Group #38754516 — subgroups

Subgroups — #38754516

Order 25216.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#27161892	C₂	1	4	yes	maximal, complemented
2	#27161351	C₂	1	4	yes	maximal, complemented
3	#27161261	C₂	1	4	yes	maximal, complemented
4	#27161197	C₂	1	2	yes	maximal, complemented
5	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263374	C₂²	1	4	yes	—
7	#20263358	—	2	8	no	—
8	#20263357	C₂²	1	8	yes	—
9	#20263349	—	2	4	no	—
10	#20263348	C₂²	1	2	yes	—
11	Q₁₆×D₁₉₇	—	2	4	no	—
12	#20263332	C₂²	1	4	yes	—
13	#20263262	C₂²	1	2	yes	—
14	#20263254	C₂²	1	4	yes	—
15	#20263245	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₉₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
19	#15691359	—	2	8	no	—
20	#15691358	—	4	4	no	—
21	#15691358	C₂³	1	4	yes	—
22	Q₈×D₁₉₇	—	2	8	no	—
23	Dic₇₈₈	D₄	1	2	yes	—
24	#15691348	—	2	8	no	—
25	#15691347	D₄	1	4	yes	—
26	#15691345	—	2	8	no	—
27	D₁₅₇₆	D₄	1	2	yes	—
28	#15691342	C₂³	1	4	yes	—
29	#15691341	—	2	4	no	—
30	C₈×D₁₉₇	—	2	4	no	—
31	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
32	Q₁₆×C₁₉₇	—	2	4	no	—
33	Q₁₆×C₁₉₇	—	2	4	no	—
34	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₉₇	—	4	2	no	—
36	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
37	Dic₃₉₄	—	4	4	no	—
38	Dic₃₉₄	C₂×D₄	1	4	yes	—
39	D₇₈₈	—	4	4	no	—
40	D₇₈₈	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₉₇	—	2	4	no	—
42	C₄×D₁₉₇	—	2	4	no	—
43	C₄×D₁₉₇	—	4	8	no	—
44	#13096802	—	2	4	no	—
45	Q₈×C₁₉₇	—	2	8	no	—
46	C₂×C₄×C₁₉₇	—	2	1	no	—
47	C₂×C₄×C₁₉₇	—	2	4	no	—
48	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
50	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
51	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₉₇	—	2	4	no	—
53	C₄×C₁₉₇	—	2	2	no	—
54	C₄×C₁₉₇	—	4	4	no	—
55	C₄×C₁₉₇	C₂²×D₄	1	2	yes	—
56	Dic₁₉₇	—	2	4	no	—
57	C₂×C₁₉₇	—	2	1	no	—
58	C₂×C₁₉₇	D₄²	1	1	yes	characteristic
59	D₁₉₇	—	4	4	no	—
60	C₁₉₇	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	197	1	no	maximal, Sylow
62	#578	—	197	4	no	—
63	#494	D₁₉₇	1	1	yes	characteristic, complemented
64	#473	—	197	4	no	—
65	#456	—	197	4	no	—
66	#443	—	197	2	no	—
67	2₋¹⁺⁴	—	197	4	no	—
68	#137	—	394	8	no	—
69	#137	—	197	8	no	—
70	#135	—	394	4	no	—
71	#135	—	197	2	no	—
72	C₂×Q₁₆	—	394	4	no	—
73	C₂×Q₁₆	C₂×D₁₉₇	1	4	yes	—
74	#131	—	197	4	no	—
75	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	—
76	#108	—	197	2	no	—
77	#104	—	197	4	no	—
78	#101	—	197	4	no	—
79	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
80	#41	—	788	4	no	—
81	#41	—	394	8	no	—
82	#41	—	197	4	no	—
83	C₂×Q₈	—	394	8	no	—
84	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	394	8	no	—
88	Q₁₆	—	197	2	no	—
89	SD₁₆	—	394	8	no	—
90	SD₁₆	—	197	4	no	—
91	D₈	—	197	2	no	—
92	M₁₆	—	394	4	no	—
93	M₁₆	—	197	4	no	—
94	C₂×C₈	—	394	4	no	—
95	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	788	4	no	—
100	Q₈	—	197	4	no	—
101	D₄	—	788	4	no	—
102	D₄	—	394	4	no	—
103	D₄	—	197	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	788	8	no	—
107	C₂×C₄	—	394	4	no	—
108	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
109	C₈	—	394	4	no	—
110	C₈	D₄×D₁₉₇	1	2	yes	—
111	C₈	D₄×D₁₉₇	1	2	yes	—
112	C₂²	—	394	4	no	—
113	C₂²	#20263371	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	394	4	no	—
117	C₄	C₂×D₄×D₁₉₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	788	4	no	—
120	C₂	#27161714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161892

