Group #27143001 — subgroups

Subgroups — #27143001

Order 12416.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20260761	C₂	1	1	yes	maximal, characteristic
2	#20260744	C₂	1	2	yes	maximal
3	C₂×#15690729	C₂	1	2	yes	maximal
4	#20260500	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	1	yes	characteristic
7	C₂×#13095495	C₂²	1	2	yes	—
8	#15690769	C₂²	1	1	yes	characteristic
9	#15690739	—	2	2	no	—
10	#15690734	—	2	4	no	—
11	#15690729	—	2	4	no	—
12	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₉₇	—	2	4	no	—
14	C₂×Q₁₆×C₉₇	—	2	2	no	—
15	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	#13095495	—	2	2	no	—
17	#13095495	—	2	2	no	—
18	#13095495	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	1	no	—
21	C₄⋊Dic₉₇	D₄	1	2	yes	—
22	C₂²⋊Dic₉₇	—	2	1	no	—
23	C₄×Dic₉₇	—	2	1	no	—
24	C₂×Q₈×C₉₇	—	2	2	no	—
25	C₂×Q₈×C₉₇	—	2	2	no	—
26	C₂×Q₈×C₉₇	—	2	2	no	—
27	C₂×Q₈×C₉₇	—	2	2	no	—
28	C₂×Q₈×C₉₇	—	2	2	no	—
29	C₂×Q₈×C₉₇	D₄	1	4	yes	—
30	C₂²×C₄×C₉₇	—	2	2	no	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	Q₁₆×C₉₇	—	2	8	no	—
33	C₂×C₈×C₉₇	—	2	2	no	—
34	C₂×C₈×C₉₇	D₄	1	2	yes	—
35	C₂³×C₉₇	#41	1	1	yes	characteristic
36	Q₈×C₉₇	—	2	4	no	—
37	Q₈×C₉₇	—	2	4	no	—
38	Q₈×C₉₇	—	4	2	no	—
39	Q₈×C₉₇	—	4	2	no	—
40	C₂×C₄×C₉₇	—	2	4	no	—
41	C₂×C₄×C₉₇	—	4	2	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂×Dic₉₇	—	2	2	no	—
48	C₈×C₉₇	—	2	2	no	—
49	#11715895	—	4	1	no	—
50	#11715895	—	4	2	no	—
51	C₂²×C₉₇	—	2	2	no	—
52	C₂²×C₉₇	#121	1	1	yes	characteristic
53	C₂²×C₉₇	#123	1	2	yes	—
54	C₄×C₉₇	—	4	4	no	—
55	C₄×C₉₇	#120	1	1	yes	characteristic
56	C₄×C₉₇	#121	1	2	yes	—
57	C₄×C₉₇	#124	1	1	yes	characteristic
58	Dic₉₇	—	4	2	no	—
59	C₂×C₉₇	—	2	2	no	—
60	C₂×C₉₇	#394	1	1	yes	characteristic
61	C₂×C₉₇	#473	1	2	yes	—
62	#2047	—	97	1	no	maximal, Sylow
63	C₉₇	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	D₉₇	1	1	yes	characteristic
65	#424	—	97	1	no	—
66	#419	—	97	2	no	—
67	C₂×#101	—	97	2	no	—
68	#341	—	97	1	no	—
69	C₂²×Q₈	C₂×D₉₇	1	2	yes	—
70	C₂×Q₁₆	—	2	4	no	—
71	C₂×Q₁₆	—	2	2	no	—
72	C₂×M₁₆	—	97	2	no	—
73	C₂×M₁₆	—	97	1	no	—
74	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
75	#117	—	97	1	no	—
76	C₄⋊C₈	—	194	2	no	—
77	#103	—	194	4	no	—
78	#101	—	194	4	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₂²⋊D₉₇	1	4	yes	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
87	Q₁₆	—	2	8	no	—
88	M₁₆	—	194	2	no	—
89	M₁₆	—	194	2	no	—
90	M₁₆	—	194	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	194	2	no	—
93	C₂×C₈	—	194	1	no	—
94	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
95	C₄⋊C₄	—	97	2	no	—
96	C₂²⋊C₄	—	194	1	no	—
97	C₄²	—	194	1	no	—
98	C₂³	#13095510	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	194	2	no	—
107	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
110	C₂×C₄	#13095510	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	388	1	no	—
113	C₈	—	388	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#15690781	1	1	yes	characteristic
116	C₂²	#15690791	1	2	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	388	2	no	—
119	C₄	#15690779	1	1	yes	characteristic
120	C₄	#15690782	1	2	yes	—
121	C₄	#15690797	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#20260649	1	1	yes	characteristic
124	C₂	#20261007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260761

C₂

yes

maximal, characteristic

#20260744

C₂

yes

maximal

C₂×#15690729

C₂

yes

maximal

#20260500

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

characteristic

C₂×#13095495

C₂²

yes

—

#15690769

C₂²

yes

—

—

—

C₂²×Q₈×C₉₇

C₂²

yes

—

C₂×Q₁₆×C₉₇

—

C₂×Q₁₆×C₉₇

—

C₂²×C₈×C₉₇

C₂²

yes

—

—

—

—

—

C₄⋊Dic₉₇

D₄

yes

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₂³×C₉₇

#41

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#121

yes

characteristic

C₂²×C₉₇

#123

yes

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

#121

yes

—

C₄×C₉₇

#124

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#394

yes

characteristic

C₂×C₉₇

#473

yes

—

#2047

—

maximal, Sylow

C₉₇

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₉₇

yes

—

—

—

—

C₂²×Q₈

C₂×D₉₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

C₄⋊C₈

—

194

—

#103

—

194

—

#101

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

Q₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

194

—

C₄²

—

194

—

C₂³

#13095510

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

194

—

107

C₂×C₄

D₄×D₉₇

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

110

C₂×C₄

#13095510

yes

—

111

C₈

—

112

C₈

—

388

—

113

C₈

—

388

—

114

C₂²

—

115

C₂²

#15690781

yes

characteristic

116

C₂²

#15690791

yes

—

117

C₄

—

118

C₄

—

388

—

119

C₄

#15690779

yes

characteristic

120

C₄

#15690782

yes

—

121

C₄

#15690797

yes

characteristic

122

C₂

—

123

C₂

#20260649

yes

characteristic

124

C₂

#20261007

yes

—

	subgroup	G/H
center	C₂²	#15690781
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂²×Q₁₆×C₉₇	C₂
socle	C₂²×C₉₇	#121
prime radical	C₂³×C₉₇	#41
hypercenter	C₂²×Q₁₆	D₉₇
hypocenter	C₉₇	#2047
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#15690781

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂²×Q₁₆×C₉₇

C₂

C₂²×C₉₇

C₂³×C₉₇

C₂²×Q₁₆

D₉₇

hypocenter

C₉₇

#2047

LCS

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
63. C₉₇	#2047	semidirect

H ≅ G/N

type

63. C₉₇

#2047

semidirect

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

subgroup	index
#20260761	2
#20260744	2
C₂×#15690729	2
#20260500	2
C₂²×Q₁₆×C₉₇	2
#2047	97

subgroup

index

C₂²×Q₁₆×C₉₇

Subgroups — #27143001

Conjugacy classes of subgroups

Complements

Maximal subgroups