Group #38743706 — subgroups

Subgroups — #38743706

Order 25216.

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

normal subgroups only

This group has 228 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	C₄×#15691342	C₂	1	1	yes	maximal, characteristic
3	#27160848	C₂	1	4	yes	maximal
4	C₂×#125×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	C₂²	1	2	yes	—
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₄⋊#13096802	—	2	2	no	—
8	#20263237	—	2	1	no	—
9	C₄×#13096802	—	2	1	no	—
10	#125×C₁₉₇	—	2	1	no	—
11	#125×C₁₉₇	—	2	1	no	—
12	#125×C₁₉₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₇	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
16	#15691342	—	2	2	no	—
17	#15691342	—	2	4	no	—
18	#15691342	Q₈	1	4	yes	—
19	C₂×#13096802	—	2	2	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₇	—	2	4	no	—
25	C₄⋊C₄×C₁₉₇	—	2	4	no	—
26	C₄⋊C₄×C₁₉₇	—	2	2	no	—
27	C₄⋊C₄×C₁₉₇	—	2	2	no	—
28	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₉₇	D₄	1	2	yes	—
31	#13096802	—	2	4	no	—
32	#13096802	—	4	2	no	—
33	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	—	2	4	no	—
35	C₂×C₄×C₁₉₇	—	4	2	no	—
36	C₂×C₄×C₁₉₇	—	4	4	no	—
37	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₉₇	#41	1	2	yes	—
43	C₂²×C₁₉₇	—	2	2	no	—
44	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₉₇	—	2	2	no	—
47	C₄×C₁₉₇	—	2	2	no	—
48	C₄×C₁₉₇	—	4	4	no	—
49	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₉₇	#124	1	1	yes	characteristic
51	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₁₉₇	—	2	2	no	—
53	C₂×C₁₉₇	#391	1	1	yes	characteristic
54	C₂×C₁₉₇	#413	1	2	yes	—
55	C₁₉₇	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	197	1	no	maximal, Sylow
57	C₂×#125	D₁₉₇	1	1	yes	characteristic
58	#423	—	197	1	no	—
59	C₄×M₁₆	—	197	1	no	—
60	#343	—	197	4	no	—
61	C₂×M₁₆	—	197	2	no	—
62	C₂×M₁₆	—	197	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₁₉₇	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
68	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
69	C₄⋊C₈	—	394	2	no	—
70	#97	—	394	1	no	—
71	C₄×C₈	—	394	1	no	—
72	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
73	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
75	M₁₆	—	394	4	no	—
76	M₁₆	—	394	2	no	—
77	M₁₆	—	197	4	no	—
78	C₂×C₈	—	394	2	no	—
79	C₂×C₈	—	394	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
87	C₂³	C₂²×Dic₁₉₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
96	C₂×C₄	#15691359	1	2	yes	—
97	C₈	—	788	2	no	—
98	C₈	—	394	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
101	C₂²	Q₈×Dic₁₉₇	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#20263302	1	2	yes	—
106	C₄	#20263315	1	1	yes	characteristic
107	C₄	#20263324	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#27160986	1	1	yes	characteristic
110	C₂	#27161063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161103

C₂

yes

maximal, characteristic

C₄×#15691342

C₂

yes

maximal, characteristic

#27160848

C₂

yes

maximal

C₂×#125×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂×#15691342

C₂²

yes

—

C₄⋊#13096802

—

#20263237

—

C₄×#13096802

—

#125×C₁₉₇

—

#125×C₁₉₇

—

#125×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

—

—

Q₈

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

#124

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

C₂×C₁₉₇

—

C₂×C₁₉₇

#391

yes

characteristic

C₂×C₁₉₇

#413

yes

—

C₁₉₇

#1998

yes

characteristic, complemented, Sylow

#1998

—

197

maximal, Sylow

C₂×#125

D₁₉₇

yes

characteristic

#423

—

197

—

C₄×M₁₆

—

197

—

#343

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

—

197

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₁₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

#97

—

394

—

C₄×C₈

—

394

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

M₁₆

—

394

—

M₁₆

—

197

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

—

C₂³

C₂²×Dic₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

Q₈×D₁₉₇

yes

—

C₂×C₄

#15691359

yes

—

C₈

—

788

—

C₈

—

394

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₁₉₇

yes

characteristic

101

C₂²

Q₈×Dic₁₉₇

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#20263302

yes

—

106

C₄

#20263315

yes

characteristic

107

C₄

#20263324

yes

characteristic

108

C₂

—

109

C₂

#27160986

yes

characteristic

110

C₂

#27161063

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₇
commutator, prime radical	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#125×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#125	D₁₉₇
hypocenter	C₁₉₇	#1998
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₇

commutator, prime radical

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#125×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
55. C₁₉₇	#1998	semidirect

H ≅ G/N

type

55. C₁₉₇

#1998

semidirect

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

subgroup	index
#27161103	2
C₄×#15691342	2
#27160848	2
C₂×#125×C₁₉₇	2
#1998	197

subgroup

index

C₂×#125×C₁₉₇

197

Subgroups — #38743706

Conjugacy classes of subgroups

Complements

Maximal subgroups