Group #38744789 — subgroups

Subgroups — #38744789

Order 25216.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#27161517	C₂	1	2	yes	maximal, complemented
2	#27161456	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161197	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160893	C₂	1	2	yes	maximal, complemented
5	#364×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263348	C₂²	1	1	yes	characteristic
7	C₂×D₁₅₇₆	—	2	1	no	—
8	C₂×D₁₅₇₆	C₂²	1	2	yes	—
9	#20263332	—	2	1	no	—
10	#20263323	—	2	1	no	—
11	#20263271	—	2	2	no	—
12	D₃₁₅₂	—	2	2	no	—
13	#20263262	C₂²	1	1	yes	characteristic
14	#20263254	—	2	1	no	—
15	M₃₂×C₁₉₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	#15691358	—	2	1	no	—
18	C₂×D₇₈₈	—	2	2	no	—
19	C₂×D₇₈₈	—	4	2	no	—
20	Dic₇₈₈	D₄	1	1	yes	characteristic, complemented
21	#15691347	—	2	1	no	—
22	D₁₅₇₆	—	2	2	no	—
23	D₁₅₇₆	—	2	2	no	—
24	D₁₅₇₆	—	4	1	no	—
25	D₁₅₇₆	—	4	1	no	—
26	D₁₅₇₆	D₄	1	1	yes	characteristic, complemented
27	#15691342	—	2	1	no	—
28	#15691341	—	4	1	no	—
29	C₈×D₁₉₇	—	4	1	no	—
30	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
33	C₁₆×C₁₉₇	—	2	2	no	—
34	C₂²×D₁₉₇	—	4	2	no	—
35	Dic₃₉₄	—	2	1	no	—
36	D₇₈₈	—	2	1	no	—
37	D₇₈₈	—	4	2	no	—
38	D₇₈₈	—	4	2	no	—
39	D₇₈₈	—	8	4	no	—
40	C₂²⋊D₁₉₇	—	4	1	no	—
41	C₄×D₁₉₇	—	4	1	no	—
42	#13096802	—	4	1	no	—
43	C₂×C₄×C₁₉₇	—	2	1	no	—
44	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₉₇	D₈	1	1	yes	characteristic
46	C₈×C₁₉₇	D₈	1	1	yes	characteristic
47	C₈×C₁₉₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₉₇	#41	1	1	yes	characteristic
49	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₉₇	—	4	1	no	—
51	C₂×D₁₉₇	—	8	4	no	—
52	C₂×D₁₉₇	—	8	2	no	—
53	C₄×C₁₉₇	—	2	2	no	—
54	C₄×C₁₉₇	#121	1	1	yes	characteristic
55	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
56	Dic₁₉₇	—	4	1	no	—
57	C₂×C₁₉₇	—	2	1	no	—
58	C₂×C₁₉₇	#459	1	1	yes	characteristic
59	D₁₉₇	—	8	1	no	—
60	D₁₉₇	—	16	2	no	—
61	C₁₉₇	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	197	1	no	maximal, Sylow
63	#509	—	197	2	no	—
64	#493	—	197	1	no	—
65	#443	—	197	1	no	—
66	#364	D₁₉₇	1	1	yes	characteristic, complemented
67	#361	—	197	2	no	—
68	#135	—	197	1	no	—
69	C₂×D₈	—	394	1	no	—
70	C₂×D₈	—	197	2	no	—
71	#131	—	394	1	no	—
72	#127	—	394	1	no	—
73	SD₃₂	—	394	2	no	—
74	D₁₆	—	394	2	no	—
75	M₃₂	C₂×D₁₉₇	1	2	yes	—
76	#108	—	197	1	no	—
77	#104	—	394	1	no	—
78	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
79	#41	—	394	1	no	—
80	C₂×D₄	—	788	2	no	—
81	C₂×D₄	—	394	2	no	—
82	Q₁₆	—	197	1	no	—
83	SD₁₆	—	394	1	no	—
84	D₈	—	788	1	no	—
85	D₈	—	788	1	no	—
86	D₈	—	394	2	no	—
87	D₈	—	394	2	no	—
88	D₈	—	197	1	no	—
89	M₁₆	—	788	1	no	—
90	M₁₆	—	394	1	no	—
91	C₂×C₈	—	788	1	no	—
92	C₂×C₈	D₇₈₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
94	C₄²	D₇₈₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	788	2	no	—
97	Q₈	—	394	1	no	—
98	D₄	—	788	1	no	—
99	D₄	—	788	2	no	—
100	D₄	—	788	2	no	—
101	D₄	—	1576	4	no	—
102	D₄	—	394	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	788	1	no	—
105	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
106	C₈	—	788	1	no	—
107	C₈	D₁₅₇₆	1	1	yes	characteristic
108	C₈	D₁₅₇₆	1	1	yes	characteristic
109	C₈	D₄×D₁₉₇	1	1	yes	characteristic
110	C₈	#15691359	1	1	yes	characteristic
111	C₂²	—	788	1	no	—
112	C₂²	—	1576	4	no	—
113	C₂²	—	1576	2	no	—
114	C₂²	#20263354	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	788	1	no	—
117	C₄	#20263301	1	1	yes	characteristic
118	C₄	C₂×D₁₅₇₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	1576	1	no	—
121	C₂	—	3152	2	no	—
122	C₂	#27161275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161517

