Group #38741487 — subgroups

Subgroups — #38741487

Order 25216.

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

normal subgroups only

This group has 168 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	#27161103	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13096802	C₂	1	1	yes	maximal, characteristic
3	#27160832	C₂	1	4	yes	maximal, complemented
4	#431×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	—	2	1	no	—
6	C₂²×#13096802	—	2	1	no	—
7	C₄⋊#13096802	—	2	1	no	—
8	C₄⋊#13096802	—	2	1	no	—
9	C₄⋊#13096802	C₂²	1	2	yes	—
10	C₄⋊#13096802	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₉₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₉₇	C₄	1	2	yes	—
14	C₄×C₈×C₁₉₇	C₂²	1	2	yes	—
15	#15691342	—	4	2	no	—
16	C₂×#13096802	—	2	2	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	4	2	no	—
19	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
23	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₉₇	Q₈	1	1	yes	characteristic
26	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
27	#13096802	—	4	2	no	—
28	#13096802	—	4	2	no	—
29	C₂³×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₇	—	2	2	no	—
31	C₂×C₄×C₁₉₇	—	4	1	no	—
32	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	D₈	1	2	yes	—
36	C₂×C₄×C₁₉₇	Q₁₆	1	2	yes	—
37	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₉₇	—	2	4	no	—
41	C₂²×C₁₉₇	—	2	1	no	—
42	C₂²×C₁₉₇	—	2	1	no	—
43	C₂²×C₁₉₇	#107	1	2	yes	—
44	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₉₇	—	2	1	no	—
46	C₄×C₁₉₇	—	2	1	no	—
47	C₄×C₁₉₇	—	2	1	no	—
48	C₄×C₁₉₇	#107	1	2	yes	—
49	C₄×C₁₉₇	#131	1	2	yes	—
50	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
51	C₄×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₁₉₇	—	2	2	no	—
53	C₂×C₁₉₇	#424	1	1	yes	characteristic
54	C₂×C₁₉₇	C₂×#107	1	1	yes	characteristic
55	C₂×C₁₉₇	#430	1	1	yes	characteristic
56	C₁₉₇	#1571	1	1	yes	characteristic, complemented, Sylow
57	#1571	—	197	1	no	maximal, Sylow
58	#431	D₁₉₇	1	1	yes	characteristic
59	#423	—	197	1	no	—
60	C₂×C₄⋊C₈	—	197	1	no	—
61	#335	—	197	4	no	—
62	C₂×M₁₆	—	394	1	no	—
63	C₂²×C₈	—	394	1	no	—
64	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
65	C₄⋊C₈	—	394	1	no	—
66	C₄⋊C₈	—	394	1	no	—
67	C₄⋊C₈	—	197	2	no	—
68	C₄⋊C₈	—	197	2	no	—
69	C₄⋊C₈	Dic₁₉₇	1	2	yes	—
70	C₂²⋊C₈	Dic₁₉₇	1	2	yes	—
71	C₄×C₈	C₂×D₁₉₇	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
73	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
74	C₂²×C₄	Dic₃₉₄	1	1	yes	characteristic
75	M₁₆	—	788	2	no	—
76	C₂×C₈	—	788	2	no	—
77	C₂×C₈	—	394	2	no	—
78	C₂×C₈	—	394	2	no	—
79	C₂×C₈	C₂×Dic₁₉₇	1	4	yes	—
80	C₄²	C₂×Dic₁₉₇	1	1	yes	characteristic
81	C₄²	D₇₈₈	1	1	yes	characteristic
82	C₄²	Dic₃₉₄	1	1	yes	characteristic
83	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₉₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
90	C₂×C₄	D₁₅₇₆	1	2	yes	—
91	C₂×C₄	Dic₇₈₈	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	788	2	no	—
97	C₈	—	788	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#20263260	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#20263260	1	2	yes	—
106	C₄	#20263333	1	2	yes	—
107	C₄	C₂×D₁₅₇₆	1	1	yes	characteristic
108	C₄	C₂×Dic₇₈₈	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#27161106	1	1	yes	characteristic
111	C₂	C₂×#20263260	1	1	yes	characteristic
112	C₂	#27161128	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161103

C₂

yes

maximal, characteristic

C₂×C₄⋊#13096802

C₂

yes

maximal, characteristic

#27160832

C₂

yes

maximal, complemented

#431×C₁₉₇

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#13096802

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₄

yes

—

C₂²⋊C₈×C₁₉₇

C₄

yes

—

C₄×C₈×C₁₉₇

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

Q₈

yes

characteristic

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

Q₈

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

characteristic

#13096802

—

#13096802

—

C₂³×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

D₈

yes

—

C₂×C₄×C₁₉₇

Q₁₆

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#107

yes

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#107

yes

—

C₄×C₁₉₇

#131

yes

—

C₄×C₁₉₇

C₂×D₈

yes

characteristic

C₄×C₁₉₇

C₂×Q₁₆

yes

characteristic

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1571

—

197

maximal, Sylow

#431

D₁₉₇

yes

characteristic

#423

—

197

—

C₂×C₄⋊C₈

—

197

—

#335

—

197

—

C₂×M₁₆

—

394

—

C₂²×C₈

—

394

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

C₄⋊C₈

—

394

—

C₄⋊C₈

—

197

—

C₄⋊C₈

—

197

—

C₄⋊C₈

Dic₁₉₇

yes

—

C₂²⋊C₈

Dic₁₉₇

yes

—

C₄×C₈

C₂×D₁₉₇

yes

—

C₂²×C₄

C₂×Dic₁₉₇

yes

characteristic

C₂²×C₄

D₇₈₈

yes

characteristic

C₂²×C₄

Dic₃₉₄

yes

characteristic

M₁₆

—

788

—

C₂×C₈

—

788

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

C₂×Dic₁₉₇

yes

—

C₄²

C₂×Dic₁₉₇

yes

characteristic

C₄²

D₇₈₈

yes

characteristic

C₄²

Dic₃₉₄

yes

characteristic

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

C₄⋊Dic₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₉₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₇

yes

characteristic

C₂×C₄

D₁₅₇₆

yes

—

C₂×C₄

Dic₇₈₈

yes

—

C₂×C₄

C₂²×Dic₁₉₇

yes

characteristic

C₂×C₄

C₂×D₇₈₈

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

C₈

—

C₈

—

788

—

C₈

—

788

—

C₂²

—

C₂²

—

100

C₂²

#20263260

yes

—

101

C₂²

C₂×C₄⋊Dic₁₉₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#20263260

yes

—

106

C₄

#20263333

yes

—

107

C₄

C₂×D₁₅₇₆

yes

characteristic

108

C₄

C₂×Dic₇₈₈

yes

characteristic

109

C₂

—

110

C₂

#27161106

yes

characteristic

111

C₂

C₂×#20263260

yes

characteristic

112

C₂

#27161128

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₈₈
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#431×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₇	C₄⋊C₄
cosocle	C₄²×C₁₉₇	C₂³
hypercenter	#431	D₁₉₇
hypocenter	C₁₉₇	#1571
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₇₈₈

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#431×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₇

C₄⋊C₄

cosocle

C₄²×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #27160832	C₂	semidirect
56. C₁₉₇	#1571	semidirect

H ≅ G/N

type

3. #27160832

C₂

semidirect

56. C₁₉₇

#1571

semidirect

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

subgroup	index
#27161103	2
C₂×C₄⋊#13096802	2
#27160832	2
#431×C₁₉₇	2
#1571	197

subgroup

index

#431×C₁₉₇

197

Subgroups — #38741487

Conjugacy classes of subgroups

Complements

Maximal subgroups