C₂

yes

maximal, complemented

#27161351

C₂

yes

maximal, complemented

#27161261

C₂

yes

maximal, complemented

#27161197

C₂

yes

maximal, complemented

#494×C₁₉₇

C₂

yes

maximal, characteristic, complemented

#20263374

C₂²

yes

—

#20263358

—

#20263357

C₂²

yes

—

#20263349

—

#20263348

C₂²

yes

—

Q₁₆×D₁₉₇

—

#20263332

C₂²

yes

—

#20263262

C₂²

yes

—

#20263254

C₂²

yes

—

#20263245

C₂²

yes

—

C₂×Q₁₆×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₄×C₈×C₁₉₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₉₇

—

Dic₇₈₈

D₄

yes

—

#15691348

—

#15691347

D₄

yes

—

#15691345

—

D₁₅₇₆

D₄

yes

—

#15691342

C₂³

yes

—

#15691341

—

C₈×D₁₉₇

—

C₂×Q₈×C₁₉₇

C₂³

yes

—

Q₁₆×C₁₉₇

—

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂³

yes

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

Dic₃₉₄

C₂×D₄

yes

—

D₇₈₈

—

D₇₈₈

C₂×D₄

yes

—

C₂²⋊D₁₉₇

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₈×C₁₉₇

C₂×D₄

yes

—

C₈×C₁₉₇

C₂×D₄

yes

—

C₂²×C₁₉₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²×D₄

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

D₄²

yes

characteristic

D₁₉₇

—

C₁₉₇

#3293

yes

characteristic, complemented, Sylow

#3293

—

197

maximal, Sylow

#578

—

197

—

#494

D₁₉₇

yes

characteristic, complemented

#473

—

197

—

#456

—

197

—

#443

—

197

—

2₋¹⁺⁴

—

197

—

#137

—

394

—

#137

—

197

—

#135

—

394

—

#135

—

197

—

C₂×Q₁₆

—

394

—

C₂×Q₁₆

C₂×D₁₉₇

yes

—

#131

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

#108

—

197

—

#104

—

197

—

#101

—

197

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

#41

—

788

—

#41

—

394

—

#41

—

197

—

C₂×Q₈

—

394

—

C₂×Q₈

C₂²×D₁₉₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

394

—

Q₁₆

—

197

—

SD₁₆

—

394

—

SD₁₆

—

197

—

D₈

—

197

—

M₁₆

—

394

—

M₁₆

—

197

—

C₂×C₈

—

394

—

C₂×C₈

C₂²×D₁₉₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₉₇

yes

characteristic

Q₈

—

Q₈

—

788

—

100

Q₈

—

197

—

101

D₄

—

788

—

102

D₄

—

394

—

103

D₄

—

197

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

788

—

107

C₂×C₄

—

394

—

108

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

109

C₈

—

394

—

110

C₈

D₄×D₁₉₇

yes

—

111

C₈

D₄×D₁₉₇

yes

—

112

C₂²

—

394

—

113

C₂²

#20263371

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

394

—

117

C₄

C₂×D₄×D₁₉₇

yes

—

118

C₂

—

119

C₂

—

788

—

120

C₂

#27161714

yes

characteristic

	subgroup	G/H
center	C₂	#27161714
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#494×C₁₉₇	C₂
socle	C₂×C₁₉₇	D₄²
hypercenter	#494	D₁₉₇
hypocenter	C₁₉₇	#3293
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#27161714

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

#494×C₁₉₇

C₂

socle

C₂×C₁₉₇

D₄²

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #27161892	C₂	semidirect
2. #27161351	C₂	semidirect
3. #27161261	C₂	semidirect
4. #27161197	C₂	semidirect
5. #494×C₁₉₇	C₂	semidirect
60. C₁₉₇	#3293	semidirect
63. #494	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161892

C₂

semidirect

2. #27161351

C₂

semidirect

3. #27161261

C₂

semidirect

4. #27161197

C₂

semidirect

5. #494×C₁₉₇

C₂

semidirect

60. C₁₉₇

#3293

semidirect

63. #494

D₁₉₇

semidirect

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

subgroup	index
#27161892	2
#27161351	2
#27161261	2
#27161197	2
#494×C₁₉₇	2
#3293	197

subgroup

index

#494×C₁₉₇

197

Subgroups — #38754516

Conjugacy classes of subgroups

Complements

Maximal subgroups