Group #27139633 — subgroups

Subgroups — #27139633

Order 12416.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20260788	C₂	1	1	yes	maximal, characteristic
2	#20260759	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	—	2	1	no	—
5	C₄⋊#11715895	—	2	1	no	—
6	C₄⋊#11715895	—	2	1	no	—
7	C₄⋊#11715895	C₄	1	2	yes	—
8	C₂²⋊#11715895	C₄	1	2	yes	—
9	C₄×#11715895	—	2	1	no	—
10	C₂²×C₈×C₉₇	—	2	1	no	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₉₇	—	2	1	no	—
13	C₄⋊C₈×C₉₇	C₄	1	2	yes	—
14	#13095495	—	4	1	no	—
15	#13095495	—	4	1	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₉₇	—	2	2	no	—
22	C₂×C₈×C₉₇	—	4	1	no	—
23	C₂×C₈×C₉₇	—	4	1	no	—
24	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₉₇	D₄	1	1	yes	characteristic
27	C₄²×C₉₇	D₄	1	1	yes	characteristic
28	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	—	4	1	no	—
31	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
38	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₉₇	Q₁₆	1	1	yes	characteristic
41	C₈×C₉₇	—	4	2	no	—
42	#11715895	—	4	2	no	—
43	#11715895	—	4	2	no	—
44	C₂²×C₉₇	—	2	1	no	—
45	C₂²×C₉₇	—	2	1	no	—
46	C₂²×C₉₇	#95	1	1	yes	characteristic
47	C₂²×C₉₇	#106	1	1	yes	characteristic
48	C₂²×C₉₇	#107	1	1	yes	characteristic
49	C₄×C₉₇	—	2	1	no	—
50	C₄×C₉₇	—	2	1	no	—
51	C₄×C₉₇	—	2	1	no	—
52	C₄×C₉₇	#102	1	1	yes	characteristic
53	C₄×C₉₇	#102	1	1	yes	characteristic
54	C₄×C₉₇	#103	1	1	yes	characteristic
55	C₄×C₉₇	#103	1	1	yes	characteristic
56	C₄×C₉₇	#108	1	2	yes	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	—	2	1	no	—
59	C₂×C₉₇	#340	1	1	yes	characteristic
60	C₂×C₉₇	#341	1	1	yes	characteristic
61	C₂×C₉₇	#344	1	1	yes	characteristic
62	#1301	—	97	1	no	maximal, Sylow
63	C₉₇	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	—	97	1	no	—
65	#423	—	97	1	no	—
66	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
67	C₂×M₁₆	—	194	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	97	2	no	—
72	C₄⋊C₈	—	194	1	no	—
73	C₄⋊C₈	—	194	1	no	—
74	C₄⋊C₈	Dic₉₇	1	2	yes	—
75	C₂²⋊C₈	—	97	2	no	—
76	C₄×C₈	—	194	1	no	—
77	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
80	M₁₆	—	388	1	no	—
81	M₁₆	—	388	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	194	2	no	—
86	C₂×C₈	—	194	2	no	—
87	C₄²	C₄×D₉₇	1	1	yes	characteristic
88	C₄²	D₃₈₈	1	1	yes	characteristic
89	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
91	C₂³	#13095490	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
95	C₂×C₄	#13095488	1	1	yes	characteristic
96	C₂×C₄	#13095488	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
98	C₂×C₄	#13095490	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
100	C₂×C₄	#13095496	1	1	yes	characteristic
101	C₂×C₄	#13095498	1	1	yes	characteristic
102	C₂×C₄	#13095499	1	1	yes	characteristic
103	C₂×C₄	#13095501	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	388	2	no	—
106	C₈	—	388	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#15690717	1	1	yes	characteristic
110	C₂²	#15690741	1	1	yes	characteristic
111	C₂²	#15690743	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#15690730	1	1	yes	characteristic
116	C₄	#15690731	1	1	yes	characteristic
117	C₄	#15690733	1	1	yes	characteristic
118	C₄	#15690734	1	1	yes	characteristic
119	C₄	#15690746	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#20260497	1	1	yes	characteristic
123	C₂	#20260500	1	1	yes	characteristic
124	C₂	#20260507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260788

C₂

yes

maximal, characteristic

#20260759

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

C₂²⋊#11715895

C₄

yes

—

C₄×#11715895

—

C₂²×C₈×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₄⋊C₈×C₉₇

C₄

yes

—

—

—

—

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

Q₈

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₂³×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

characteristic

C₂×C₄×C₉₇

Q₁₆

yes

characteristic

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

D₉₇

yes

characteristic

C₂×M₁₆

—

194

—

C₂²×C₈

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

C₄⋊C₈

Dic₉₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

194

—

C₂²×C₄

Dic₁₉₄

yes

characteristic

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

M₁₆

—

388

—

M₁₆

—

388

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

C₄×D₉₇

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

characteristic

C₂³

#13095490

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

C₂×C₄

#13095490

yes

characteristic

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

100

C₂×C₄

#13095496

yes

characteristic

101

C₂×C₄

#13095498

yes

characteristic

102

C₂×C₄

#13095499

yes

characteristic

103

C₂×C₄

#13095501

yes

characteristic

104

C₈

—

105

C₈

—

388

—

106

C₈

—

388

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15690717

yes

characteristic

110

C₂²

#15690741

yes

characteristic

111

C₂²

#15690743

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#15690730

yes

characteristic

116

C₄

#15690731

yes

characteristic

117

C₄

#15690733

yes

characteristic

118

C₄

#15690734

yes

characteristic

119

C₄

#15690746

yes

—

120

C₂

—

121

C₂

—

122

C₂

#20260497

yes

characteristic

123

C₂

#20260500

yes

characteristic

124

C₂

#20260507

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13095488
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	#1301
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#13095488

commutator

C₂×C₄×C₉₇

C₄²

Frattini

C₂×C₄²

C₂×D₉₇

Fitting

C₂×C₄⋊C₈×C₉₇

C₂

socle

C₂²×C₉₇

#95

prime radical

C₂³×C₉₇

C₄⋊C₄

cosocle

C₂×C₄²×C₉₇

C₂²

hypercenter

C₂×C₄⋊C₈

D₉₇

hypocenter

C₉₇

#1301

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
63. C₉₇	#1301	semidirect

H ≅ G/N

type

63. C₉₇

#1301

semidirect

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

subgroup	index
#20260788	2
#20260759	2
C₂×C₄⋊C₈×C₉₇	2
#1301	97

subgroup

index

#20260788

#20260759

C₂×C₄⋊C₈×C₉₇

#1301

Subgroups — #27139633

Conjugacy classes of subgroups

Complements

Maximal subgroups