Group #27158153 — subgroups

Subgroups — #27158153

Order 12416.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20261656	C₂	1	1	yes	maximal, characteristic
2	#20261656	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13095514	C₂	1	1	yes	maximal, characteristic
4	#20261624	C₂	1	1	yes	maximal, characteristic
5	#20261623	C₂	1	1	yes	maximal, characteristic
6	#20261621	C₂	1	1	yes	maximal, characteristic
7	#20260859	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095514	C₂²	1	1	yes	characteristic
9	C₂×#13095514	C₂²	1	1	yes	characteristic
10	C₂×#13095514	C₂²	1	1	yes	characteristic
11	C₂×#13095514	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
13	#15690739	C₄	1	1	yes	characteristic
14	#15690739	C₄	1	1	yes	characteristic
15	C₄×#11715895	C₂²	1	1	yes	characteristic
16	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
17	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₉₇	C₄	1	1	yes	characteristic
19	#13095514	—	2	1	no	—
20	#13095514	—	2	1	no	—
21	#13095514	—	2	1	no	—
22	#13095514	Q₈	1	2	yes	complemented
23	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
24	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
25	C₂×#11715895	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
27	#13095490	C₈	1	1	yes	characteristic
28	#13095490	C₈	1	1	yes	characteristic
29	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
41	Dic₁₉₄	—	2	2	no	—
42	C₈×C₉₇	—	2	1	no	—
43	C₈×C₉₇	—	2	1	no	—
44	#11715895	—	2	1	no	—
45	#11715895	C₂×Q₈	1	1	yes	characteristic
46	#11715895	#41	1	1	yes	characteristic
47	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₉₇	—	2	1	no	—
49	C₄×C₉₇	—	2	1	no	—
50	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₉₇	#131	1	1	yes	characteristic
52	Dic₉₇	—	2	1	no	—
53	Dic₉₇	—	2	1	no	—
54	Dic₉₇	—	2	1	no	—
55	Dic₉₇	M₃₂	1	2	yes	—
56	C₂×C₉₇	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₉₇	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₉₇	#504	1	1	yes	characteristic
59	#2183	—	97	1	no	maximal, Sylow
60	C₉₇	#2183	1	1	yes	characteristic, complemented, Sylow
61	C₈×Q₈	—	97	1	no	—
62	C₄⋊C₁₆	—	97	1	no	—
63	C₄⋊C₁₆	—	97	1	no	—
64	C₄⋊C₁₆	—	97	1	no	—
65	#346	—	97	1	no	—
66	#346	—	97	1	no	—
67	C₄×C₁₆	—	97	1	no	—
68	C₄×Q₈	—	97	1	no	—
69	C₂×C₁₆	—	97	1	no	—
70	C₂×C₁₆	—	97	1	no	—
71	C₂×C₁₆	—	97	1	no	—
72	C₂×C₁₆	—	97	1	no	—
73	C₄⋊C₈	—	97	1	no	—
74	C₄⋊C₈	—	97	1	no	—
75	C₄⋊C₈	#87604	1	1	yes	characteristic
76	C₄×C₈	—	97	1	no	—
77	C₄×C₈	—	97	1	no	—
78	C₄×C₈	—	97	1	no	—
79	C₂×Q₈	—	97	1	no	—
80	C₂×C₈	—	97	1	no	—
81	C₂×C₈	—	97	1	no	—
82	C₂×C₈	C₂×#87604	1	1	yes	characteristic
83	C₂×C₈	C₂×#87604	1	1	yes	characteristic
84	C₄⋊C₄	—	97	1	no	—
85	C₄⋊C₄	—	97	1	no	—
86	C₄⋊C₄	—	97	1	no	—
87	C₄²	—	97	1	no	—
88	C₄²	—	97	1	no	—
89	C₄²	C₂×#87604	1	1	yes	characteristic
90	C₁₆	—	97	2	no	—
91	C₁₆	—	194	1	no	—
92	C₁₆	—	194	1	no	—
93	C₁₆	—	194	1	no	—
94	Q₈	—	194	2	no	—
95	C₂×C₄	—	97	1	no	—
96	C₂×C₄	—	97	1	no	—
97	C₂×C₄	—	97	1	no	—
98	C₂×C₄	—	97	1	no	—
99	C₂×C₄	#13095519	1	1	yes	characteristic
100	C₂×C₄	#13095519	1	1	yes	characteristic
101	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	97	1	no	—
105	C₈	—	97	1	no	—
106	C₈	—	194	1	no	—
107	C₂²	C₂×#13095519	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	97	2	no	—
111	C₄	—	194	1	no	—
112	C₄	—	194	1	no	—
113	C₄	—	194	1	no	—
114	C₄	Q₈×#87604	1	1	yes	characteristic
115	C₄	#15690897	1	1	yes	characteristic
116	C₂	#20261784	1	1	yes	characteristic
117	C₂	#20261791	1	1	yes	characteristic
118	C₂	#20261794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261656

