Group #38743461 — subgroups

Subgroups — #38743461

Order 25216.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#20263262	C₂	1	1	yes	maximal, characteristic
2	#27161106	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#27160838	C₂	1	2	yes	maximal
5	#427×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263262	—	2	2	no	—
10	C₄⋊#13096802	—	2	4	no	—
11	C₈×Dic₁₉₇	—	2	2	no	—
12	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	#117×C₁₉₇	C₂²	1	2	yes	—
14	#107×C₁₉₇	C₄	1	2	yes	complemented
15	#106×C₁₉₇	C₄	1	2	yes	complemented
16	C₂²×Dic₁₉₇	—	2	1	no	—
17	#15691342	—	2	2	no	—
18	#15691342	—	2	2	no	—
19	C₂×#13096802	—	2	1	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₂×#13096802	—	2	1	no	—
22	C₂×#13096802	D₄	1	2	yes	—
23	C₄×Dic₁₉₇	—	2	1	no	—
24	C₄×Dic₁₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
32	C₄²×C₁₉₇	—	2	2	no	—
33	C₂×Dic₁₉₇	—	2	4	no	—
34	C₂×Dic₁₉₇	—	2	2	no	—
35	#13096802	—	2	2	no	—
36	#13096802	—	4	2	no	—
37	C₂³×C₁₉₇	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₉₇	—	2	4	no	—
39	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
45	C₈×C₁₉₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₉₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₉₇	—	2	2	no	—
48	C₂²×C₁₉₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₉₇	#125	1	1	yes	characteristic
51	C₄×C₁₉₇	—	4	4	no	—
52	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₉₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₉₇	#121	1	1	yes	characteristic
55	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₉₇	—	2	4	no	—
57	C₂×C₁₉₇	—	2	1	no	—
58	C₂×C₁₉₇	—	2	1	no	—
59	C₂×C₁₉₇	#389	1	1	yes	characteristic
60	C₂×C₁₉₇	#443	1	2	yes	—
61	C₁₉₇	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	197	1	no	maximal, Sylow
63	C₂×#108	—	197	1	no	—
64	#427	D₁₉₇	1	1	yes	characteristic
65	#424	—	197	2	no	—
66	C₂×C₄×C₈	—	197	1	no	—
67	#339	—	197	2	no	—
68	C₂×M₁₆	—	197	2	no	—
69	C₂²×C₈	—	197	1	no	—
70	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
71	#117	C₂×D₁₉₇	1	2	yes	—
72	C₂×C₄²	—	197	1	no	—
73	#108	—	394	2	no	—
74	#107	Dic₁₉₇	1	2	yes	complemented
75	#106	Dic₁₉₇	1	2	yes	complemented
76	C₄⋊C₈	—	394	4	no	—
77	C₄×C₈	—	394	2	no	—
78	C₂²×C₄	—	394	1	no	—
79	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
80	M₁₆	—	394	2	no	—
81	M₁₆	—	394	2	no	—
82	C₂×C₈	—	394	2	no	—
83	C₂×C₈	—	394	1	no	—
84	C₂×C₈	—	394	1	no	—
85	C₂×C₈	—	197	2	no	—
86	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
88	C₂×C₈	D₇₈₈	1	2	yes	—
89	C₂×C₈	Dic₃₉₄	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	394	1	no	—
94	C₄²	—	197	2	no	—
95	C₂³	#15691357	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	394	4	no	—
98	C₂×C₄	—	394	2	no	—
99	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
104	C₂×C₄	#15691359	1	1	yes	characteristic
105	C₈	—	788	2	no	—
106	C₈	—	394	2	no	—
107	C₈	C₄⋊Dic₁₉₇	1	2	yes	—
108	C₈	C₄⋊Dic₁₉₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₉₇	1	1	yes	characteristic
111	C₂²	#20263306	1	1	yes	characteristic
112	C₂²	#20263317	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	394	4	no	—
115	C₄	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₉₇	1	1	yes	characteristic
117	C₄	#20263301	1	1	yes	characteristic
118	C₄	#20263325	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#27160978	1	1	yes	characteristic
122	C₂	#27161196	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263262

