Group #38742347 — subgroups

Subgroups — #38742347

Order 25216.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#27161477	C₂	1	1	yes	maximal, characteristic
2	#27161475	C₂	1	1	yes	maximal, characteristic
3	#27161475	C₂	1	1	yes	maximal, characteristic
4	#27161458	C₂	1	1	yes	maximal, characteristic
5	#27161431	C₂	1	1	yes	maximal, characteristic
6	#27160801	C₂	1	1	yes	maximal, characteristic
7	#487×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#15691348	—	2	2	no	—
9	#20263326	C₂²	1	1	yes	characteristic
10	#20263326	C₂²	1	1	yes	characteristic
11	#20263258	—	2	2	no	—
12	#20263257	—	2	2	no	—
13	#20263257	—	2	2	no	—
14	#20263249	—	2	2	no	—
15	C₄×#13096802	C₂²	1	1	yes	characteristic
16	C₄×#13096802	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
19	#103×C₁₉₇	—	2	2	no	—
20	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₉₄	—	2	1	no	—
22	C₂×Dic₃₉₄	—	2	1	no	—
23	#15691348	—	4	4	no	—
24	C₂×#13096802	D₄	1	2	yes	—
25	C₂×#13096802	D₄	1	2	yes	—
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	C₂×Q₈×C₁₉₇	—	2	1	no	—
31	C₂×Q₈×C₁₉₇	—	2	1	no	—
32	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₉₇	—	2	1	no	—
34	C₄⋊C₄×C₁₉₇	—	2	1	no	—
35	C₄⋊C₄×C₁₉₇	—	2	1	no	—
36	C₄⋊C₄×C₁₉₇	—	2	1	no	—
37	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
38	Dic₃₉₄	—	4	2	no	—
39	Dic₃₉₄	—	4	2	no	—
40	C₂×Dic₁₉₇	—	4	1	no	—
41	C₂×Dic₁₉₇	—	4	1	no	—
42	#13096802	—	2	2	no	—
43	#13096802	SD₁₆	1	4	yes	—
44	Q₈×C₁₉₇	—	4	2	no	—
45	Q₈×C₁₉₇	—	4	2	no	—
46	C₂×C₄×C₁₉₇	—	4	1	no	—
47	C₂×C₄×C₁₉₇	—	4	1	no	—
48	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₉₇	—	2	2	no	—
52	C₂²×C₁₉₇	#127	1	1	yes	characteristic
53	C₄×C₁₉₇	—	8	1	no	—
54	C₄×C₁₉₇	—	8	1	no	—
55	C₄×C₁₉₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₉₇	#137	1	2	yes	—
57	C₄×C₁₉₇	#137	1	2	yes	—
58	Dic₁₉₇	—	8	1	no	—
59	Dic₁₉₇	—	8	1	no	—
60	C₂×C₁₉₇	#492	1	1	yes	characteristic
61	C₂×C₁₉₇	#497	1	1	yes	characteristic
62	C₂×C₁₉₇	#497	1	1	yes	characteristic
63	C₁₉₇	#1716	1	1	yes	characteristic, complemented, Sylow
64	#1716	—	197	1	no	maximal, Sylow
65	#498	—	197	1	no	—
66	#498	—	197	1	no	—
67	#498	—	197	1	no	—
68	#494	—	197	1	no	—
69	#487	—	197	1	no	—
70	#487	D₁₉₇	1	1	yes	characteristic
71	#322	—	197	1	no	—
72	C₂×Q₁₆	—	394	2	no	—
73	C₄⋊Q₈	—	197	1	no	—
74	C₄⋊Q₈	—	197	1	no	—
75	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
77	#106	—	394	2	no	—
78	#106	—	394	2	no	—
79	#106	—	394	2	no	—
80	#103	—	2	2	no	—
81	#103	—	394	2	no	—
82	C₄×C₈	—	197	1	no	—
83	C₄×C₈	—	197	1	no	—
84	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	394	1	no	—
88	C₂×Q₈	—	394	1	no	—
89	Q₁₆	—	788	4	no	—
90	C₂×C₈	—	197	2	no	—
91	C₂×C₈	—	197	2	no	—
92	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	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	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	788	2	no	—
105	Q₈	—	788	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	788	1	no	—
109	C₂×C₄	—	788	1	no	—
110	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	394	2	no	—
115	C₈	—	197	4	no	—
116	C₂²	#20263322	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1576	1	no	—
120	C₄	—	1576	1	no	—
121	C₄	SD₁₆×D₁₉₇	1	2	yes	—
122	C₄	#20263356	1	2	yes	—
123	C₄	#20263358	1	2	yes	—
124	C₂	#27161450	1	1	yes	characteristic
125	C₂	#27161472	1	1	yes	characteristic
126	C₂	#27161474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161477

C₂

yes

maximal, characteristic

#27161475

C₂

yes

maximal, characteristic

#27161475

C₂

yes

maximal, characteristic

#27161458

C₂

yes

maximal, characteristic

#27161431

C₂

yes

maximal, characteristic

#27160801

C₂

yes

maximal, characteristic

#487×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691348

—

#20263326

C₂²

yes

characteristic

#20263326

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#13096802

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

#103×C₁₉₇

—

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

#15691348

—

C₂×#13096802

D₄

yes

—

C₂×#13096802

D₄

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

SD₁₆

yes

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

#127

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×SD₁₆

yes

—

C₄×C₁₉₇

#137

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1716

—

197

maximal, Sylow

#498

—

197

—

#498

—

197

—

#498

—

197

—

#494

—

197

—

#487

—

197

—

#487

D₁₉₇

yes

characteristic

#322

—

197

—

C₂×Q₁₆

—

394

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#106

—

394

—

#106

—

394

—

#106

—

394

—

#103

—

#103

—

394

—

C₄×C₈

—

197

—

C₄×C₈

—

197

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

Q₁₆

—

788

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

C₂²⋊D₁₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

100

C₄⋊C₄

—

394

—

101

C₄²

C₂²×D₁₉₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

788

—

105

Q₈

—

788

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

788

—

109

C₂×C₄

—

788

—

110

C₂×C₄

D₄×D₁₉₇

yes

characteristic

111

C₂×C₄

D₄×D₁₉₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

113

C₈

—

114

C₈

—

394

—

115

C₈

—

197

—

116

C₂²

#20263322

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1576

—

120

C₄

—

1576

—

121

C₄

SD₁₆×D₁₉₇

yes

—

122

C₄

#20263356

yes

—

123

C₄

#20263358

yes

—

124

C₂

#27161450

yes

characteristic

125

C₂

#27161472

yes

characteristic

126

C₂

#27161474

yes

characteristic

	subgroup	G/H
center	C₂²	#20263322
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#487×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#127
hypercenter	#487	D₁₉₇
hypocenter	C₁₉₇	#1716
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#20263322

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#487×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
63. C₁₉₇	#1716	semidirect

H ≅ G/N

type

63. C₁₉₇

#1716

semidirect

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

subgroup	index
#27161477	2
#27161475	2
#27161458	2
#27161431	2
#27160801	2
#487×C₁₉₇	2
#1716	197

subgroup

index

#487×C₁₉₇

197

Subgroups — #38742347

Conjugacy classes of subgroups

Complements

Maximal subgroups