Group #38741771 — subgroups

Subgroups — #38741771

Order 25216.

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

normal subgroups only

This group has 240 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	#27161455	C₂	1	1	yes	maximal, characteristic
2	#27161279	C₂	1	2	yes	maximal, complemented
3	#27160823	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#13096802	C₂	1	2	yes	maximal, complemented
5	#559×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691345	—	2	2	no	—
7	C₂×#15691343	—	2	2	no	—
8	#20263323	C₂²	1	1	yes	characteristic
9	C₄⋊#13096802	C₂²	1	2	yes	—
10	#20263246	—	2	2	no	—
11	C₄×#13096802	C₂²	1	1	yes	characteristic
12	#127×C₁₉₇	C₂²	1	1	yes	characteristic
13	#124×C₁₉₇	—	2	2	no	—
14	#123×C₁₉₇	—	2	2	no	—
15	C₄×Q₈×C₁₉₇	C₂²	1	2	yes	—
16	C₄×D₄×C₁₉₇	—	2	2	no	—
17	C₂×D₇₈₈	—	2	2	no	—
18	C₂×D₇₈₈	—	2	1	no	—
19	#15691345	—	4	4	no	—
20	#15691343	—	4	4	no	—
21	C₂×#13096802	—	2	1	no	—
22	C₂×#13096802	—	2	2	no	—
23	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
24	C₂×D₄×C₁₉₇	—	2	2	no	—
25	C₂×D₄×C₁₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉₇	—	2	2	no	—
27	C₄⋊C₄×C₁₉₇	—	2	2	no	—
28	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
30	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	—	2	2	no	—
32	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₇	—	4	1	no	—
34	D₇₈₈	—	4	2	no	—
35	D₇₈₈	—	4	2	no	—
36	D₇₈₈	—	4	2	no	—
37	#13096802	—	2	2	no	—
38	#13096802	—	4	2	no	—
39	C₂³×C₁₉₇	—	2	2	no	—
40	Q₈×C₁₉₇	—	2	2	no	—
41	Q₈×C₁₉₇	—	2	2	no	—
42	D₄×C₁₉₇	—	4	2	no	—
43	D₄×C₁₉₇	—	4	2	no	—
44	D₄×C₁₉₇	—	4	2	no	—
45	C₂×C₄×C₁₉₇	—	2	2	no	—
46	C₂×C₄×C₁₉₇	—	2	2	no	—
47	C₂×C₄×C₁₉₇	—	8	2	no	—
48	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
50	C₂²×C₁₉₇	—	4	4	no	—
51	C₂²×C₁₉₇	—	4	2	no	—
52	C₂²×C₁₉₇	#120	1	1	yes	characteristic
53	C₂×D₁₉₇	—	8	2	no	—
54	C₂×D₁₉₇	—	8	1	no	—
55	C₄×C₁₉₇	—	2	1	no	—
56	C₄×C₁₉₇	—	4	2	no	—
57	C₄×C₁₉₇	—	4	2	no	—
58	C₄×C₁₉₇	#135	1	2	yes	—
59	C₄×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₉₇	—	8	2	no	—
61	C₂×C₁₉₇	#449	1	2	yes	—
62	C₂×C₁₉₇	#453	1	1	yes	characteristic
63	D₁₉₇	—	16	1	no	—
64	C₁₉₇	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	197	1	no	maximal, Sylow
66	#559	D₁₉₇	1	1	yes	characteristic, complemented
67	#493	—	197	1	no	—
68	#460	—	197	2	no	—
69	#331	—	197	1	no	—
70	Q₈⋊C₈	—	197	2	no	—
71	C₂×SD₁₆	—	394	2	no	—
72	C₂×D₈	—	394	2	no	—
73	#127	—	197	1	no	—
74	#127	C₂×D₁₉₇	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₉₇	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	197	2	no	—
80	#102	—	394	2	no	—
81	C₄×C₈	—	197	1	no	—
82	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	394	2	no	—
85	C₂×D₄	—	394	1	no	—
86	C₂×D₄	C₂²⋊D₁₉₇	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	788	4	no	—
89	D₈	—	788	4	no	—
90	C₂×C₈	—	394	2	no	—
91	C₂×C₈	—	394	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	788	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	788	2	no	—
106	D₄	—	788	2	no	—
107	D₄	—	788	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
113	C₈	—	788	2	no	—
114	C₈	—	394	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	1576	2	no	—
118	C₂²	—	1576	1	no	—
119	C₂²	#20263295	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#20263346	1	2	yes	—
124	C₄	#20263351	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	3152	1	no	—
127	C₂	#27161224	1	2	yes	—
128	C₂	#27161247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161455