C₂

yes

maximal, characteristic

#27161106

C₂

yes

maximal

C₂×C₈×Dic₁₉₇

C₂

yes

maximal, characteristic

#27160838

C₂

yes

maximal

#427×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263262

—

C₄⋊#13096802

—

C₈×Dic₁₉₇

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

#117×C₁₉₇

C₂²

yes

—

#107×C₁₉₇

C₄

yes

complemented

#106×C₁₉₇

C₄

yes

complemented

C₂²×Dic₁₉₇

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

Q₈

yes

—

C₄⋊C₄×C₁₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₈×C₁₉₇

C₄⋊C₄

yes

—

C₈×C₁₉₇

C₄⋊C₄

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₄×D₄

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#125

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₇

C₄×Q₈

yes

characteristic

C₄×C₁₉₇

#121

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#389

yes

characteristic

C₂×C₁₉₇

#443

yes

—

C₁₉₇

#1945

yes

characteristic, complemented, Sylow

#1945

—

197

maximal, Sylow

C₂×#108

—

197

—

#427

D₁₉₇

yes

characteristic

#424

—

197

—

C₂×C₄×C₈

—

197

—

#339

—

197

—

C₂×M₁₆

—

197

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

#117

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

#108

—

394

—

#107

Dic₁₉₇

yes

complemented

#106

Dic₁₉₇

yes

complemented

C₄⋊C₈

—

394

—

C₄×C₈

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

C₂×Dic₁₉₇

yes

characteristic

C₂×C₈

C₂×Dic₁₉₇

yes

characteristic

C₂×C₈

D₇₈₈

yes

—

C₂×C₈

Dic₃₉₄

yes

—

C₄⋊C₄

C₂×Dic₁₉₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

394

—

C₄²

—

197

—

C₂³

#15691357

yes

characteristic

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

C₂²×Dic₁₉₇

yes

characteristic

100

C₂×C₄

D₄×D₁₉₇

yes

characteristic

101

C₂×C₄

C₂×D₇₈₈

yes

characteristic

102

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

103

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

104

C₂×C₄

#15691359

yes

characteristic

105

C₈

—

788

—

106

C₈

—

394

—

107

C₈

C₄⋊Dic₁₉₇

yes

—

108

C₈

C₄⋊Dic₁₉₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₉₇

yes

characteristic

111

C₂²

#20263306

yes

characteristic

112

C₂²

#20263317

yes

characteristic

113

C₄

—

114

C₄

—

394

—

115

C₄

C₂×C₄⋊Dic₁₉₇

yes

characteristic

116

C₄

Q₈×Dic₁₉₇

yes

characteristic

117

C₄

#20263301

yes

characteristic

118

C₄

#20263325

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#27160978

yes

characteristic

122

C₂

#27161196

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₇
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#427×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₄×D₄
prime radical	C₂³×C₁₉₇	#41
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#427	D₁₉₇
hypocenter	C₁₉₇	#1945
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₇

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#427×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₄×D₄

prime radical

C₂³×C₁₉₇

#41

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₉₇	C₄	semidirect
15. #106×C₁₉₇	C₄	semidirect
61. C₁₉₇	#1945	semidirect
74. #107	Dic₁₉₇	semidirect
75. #106	Dic₁₉₇	semidirect

H ≅ G/N

type

14. #107×C₁₉₇

C₄

semidirect

15. #106×C₁₉₇

C₄

semidirect

61. C₁₉₇

#1945

semidirect

74. #107

Dic₁₉₇

semidirect

75. #106

Dic₁₉₇

semidirect

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

subgroup	index
C₂×#20263262	2
#27161106	2
C₂×C₈×Dic₁₉₇	2
#27160838	2
#427×C₁₉₇	2
#1945	197

subgroup

index

C₂×C₈×Dic₁₉₇

#427×C₁₉₇

197

Subgroups — #38743461

Conjugacy classes of subgroups

Complements

Maximal subgroups