C₁₉₉	C₂×D₈	1	2	yes	—
48	C₄×C₁₉₉	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₉₉	—	8	1	no	—
50	C₂×C₁₉₉	#493	1	1	yes	characteristic
51	C₂×C₁₉₉	#496	1	2	yes	—
52	D₁₉₉	—	16	2	no	—
53	D₁₉₉	—	16	1	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	—	199	1	no	—
59	#486	—	199	2	no	—
60	#322	D₁₉₉	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	398	2	no	—
62	C₂×D₈	—	398	4	no	—
63	C₂×D₈	—	398	2	no	—
64	C₄⋊Q₈	—	199	1	no	—
65	#127	—	199	2	no	—
66	#127	—	199	1	no	—
67	#102	—	398	4	no	—
68	C₄×C₈	C₂×D₁₉₉	1	2	yes	—
69	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
70	C₂×Q₈	—	398	1	no	—
71	C₂×D₄	—	398	2	no	—
72	C₂×D₄	—	398	2	no	—
73	C₂×D₄	—	398	2	no	—
74	C₂×D₄	—	398	2	no	—
75	C₂×D₄	—	398	1	no	—
76	SD₁₆	—	796	4	no	—
77	D₈	—	796	8	no	—
78	D₈	—	796	4	no	—
79	C₂×C₈	D₇₉₆	1	4	yes	—
80	C₂×C₈	D₇₉₆	1	2	yes	—
81	C₄⋊C₄	—	398	2	no	—
82	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
83	C₂³	—	796	2	no	—
84	C₂³	—	796	1	no	—
85	Q₈	—	796	2	no	—
86	D₄	—	796	4	no	—
87	D₄	—	796	2	no	—
88	D₄	—	796	4	no	—
89	D₄	—	796	4	no	—
90	D₄	—	796	4	no	—
91	C₂×C₄	—	796	1	no	—
92	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₇₉₆	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₁₅₉₂	1	4	yes	complemented
96	C₂²	—	1592	2	no	—
97	C₂²	—	1592	1	no	—
98	C₂²	—	1592	2	no	—
99	C₂²	—	1592	2	no	—
100	C₂²	—	1592	2	no	—
101	C₂²	#20263952	1	1	yes	characteristic
102	C₄	—	1592	1	no	—
103	C₄	C₂×D₁₅₉₂	1	2	yes	—
104	C₄	#20263983	1	4	yes	—
105	C₂	—	3184	2	no	—
106	C₂	—	3184	1	no	—
107	C₂	#27164395	1	1	yes	characteristic
108	C₂	#27164407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164395

C₂

yes

maximal, characteristic, complemented

#27164395

C₂

yes

maximal, complemented

#27164391

C₂

yes

maximal, characteristic, complemented

#27164365

C₂

yes

maximal, complemented

#322×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691875

—

C₂×D₁₅₉₂

—

C₂×D₁₅₉₂

—

#20263955

C₂²

yes

characteristic

#20263952

C₂²

yes

—

#20263952

C₂²

yes

characteristic

#20263877

—

C₄×C₈×C₁₉₉

C₂²

yes

—

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

#15691875

—

D₁₅₉₂

—

D₁₅₉₂

—

C₄⋊Dic₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

Dic₃₉₈

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

C₂×Dic₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₈×C₁₉₉

—

C₈×C₁₉₉

D₈

yes

complemented

C₂²×C₁₉₉

#127

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

C₂×D₈

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

#493

yes

characteristic

C₂×C₁₉₉

#496

yes

—

D₁₉₉

—

D₁₉₉

—

C₁₉₉

#1721

yes

characteristic, complemented, Sylow

#1721

—

199

maximal, Sylow

#493

—

199

—

#493

—

199

—

#492

—

199

—

#486

—

199

—

#322

D₁₉₉

yes

characteristic, complemented

C₂×SD₁₆

—

398

—

C₂×D₈

—

398

—

C₂×D₈

—

398

—

C₄⋊Q₈

—

199

—

#127

—

199

—

#127

—

199

—

#102

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

SD₁₆

—

796

—

D₈

—

796

—

D₈

—

796

—

C₂×C₈

D₇₉₆

yes

—

C₂×C₈

D₇₉₆

yes

—

C₄⋊C₄

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

796

—

C₂³

—

796

—

Q₈

—

796

—

D₄

—

796

—

D₄

—

796

—

D₄

—

796

—

D₄

—

796

—

D₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

C₂×D₇₉₆

yes

characteristic

C₂×C₄

C₂×D₇₉₆

yes

—

C₈

—

C₈

D₁₅₉₂

yes

complemented

C₂²

—

1592

—

C₂²

—

1592

—

C₂²

—

1592

—

C₂²

—

1592

—

100

C₂²

—

1592

—

101

C₂²

#20263952

yes

characteristic

102

C₄

—

1592

—

103

C₄

C₂×D₁₅₉₂

yes

—

104

C₄

#20263983

yes

—

105

C₂

—

3184

—

106

C₂

—

3184

—

107

C₂

#27164395

yes

characteristic

108

C₂

#27164407

yes

—

	subgroup	G/H
center	C₂²	#20263952
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#322×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#127
hypercenter	#322	D₁₉₉
hypocenter	C₁₉₉	#1721
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#20263952

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#322×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #27164395	C₂	semidirect
2. #27164395	C₂	semidirect
3. #27164391	C₂	semidirect
4. #27164365	C₂	semidirect
5. #322×C₁₉₉	C₂	semidirect
40. C₈×C₁₉₉	D₈	semidirect
54. C₁₉₉	#1721	semidirect
60. #322	D₁₉₉	semidirect
95. C₈	D₁₅₉₂	semidirect

H ≅ G/N

type

1. #27164395

C₂

semidirect

2. #27164395

C₂

semidirect

3. #27164391

C₂

semidirect

4. #27164365

C₂

semidirect

5. #322×C₁₉₉

C₂

semidirect

40. C₈×C₁₉₉

D₈

semidirect

54. C₁₉₉

#1721

semidirect

60. #322

D₁₉₉

semidirect

95. C₈

D₁₅₉₂

semidirect

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

subgroup	index
#27164395	2
#27164391	2
#27164365	2
#322×C₁₉₉	2
#1721	199

subgroup

index

#322×C₁₉₉

199

Subgroups — #38768516

Conjugacy classes of subgroups

Complements

Maximal subgroups