Group #38742389 — subgroups

Subgroups — #38742389

Order 25216.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27161484	C₂	1	1	yes	maximal, characteristic
2	#27161484	C₂	1	1	yes	maximal, characteristic
3	#27161477	C₂	1	1	yes	maximal, characteristic
4	#27161459	C₂	1	1	yes	maximal, characteristic
5	#27161430	C₂	1	1	yes	maximal, characteristic
6	#27161430	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₈₈	—	2	2	no	—
9	#20263326	C₂²	1	1	yes	characteristic
10	#20263326	C₂²	1	1	yes	characteristic
11	#20263326	C₂²	1	1	yes	characteristic
12	#20263326	C₂²	1	1	yes	characteristic
13	#20263260	—	2	2	no	—
14	#20263260	—	2	2	no	—
15	#20263258	—	2	2	no	—
16	#20263251	—	2	2	no	—
17	#20263251	—	2	2	no	—
18	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₉₄	—	2	1	no	—
22	C₂×Dic₃₉₄	—	2	1	no	—
23	C₂×Dic₃₉₄	—	2	1	no	—
24	C₂×Dic₃₉₄	—	2	1	no	—
25	Dic₇₈₈	—	4	4	no	—
26	C₄⋊Dic₁₉₇	—	2	1	no	—
27	C₄⋊Dic₁₉₇	—	2	1	no	—
28	C₄⋊Dic₁₉₇	—	2	1	no	—
29	C₄⋊Dic₁₉₇	—	2	1	no	—
30	C₄⋊Dic₁₉₇	—	2	1	no	—
31	C₄⋊Dic₁₉₇	—	2	1	no	—
32	C₄⋊Dic₁₉₇	—	2	1	no	—
33	C₄⋊Dic₁₉₇	—	2	1	no	—
34	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
35	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
36	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
37	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
38	Dic₃₉₄	—	4	2	no	—
39	Dic₃₉₄	—	4	2	no	—
40	Dic₃₉₄	—	4	2	no	—
41	Dic₃₉₄	—	4	2	no	—
42	C₂×Dic₁₉₇	—	4	1	no	—
43	C₂×Dic₁₉₇	—	4	1	no	—
44	C₂×Dic₁₉₇	—	4	1	no	—
45	C₂×Dic₁₉₇	—	4	1	no	—
46	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₇	—	2	2	no	—
50	C₈×C₁₉₇	—	2	2	no	—
51	C₈×C₁₉₇	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₉₇	#127	1	1	yes	characteristic
53	C₄×C₁₉₇	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₉₇	#137	1	2	yes	—
56	Dic₁₉₇	—	8	1	no	—
57	Dic₁₉₇	—	8	1	no	—
58	Dic₁₉₇	—	8	1	no	—
59	Dic₁₉₇	—	8	1	no	—
60	C₂×C₁₉₇	#494	1	1	yes	characteristic
61	C₂×C₁₉₇	#496	1	1	yes	characteristic
62	C₂×C₁₉₇	#497	1	1	yes	characteristic
63	C₁₉₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	197	1	no	maximal, Sylow
65	#500	—	197	1	no	—
66	#500	—	197	1	no	—
67	#498	—	197	1	no	—
68	#494	—	197	1	no	—
69	#487	—	197	1	no	—
70	#487	—	197	1	no	—
71	#322	D₁₉₇	1	1	yes	characteristic
72	C₂×Q₁₆	—	394	2	no	—
73	C₄⋊Q₈	—	197	1	no	—
74	C₄⋊Q₈	—	197	1	no	—
75	C₄⋊Q₈	—	197	1	no	—
76	C₄⋊Q₈	—	197	1	no	—
77	#107	—	394	2	no	—
78	#107	—	394	2	no	—
79	#106	—	394	2	no	—
80	#103	—	394	2	no	—
81	#103	—	394	2	no	—
82	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
85	C₂×Q₈	—	394	1	no	—
86	C₂×Q₈	—	394	1	no	—
87	C₂×Q₈	—	394	1	no	—
88	C₂×Q₈	—	394	1	no	—
89	Q₁₆	—	788	4	no	—
90	C₂×C₈	D₇₈₈	1	2	yes	—
91	C₂×C₈	D₇₈₈	1	2	yes	—
92	C₂×C₈	D₇₈₈	1	2	yes	—
93	C₄⋊C₄	—	394	1	no	—
94	C₄⋊C₄	—	394	1	no	—
95	C₄⋊C₄	—	394	1	no	—
96	C₄⋊C₄	—	394	1	no	—
97	C₄⋊C₄	—	394	1	no	—
98	C₄⋊C₄	—	394	1	no	—
99	C₄⋊C₄	—	394	1	no	—
100	C₄⋊C₄	—	394	1	no	—
101	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
102	Q₈	—	788	2	no	—
103	Q₈	—	788	2	no	—
104	Q₈	—	788	2	no	—
105	Q₈	—	788	2	no	—
106	C₂×C₄	—	788	1	no	—
107	C₂×C₄	—	788	1	no	—
108	C₂×C₄	—	788	1	no	—
109	C₂×C₄	—	788	1	no	—
110	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
111	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
112	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₇₈₈	1	4	yes	complemented
116	C₂²	#20263323	1	1	yes	characteristic
117	C₄	—	1576	1	no	—
118	C₄	—	1576	1	no	—
119	C₄	—	1576	1	no	—
120	C₄	—	1576	1	no	—
121	C₄	C₂×Dic₇₈₈	1	2	yes	—
122	C₄	#20263354	1	2	yes	—
123	C₄	#20263359	1	2	yes	—
124	C₂	#27161459	1	1	yes	characteristic
125	C₂	#27161468	1	1	yes	characteristic
126	C₂	#27161473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161484