C₂

yes

maximal, complemented

#27161456

C₂

yes

maximal, characteristic, complemented

#27161197

C₂

yes

maximal, characteristic, complemented

#27160893

C₂

yes

maximal, complemented

#364×C₁₉₇

C₂

yes

maximal, characteristic, complemented

#20263348

C₂²

yes

characteristic

C₂×D₁₅₇₆

—

C₂×D₁₅₇₆

C₂²

yes

—

#20263332

—

#20263323

—

#20263271

—

D₃₁₅₂

—

#20263262

C₂²

yes

characteristic

#20263254

—

M₃₂×C₁₉₇

C₂²

yes

—

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

#15691358

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

Dic₇₈₈

D₄

yes

characteristic, complemented

#15691347

—

D₁₅₇₆

—

D₁₅₇₆

—

D₁₅₇₆

—

D₁₅₇₆

—

D₁₅₇₆

D₄

yes

characteristic, complemented

#15691342

—

#15691341

—

C₈×D₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

characteristic

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₁₆×C₁₉₇

—

C₂²×D₁₉₇

—

Dic₃₉₄

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂²⋊D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

D₈

yes

characteristic

C₈×C₁₉₇

D₈

yes

characteristic

C₈×C₁₉₇

C₂×D₄

yes

C₈×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#121

yes

characteristic

C₄×C₁₉₇

C₂×D₈

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#459

yes

characteristic

D₁₉₇

—

D₁₉₇

—

C₁₉₇

#2213

yes

characteristic, complemented, Sylow

#2213

—

197

maximal, Sylow

#509

—

197

—

#493

—

197

—

#443

—

197

—

#364

D₁₉₇

yes

characteristic, complemented

#361

—

197

—

#135

—

197

—

C₂×D₈

—

394

—

C₂×D₈

—

197

—

#131

—

394

—

#127

—

394

—

SD₃₂

—

394

—

D₁₆

—

394

—

M₃₂

C₂×D₁₉₇

yes

—

#108

—

197

—

#104

—

394

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

#41

—

394

—

C₂×D₄

—

788

—

C₂×D₄

—

394

—

Q₁₆

—

197

—

SD₁₆

—

394

—

D₈

—

788

—

D₈

—

788

—

D₈

—

394

—

D₈

—

394

—

D₈

—

197

—

M₁₆

—

788

—

M₁₆

—

394

—

C₂×C₈

—

788

—

C₂×C₈

D₇₈₈

yes

characteristic

C₂×C₈

C₂²×D₁₉₇

yes

characteristic

C₄²

D₇₈₈

yes

characteristic

C₁₆

—

C₂³

—

788

—

Q₈

—

394

—

D₄

—

788

—

D₄

—

788

—

100

D₄

—

788

—

101

D₄

—

1576

—

102

D₄

—

394

—

103

C₂×C₄

—

104

C₂×C₄

—

788

—

105

C₂×C₄

C₂×D₇₈₈

yes

characteristic

106

C₈

—

788

—

107

C₈

D₁₅₇₆

yes

characteristic

108

C₈

D₁₅₇₆

yes

characteristic

109

C₈

D₄×D₁₉₇

yes

characteristic

110

C₈

#15691359

yes

characteristic

111

C₂²

—

788

—

112

C₂²

—

1576

—

113

C₂²

—

1576

—

114

C₂²

#20263354

yes

characteristic

115

C₄

—

116

C₄

—

788

—

117

C₄

#20263301

yes

characteristic

118

C₄

C₂×D₁₅₇₆

yes

characteristic

119

C₂

—

120

C₂

—

1576

—

121

C₂

—

3152

—

122

C₂

#27161275

yes

characteristic

	subgroup	G/H
center	C₂	#27161275
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#364×C₁₉₇	C₂
socle	C₂×C₁₉₇	#459
hypercenter	#364	D₁₉₇
hypocenter	C₁₉₇	#2213
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#27161275

commutator, cosocle

C₂×C₈×C₁₉₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₇

Fitting

#364×C₁₉₇

C₂

C₂×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #27161517	C₂	semidirect
2. #27161456	C₂	semidirect
3. #27161197	C₂	semidirect
4. #27160893	C₂	semidirect
5. #364×C₁₉₇	C₂	semidirect
20. Dic₇₈₈	D₄	semidirect
26. D₁₅₇₆	D₄	semidirect
61. C₁₉₇	#2213	semidirect
66. #364	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161517

C₂

semidirect

2. #27161456

C₂

semidirect

3. #27161197

C₂

semidirect

4. #27160893

C₂

semidirect

5. #364×C₁₉₇

C₂

semidirect

20. Dic₇₈₈

D₄

semidirect

26. D₁₅₇₆

D₄

semidirect

61. C₁₉₇

#2213

semidirect

66. #364

D₁₉₇

semidirect

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

subgroup	index
#27161517	2
#27161456	2
#27161197	2
#27160893	2
#364×C₁₉₇	2
#2213	197

subgroup

index

#364×C₁₉₇

197

Subgroups — #38744789

Conjugacy classes of subgroups

Complements

Maximal subgroups