C₂

yes

maximal, characteristic

#27161279

C₂

yes

maximal, complemented

#27160823

C₂

yes

maximal, characteristic, complemented

Q₈⋊#13096802

C₂

yes

maximal, complemented

#559×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691345

—

C₂×#15691343

—

#20263323

C₂²

yes

characteristic

C₄⋊#13096802

C₂²

yes

—

#20263246

—

C₄×#13096802

C₂²

yes

characteristic

#127×C₁₉₇

C₂²

yes

characteristic

#124×C₁₉₇

—

#123×C₁₉₇

—

C₄×Q₈×C₁₉₇

C₂²

yes

—

C₄×D₄×C₁₉₇

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

—

—

—

—

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

C₂²×D₁₉₇

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×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

#120

yes

characteristic

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#135

yes

—

C₄×C₁₉₇

AGL(1,ℤ/8)

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#449

yes

—

C₂×C₁₉₇

#453

yes

characteristic

D₁₉₇

—

C₁₉₇

#1620

yes

characteristic, complemented, Sylow

#1620

—

197

maximal, Sylow

#559

D₁₉₇

yes

characteristic, complemented

#493

—

197

—

#460

—

197

—

#331

—

197

—

Q₈⋊C₈

—

197

—

C₂×SD₁₆

—

394

—

C₂×D₈

—

394

—

#127

—

197

—

#127

C₂×D₁₉₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₉₇

yes

—

C₄×D₄

—

C₄⋊C₈

—

197

—

#102

—

394

—

C₄×C₈

—

197

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

C₂×D₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

—

SD₁₆

—

788

—

D₈

—

788

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

C₂³

—

788

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

788

—

106

D₄

—

788

—

107

D₄

—

788

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

113

C₈

—

788

—

114

C₈

—

394

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

1576

—

118

C₂²

—

1576

—

119

C₂²

#20263295

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#20263346

yes

—

124

C₄

#20263351

yes

—

125

C₂

—

126

C₂

—

3152

—

127

C₂

#27161224

yes

—

128

C₂

#27161247

yes

characteristic

	subgroup	G/H
center	C₂²	#20263295
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#559×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	#27161455	C₂
hypercenter	#559	D₁₉₇
hypocenter	C₁₉₇	#1620
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#20263295

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#559×C₁₉₇

C₂

C₂²×C₁₉₇

C₂

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #27161279	C₂	semidirect
3. #27160823	C₂	semidirect
4. Q₈⋊#13096802	C₂	semidirect
5. #559×C₁₉₇	C₂	semidirect
64. C₁₉₇	#1620	semidirect
66. #559	D₁₉₇	semidirect

H ≅ G/N

type

2. #27161279

C₂

semidirect

3. #27160823

C₂

semidirect

4. Q₈⋊#13096802

C₂

semidirect

5. #559×C₁₉₇

C₂

semidirect

64. C₁₉₇

#1620

semidirect

66. #559

D₁₉₇

semidirect

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

subgroup	index
#27161455	2
#27161279	2
#27160823	2
Q₈⋊#13096802	2
#559×C₁₉₇	2
#1620	197

subgroup

index

#559×C₁₉₇

197

Subgroups — #38741771

Conjugacy classes of subgroups

Complements

Maximal subgroups