Group #38741857 — subgroups

Subgroups — #38741857

Order 25216.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27161458	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161369	C₂	1	2	yes	maximal, complemented
3	#27160828	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#13096802	C₂	1	2	yes	maximal, complemented
5	#555×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691348	—	2	2	no	—
7	#20263326	C₂²	1	1	yes	characteristic
8	#20263257	—	2	2	no	—
9	C₄⋊#13096802	C₂²	1	2	yes	—
10	#20263247	—	2	2	no	—
11	C₄×#13096802	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	#124×C₁₉₇	—	2	2	no	—
14	#121×C₁₉₇	—	2	2	no	—
15	C₄×D₄×C₁₉₇	C₂²	1	2	yes	—
16	#117×C₁₉₇	—	2	2	no	—
17	C₂×Dic₃₉₄	—	2	1	no	—
18	#15691348	—	4	4	no	—
19	C₂×#13096802	—	2	1	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₄⋊Dic₁₉₇	—	2	2	no	—
22	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
23	C₂×D₄×C₁₉₇	—	2	2	no	—
24	C₂×D₄×C₁₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₇	—	2	2	no	—
26	C₄⋊C₄×C₁₉₇	—	2	2	no	—
27	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
29	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
30	C₄²×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
32	Dic₃₉₄	—	4	2	no	—
33	C₂×Dic₁₉₇	—	4	1	no	—
34	#13096802	—	2	2	no	—
35	#13096802	—	4	2	no	—
36	C₂³×C₁₉₇	—	2	2	no	—
37	Q₈×C₁₉₇	—	4	2	no	—
38	D₄×C₁₉₇	—	2	2	no	—
39	D₄×C₁₉₇	—	2	2	no	—
40	D₄×C₁₉₇	—	8	2	no	—
41	C₂×C₄×C₁₉₇	—	2	2	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	—	4	2	no	—
44	C₂×C₄×C₁₉₇	—	4	2	no	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
47	C₂²×C₁₉₇	—	4	2	no	—
48	C₂²×C₁₉₇	—	4	2	no	—
49	C₂²×C₁₉₇	—	4	2	no	—
50	C₂²×C₁₉₇	#120	1	1	yes	characteristic
51	C₄×C₁₉₇	—	2	1	no	—
52	C₄×C₁₉₇	—	4	2	no	—
53	C₄×C₁₉₇	—	4	2	no	—
54	C₄×C₁₉₇	#135	1	2	yes	—
55	C₄×C₁₉₇	#137	1	2	yes	—
56	Dic₁₉₇	—	8	1	no	—
57	C₂×C₁₉₇	—	8	2	no	—
58	C₂×C₁₉₇	#452	1	2	yes	—
59	C₂×C₁₉₇	#453	1	1	yes	characteristic
60	C₁₉₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	197	1	no	maximal, Sylow
62	#555	D₁₉₇	1	1	yes	characteristic
63	#494	—	197	1	no	—
64	#476	—	197	2	no	—
65	#333	—	197	1	no	—
66	D₄⋊C₈	—	197	2	no	—
67	C₂×Q₁₆	—	394	2	no	—
68	C₄⋊Q₈	—	197	1	no	—
69	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₉₇	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	394	2	no	—
75	C₄⋊C₈	—	197	2	no	—
76	#102	—	394	2	no	—
77	C₄×C₈	—	197	1	no	—
78	C₂×Q₈	—	394	1	no	—
79	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₉₇	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	788	4	no	—
84	C₂×C₈	—	394	2	no	—
85	C₂×C₈	—	394	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	394	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	788	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	788	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
106	C₈	—	788	2	no	—
107	C₈	—	394	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#20263295	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1576	1	no	—
116	C₄	#20263346	1	2	yes	—
117	C₄	#20263356	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#27161241	1	2	yes	—
120	C₂	#27161247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161458

C₂

yes

maximal, characteristic, complemented

#27161369

C₂

yes

maximal, complemented

#27160828

C₂

yes

maximal, characteristic, complemented

D₄⋊#13096802

C₂

yes

maximal, complemented

#555×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691348

—

#20263326

C₂²

yes

characteristic

#20263257

—

C₄⋊#13096802

C₂²

yes

—

#20263247

—

C₄×#13096802

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

#124×C₁₉₇

—

#121×C₁₉₇

—

C₄×D₄×C₁₉₇

C₂²

yes

—

#117×C₁₉₇

—

C₂×Dic₃₉₄

—

#15691348

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₂²⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

—

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#135

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#452

yes

—

C₂×C₁₉₇

#453

yes

characteristic

C₁₉₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

197

maximal, Sylow

#555

D₁₉₇

yes

characteristic

#494

—

197

—

#476

—

197

—

#333

—

197

—

D₄⋊C₈

—

197

—

C₂×Q₁₆

—

394

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₉₇

yes

—

#117

—

#106

—

394

—

C₄⋊C₈

—

197

—

#102

—

394

—

C₄×C₈

—

197

—

C₂×Q₈

—

394

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

—

Q₁₆

—

788

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

788

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

788

—

104

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

106

C₈

—

788

—

107

C₈

—

394

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#20263295

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1576

—

116

C₄

#20263346

yes

—

117

C₄

#20263356

yes

—

118

C₂

—

119

C₂

#27161241

yes

—

120

C₂

#27161247

yes

characteristic

	subgroup	G/H
center	C₂²	#20263295
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting, prime radical	#555×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
hypercenter	#555	D₁₉₇
hypocenter	C₁₉₇	#1634
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#20263295

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting, prime radical

#555×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #27161458	C₂	semidirect
2. #27161369	C₂	semidirect
3. #27160828	C₂	semidirect
4. D₄⋊#13096802	C₂	semidirect
60. C₁₉₇	#1634	semidirect

H ≅ G/N

type

1. #27161458

C₂

semidirect

2. #27161369

C₂

semidirect

3. #27160828

C₂

semidirect

4. D₄⋊#13096802

C₂

semidirect

60. C₁₉₇

#1634

semidirect

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

subgroup	index
#27161458	2
#27161369	2
#27160828	2
D₄⋊#13096802	2
#555×C₁₉₇	2
#1634	197

subgroup

index

#555×C₁₉₇

197

Subgroups — #38741857

Conjugacy classes of subgroups

Complements

Maximal subgroups