C₂

yes

maximal, characteristic

#20261656

C₂

yes

maximal, characteristic

C₄⋊#13095514

C₂

yes

maximal, characteristic

#20261624

C₂

yes

maximal, characteristic

#20261623

C₂

yes

maximal, characteristic

#20261621

C₂

yes

maximal, characteristic

#20260859

C₂

yes

maximal, characteristic

C₂×#13095514

C₂²

yes

characteristic

C₂×#13095514

C₂²

yes

characteristic

C₂×#13095514

C₂²

yes

characteristic

C₂×#13095514

C₂²

yes

characteristic

C₄×Dic₁₉₄

C₄

yes

characteristic

#15690739

C₄

yes

characteristic

#15690739

C₄

yes

characteristic

C₄×#11715895

C₂²

yes

characteristic

C₈×Dic₉₇

C₂²

yes

characteristic

C₈×Dic₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₄

yes

—

—

—

Q₈

yes

complemented

C₂×Dic₁₉₄

C₈

yes

characteristic

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

C₂³

yes

characteristic

C₄⋊Dic₉₇

C₈

yes

characteristic

#13095490

C₈

yes

characteristic

#13095490

C₈

yes

characteristic

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×C₈

yes

characteristic

C₂×C₄×C₉₇

C₂×C₈

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

C₂×C₈

yes

characteristic

C₂×Dic₉₇

C₂×C₈

yes

characteristic

C₂×Dic₉₇

C₂×C₈

yes

characteristic

C₂×Dic₉₇

C₂×C₈

yes

characteristic

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

C₂×Q₈

yes

yes

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

characteristic

C₄×C₉₇

#131

yes

characteristic

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

M₃₂

yes

—

C₂×C₉₇

C₈×Q₈

yes

characteristic

C₂×C₉₇

C₂×M₃₂

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#87604

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#87604

yes

characteristic

C₁₆

—

C₁₆

—

194

—

C₁₆

—

194

—

C₁₆

—

194

—

Q₈

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13095519

yes

characteristic

100

C₂×C₄

#13095519

yes

characteristic

101

C₂×C₄

C₂²×#87604

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

194

—

107

C₂²

C₂×#13095519

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

194

—

112

C₄

—

194

—

113

C₄

—

194

—

114

C₄

Q₈×#87604

yes

characteristic

115

C₄

#15690897

yes

characteristic

116

C₂

#20261784

yes

characteristic

117

C₂

#20261791

yes

characteristic

118

C₂

#20261794

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87604
commutator, socle, prime radical	C₂²×C₉₇	C₂²×C₈
Fitting	C₄⋊C₈×C₉₇	C₄
cosocle	C₂×#11715895	C₂³
hypercenter	C₄⋊C₈	#87604
hypocenter	C₉₇	#2183
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#87604

commutator, socle, prime radical

C₂²×C₉₇

C₂²×C₈

Fitting

C₄⋊C₈×C₉₇

C₄

C₂³

C₄⋊C₈

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
22. #13095514	Q₈	semidirect
60. C₉₇	#2183	semidirect

H ≅ G/N

type

22. #13095514

Q₈

semidirect

60. C₉₇

#2183

semidirect

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

subgroup	index
#20261656	2
C₄⋊#13095514	2
#20261624	2
#20261623	2
#20261621	2
#20260859	2
#2183	97

subgroup

index

Subgroups — #27158153

Conjugacy classes of subgroups

Complements

Maximal subgroups