Group #27150011 — subgroups

Subgroups — #27150011

Order 12416.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20260669	C₂	1	4	yes	maximal
2	#20260668	C₂	1	4	yes	maximal
3	#20260667	C₂	1	4	yes	maximal
4	#20260601	C₂	1	2	yes	maximal
5	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×#13095490	C₂²	1	4	yes	—
10	C₂×#13095490	C₂²	1	4	yes	—
11	#15690717	C₂²	1	4	yes	—
12	#15690717	C₂²	1	4	yes	—
13	#15690717	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#95×C₉₇	C₂²	1	4	yes	—
17	C₂²×Dic₉₇	C₂³	1	4	yes	—
18	C₂²×Dic₉₇	C₂³	1	4	yes	—
19	C₄⋊Dic₉₇	—	4	2	no	—
20	C₄⋊Dic₉₇	—	4	4	no	—
21	C₄⋊Dic₉₇	—	4	2	no	—
22	#13095490	—	4	4	no	—
23	#13095490	—	4	4	no	—
24	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₉₇	—	4	2	no	—
28	C₄²×C₉₇	Q₈	1	4	yes	—
29	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₉₇	—	2	1	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	1	no	—
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
38	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
39	C₂×Dic₉₇	—	2	4	no	—
40	C₂×Dic₉₇	—	2	4	no	—
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	2	4	no	—
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	2	4	no	—
45	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
46	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₉₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
50	C₄×C₉₇	—	2	4	no	—
51	C₄×C₉₇	—	2	2	no	—
52	C₄×C₉₇	—	4	4	no	—
53	Dic₉₇	—	4	4	no	—
54	Dic₉₇	—	4	4	no	—
55	C₂×C₉₇	#544	1	2	yes	—
56	C₂×C₉₇	#544	1	1	yes	characteristic
57	C₂×C₉₇	#562	1	2	yes	—
58	C₂×C₉₇	#562	1	1	yes	characteristic
59	C₂×C₉₇	#564	1	1	yes	characteristic
60	#2845	—	97	1	no	maximal, Sylow
61	C₉₇	#2845	1	1	yes	characteristic, complemented, Sylow
62	#398	—	97	4	no	—
63	#398	—	97	4	no	—
64	#398	—	97	4	no	—
65	#382	—	97	2	no	—
66	#382	D₉₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	97	4	no	—
68	C₂×C₄⋊C₄	—	97	4	no	—
69	C₂×C₄⋊C₄	—	97	2	no	—
70	C₂×C₄⋊C₄	—	97	2	no	—
71	C₂×C₄⋊C₄	—	97	4	no	—
72	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
73	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
74	#95	—	97	4	no	—
75	#95	—	97	4	no	—
76	#95	—	97	4	no	—
77	#95	C₂×D₉₇	1	4	yes	—
78	C₂²×C₄	—	97	4	no	—
79	C₂²×C₄	—	97	4	no	—
80	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
81	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	388	4	no	—
85	C₄⋊C₄	—	388	4	no	—
86	C₄⋊C₄	—	388	2	no	—
87	C₄⋊C₄	—	388	2	no	—
88	C₄⋊C₄	—	388	4	no	—
89	C₄²	Dic₁₉₄	1	4	yes	—
90	C₂³	C₂³×D₉₇	1	1	yes	characteristic
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	194	4	no	—
99	C₂×C₄	—	194	4	no	—
100	C₂×C₄	—	194	4	no	—
101	C₂×C₄	—	194	4	no	—
102	C₂×C₄	—	194	4	no	—
103	C₂×C₄	—	194	4	no	—
104	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
105	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
106	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
107	C₂²	#15690855	1	1	yes	characteristic
108	C₂²	#15690856	1	2	yes	—
109	C₂²	#15690857	1	2	yes	—
110	C₂²	#15690858	1	1	yes	characteristic
111	C₄	—	2	4	no	—
112	C₄	—	2	2	no	—
113	C₄	—	4	4	no	—
114	C₄	—	388	4	no	—
115	C₄	—	388	4	no	—
116	C₂	#20261363	1	2	yes	—
117	C₂	#20261366	1	1	yes	characteristic
118	C₂	#20261479	1	2	yes	—
119	C₂	#20261483	1	1	yes	characteristic
120	C₂	C₉₇⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260669

C₂

yes

maximal

#20260668

C₂

yes

maximal

#20260667

C₂

yes

maximal

#20260601

C₂

yes

maximal

#382×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

C₂²

yes

—

C₂×#13095490

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₂²

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

#13095490

—

#13095490

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂²×C₉₇

C₂²×Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#544

yes

—

C₂×C₉₇

#544

yes

characteristic

C₂×C₉₇

#562

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

—

—

—

C₂×D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄²

Dic₁₉₄

yes

—

C₂³

C₂³×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

C₂×Dic₁₉₄

yes

—

105

C₂×C₄

C₂×Dic₁₉₄

yes

—

106

C₂²

C₂²×Dic₁₉₄

yes

characteristic

107

C₂²

#15690855

yes

characteristic

108

C₂²

#15690856

yes

—

109

C₂²

#15690857

yes

—

110

C₂²

#15690858

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

388

—

115

C₄

—

388

—

116

C₂

#20261363

yes

—

117

C₂

#20261366

yes

characteristic

118

C₂

#20261479

yes

—

119

C₂

#20261483

yes

characteristic

120

C₂

C₉₇⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#382×C₉₇	C₂
hypercenter	#382	D₉₇
hypocenter	C₉₇	#2845
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, prime radical, cosocle

C₂³×C₉₇

C₂⁴

Fitting

#382×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
61. C₉₇	#2845	semidirect

H ≅ G/N

type

61. C₉₇

#2845

semidirect

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

subgroup	index
#20260669	2
#20260668	2
#20260667	2
#20260601	2
#382×C₉₇	2
#2845	97

subgroup

index

#382×C₉₇

Subgroups — #27150011

Conjugacy classes of subgroups

Complements

Maximal subgroups