C₂

yes

maximal, characteristic

#27161484

C₂

yes

maximal, characteristic

#27161477

C₂

yes

maximal, characteristic

#27161459

C₂

yes

maximal, characteristic

#27161430

C₂

yes

maximal, characteristic

#27161430

C₂

yes

maximal, characteristic

#322×C₁₉₇

C₂

yes

maximal, characteristic

C₂×Dic₇₈₈

—

#20263326

C₂²

yes

characteristic

#20263326

C₂²

yes

characteristic

#20263326

C₂²

yes

characteristic

#20263326

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

Dic₇₈₈

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₈×C₁₉₇

Q₁₆

yes

complemented

C₂²×C₁₉₇

#127

yes

characteristic

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1723

—

197

maximal, Sylow

#500

—

197

—

#500

—

197

—

#498

—

197

—

#494

—

197

—

#487

—

197

—

#487

—

197

—

#322

D₁₉₇

yes

characteristic

C₂×Q₁₆

—

394

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

—

197

—

#107

—

394

—

#107

—

394

—

#106

—

394

—

#103

—

394

—

#103

—

394

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

Q₁₆

—

788

—

C₂×C₈

D₇₈₈

yes

—

C₂×C₈

D₇₈₈

yes

—

C₂×C₈

D₇₈₈

yes

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

100

C₄⋊C₄

—

394

—

101

C₄²

C₂²×D₁₉₇

yes

characteristic

102

Q₈

—

788

—

103

Q₈

—

788

—

104

Q₈

—

788

—

105

Q₈

—

788

—

106

C₂×C₄

—

788

—

107

C₂×C₄

—

788

—

108

C₂×C₄

—

788

—

109

C₂×C₄

—

788

—

110

C₂×C₄

C₂×D₇₈₈

yes

characteristic

111

C₂×C₄

C₂×D₇₈₈

yes

characteristic

112

C₂×C₄

C₂×D₇₈₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₇₈₈

yes

complemented

116

C₂²

#20263323

yes

characteristic

117

C₄

—

1576

—

118

C₄

—

1576

—

119

C₄

—

1576

—

120

C₄

—

1576

—

121

C₄

C₂×Dic₇₈₈

yes

—

122

C₄

#20263354

yes

—

123

C₄

#20263359

yes

—

124

C₂

#27161459

yes

characteristic

125

C₂

#27161468

yes

characteristic

126

C₂

#27161473

yes

characteristic

	subgroup	G/H
center	C₂²	#20263323
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#322×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#127
hypercenter	#322	D₁₉₇
hypocenter	C₁₉₇	#1723
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#20263323

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#322×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₉₇	Q₁₆	semidirect
63. C₁₉₇	#1723	semidirect
115. C₈	Dic₇₈₈	semidirect

H ≅ G/N

type

51. C₈×C₁₉₇

Q₁₆

semidirect

63. C₁₉₇

#1723

semidirect

115. C₈

Dic₇₈₈

semidirect

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

subgroup	index
#27161484	2
#27161477	2
#27161459	2
#27161430	2
#322×C₁₉₇	2
#1723	197

subgroup

index

#322×C₁₉₇

197

Subgroups — #38742389

Conjugacy classes of subgroups

Complements

Maximal subgroups