Group #27141848 — subgroups

Subgroups — #27141848

Order 12416.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#20261213	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690729	C₂	1	1	yes	maximal, characteristic
3	#20260507	C₂	1	2	yes	maximal
4	#20260506	C₂	1	2	yes	maximal
5	C₂×#101×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	Q₈×Dic₉₇	—	2	2	no	—
8	Q₈×Dic₉₇	—	2	2	no	—
9	#15690769	C₄	1	4	yes	—
10	#15690769	C₂²	1	1	yes	characteristic
11	#15690769	C₂²	1	1	yes	characteristic
12	#15690729	—	2	2	no	—
13	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
15	#101×C₉₇	—	2	2	no	—
16	#13095495	—	2	2	no	—
17	#13095495	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄⋊Dic₉₇	—	2	4	no	—
20	C₄⋊Dic₉₇	—	2	1	no	—
21	C₄⋊Dic₉₇	—	2	1	no	—
22	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₉₇	—	2	4	no	—
25	C₄×Dic₉₇	—	2	1	no	—
26	C₄×Dic₉₇	—	2	1	no	—
27	C₂×Q₈×C₉₇	—	2	2	no	—
28	C₂×Q₈×C₉₇	—	2	2	no	—
29	C₂×Q₈×C₉₇	D₄	1	2	yes	—
30	C₂×Q₈×C₉₇	D₄	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₉₇	—	2	2	no	—
34	M₁₆×C₉₇	—	2	2	no	—
35	C₂×C₈×C₉₇	—	2	2	no	—
36	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₉₇	—	2	4	no	—
38	Q₈×C₉₇	—	2	4	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	#41	1	2	yes	—
45	C₂×C₄×C₉₇	#41	1	2	yes	—
46	C₂×Dic₉₇	—	2	2	no	—
47	C₂×Dic₉₇	—	2	2	no	—
48	C₈×C₉₇	—	4	2	no	—
49	#11715895	—	4	2	no	—
50	C₂²×C₉₇	—	4	1	no	—
51	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₉₇	#117	1	1	yes	characteristic
53	C₂²×C₉₇	#117	1	1	yes	characteristic
54	C₄×C₉₇	—	2	4	no	—
55	C₄×C₉₇	#123	1	2	yes	—
56	C₄×C₉₇	#123	1	2	yes	—
57	Dic₉₇	—	4	2	no	—
58	Dic₉₇	—	4	2	no	—
59	C₂×C₉₇	—	2	1	no	—
60	C₂×C₉₇	—	2	1	no	—
61	C₂×C₉₇	—	2	1	no	—
62	C₂×C₉₇	#381	1	1	yes	characteristic
63	#1830	—	97	1	no	maximal, Sylow
64	C₉₇	#1830	1	1	yes	characteristic, complemented, Sylow
65	#519	—	97	1	no	—
66	C₂×#101	—	97	1	no	—
67	C₂×#101	D₉₇	1	1	yes	characteristic
68	#344	—	97	2	no	—
69	#344	—	97	2	no	—
70	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
71	C₂×M₁₆	—	97	2	no	—
72	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
73	C₄×Q₈	—	194	2	no	—
74	C₄×Q₈	—	194	2	no	—
75	#117	—	97	4	no	—
76	#117	—	97	1	no	—
77	#117	—	97	1	no	—
78	#101	—	2	2	no	—
79	#101	—	194	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	D₃₈₈	1	2	yes	—
83	C₂×Q₈	C₂²⋊D₉₇	1	2	yes	—
84	C₂²×C₄	C₄×D₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	194	2	no	—
89	M₁₆	—	194	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	194	2	no	—
92	C₄⋊C₄	—	194	4	no	—
93	C₄⋊C₄	—	194	1	no	—
94	C₄⋊C₄	—	194	1	no	—
95	C₂²⋊C₄	—	97	2	no	—
96	C₂²⋊C₄	—	97	2	no	—
97	C₄²	—	194	4	no	—
98	C₄²	—	194	1	no	—
99	C₄²	—	194	1	no	—
100	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	194	2	no	—
106	C₂×C₄	—	194	2	no	—
107	C₂×C₄	#13095488	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
110	C₂×C₄	#13095510	1	2	yes	—
111	C₂×C₄	#13095510	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	388	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#13095488	1	1	yes	characteristic
116	C₂²	#15690766	1	1	yes	characteristic
117	C₂²	#15690766	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	388	2	no	—
120	C₄	—	388	2	no	—
121	C₄	#15690791	1	2	yes	—
122	C₄	#15690794	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#20260596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261213

C₂

yes

maximal, characteristic

C₂×#15690729

C₂

yes

maximal, characteristic

#20260507

C₂

yes

maximal

#20260506

C₂

yes

maximal

C₂×#101×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

Q₈×Dic₉₇

—

Q₈×Dic₉₇

—

#15690769

C₄

yes

—

#15690769

C₂²

yes

characteristic

#15690769

C₂²

yes

characteristic

#15690729

—

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

—

#101×C₉₇

—

#13095495

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

C₂×C₄

yes

—

C₂²⋊Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

#123

yes

—

C₄×C₉₇

#123

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#519

—

C₂×#101

—

C₂×#101

D₉₇

yes

characteristic

#344

—

#344

—

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₄×Q₈

—

194

—

C₄×Q₈

—

194

—

—

—

—

—

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₃₈₈

yes

—

C₂×Q₈

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

—

194

—

100

C₂³

C₂×C₄×D₉₇

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

194

—

106

C₂×C₄

—

194

—

107

C₂×C₄

#13095488

yes

—

108

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

109

C₂×C₄

C₂×D₃₈₈

yes

characteristic

110

C₂×C₄

#13095510

yes

—

111

C₂×C₄

#13095510

yes

—

112

C₈

—

113

C₈

—

388

—

114

C₂²

—

115

C₂²

C₂×#13095488

yes

characteristic

116

C₂²

#15690766

yes

characteristic

117

C₂²

#15690766

yes

characteristic

118

C₄

—

119

C₄

—

388

—

120

C₄

—

388

—

121

C₄

#15690791

yes

—

122

C₄

#15690794

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#20260596

yes

characteristic

	subgroup	G/H
center	C₂	#20260596
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#101×C₉₇	C₂
socle	C₂×C₉₇	#381
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#101	D₉₇
hypocenter	C₉₇	#1830
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#20260596

commutator, prime radical

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#101×C₉₇

C₂

socle

C₂×C₉₇

#381

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
64. C₉₇	#1830	semidirect

H ≅ G/N

type

64. C₉₇

#1830

semidirect

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

subgroup	index
#20261213	2
C₂×#15690729	2
#20260507	2
#20260506	2
C₂×#101×C₉₇	2
#1830	97

subgroup

index

C₂×#101×C₉₇

Subgroups — #27141848

Conjugacy classes of subgroups

Complements

Maximal subgroups