Group #27150000 — subgroups

Subgroups — #27150000

Order 12416.

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

normal subgroups only

This group has 548 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	#20260671	C₂	1	4	yes	maximal, complemented
2	#20260658	C₂	1	8	yes	maximal, complemented
3	#20260591	C₂	1	2	yes	maximal
4	#379×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	complemented
6	C₂×#13095490	C₂²	1	8	yes	complemented
7	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	—
10	#15690717	C₂²	1	4	yes	complemented
11	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₉₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
14	C₂²×Dic₉₇	C₂³	1	8	yes	—
15	C₄⋊Dic₉₇	—	4	4	no	—
16	#13095490	—	4	8	no	—
17	C₂²⋊Dic₉₇	—	4	4	no	—
18	C₂²⋊Dic₉₇	—	4	4	no	—
19	C₂²⋊Dic₉₇	—	4	4	no	—
20	C₂⁴×C₉₇	—	4	4	no	—
21	C₂⁴×C₉₇	—	4	2	no	—
22	C₂⁴×C₉₇	Q₈	1	4	yes	—
23	C₂⁴×C₉₇	C₂³	1	2	yes	—
24	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
26	C₂²⋊C₄×C₉₇	—	4	4	no	—
27	C₂²⋊C₄×C₉₇	—	4	2	no	—
28	C₂³×C₉₇	—	2	4	no	—
29	C₂³×C₉₇	—	2	2	no	—
30	C₂³×C₉₇	—	2	4	no	—
31	C₂³×C₉₇	—	2	4	no	—
32	C₂³×C₉₇	—	2	4	no	—
33	C₂³×C₉₇	—	4	8	no	—
34	C₂³×C₉₇	—	4	4	no	—
35	C₂³×C₉₇	—	4	16	no	—
36	C₂³×C₉₇	C₂×Q₈	1	4	yes	—
37	C₂³×C₉₇	C₂×Q₈	1	2	yes	—
38	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₉₇	—	2	8	no	—
40	C₂×C₄×C₉₇	—	2	4	no	—
41	C₂×Dic₉₇	—	2	8	no	—
42	C₂×Dic₉₇	—	2	8	no	—
43	C₂×Dic₉₇	—	2	8	no	—
44	C₂²×C₉₇	—	2	4	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	—	2	4	no	—
47	C₂²×C₉₇	—	2	4	no	—
48	C₂²×C₉₇	—	2	4	no	—
49	C₂²×C₉₇	—	4	16	no	—
50	C₂²×C₉₇	—	4	8	no	—
51	C₂²×C₉₇	—	4	4	no	—
52	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₉₇	H(5,𝔽₂)	1	4	yes	—
54	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
55	C₄×C₉₇	—	4	4	no	—
56	Dic₉₇	—	4	8	no	—
57	C₂×C₉₇	—	4	4	no	—
58	C₂×C₉₇	—	4	2	no	—
59	C₂×C₉₇	#543	1	2	yes	—
60	C₂×C₉₇	#543	1	1	yes	characteristic
61	C₂×C₉₇	#560	1	4	yes	—
62	#2842	—	97	1	no	maximal, Sylow
63	C₉₇	#2842	1	1	yes	characteristic, complemented, Sylow
64	#399	—	97	4	no	—
65	#396	—	97	8	no	—
66	#379	—	97	2	no	—
67	#379	D₉₇	1	1	yes	characteristic
68	C₂⁵	C₂×D₉₇	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	97	8	no	—
70	C₂×C₄⋊C₄	—	97	4	no	—
71	C₂×C₂²⋊C₄	—	97	4	no	—
72	C₂×C₂²⋊C₄	—	97	4	no	—
73	C₂×C₂²⋊C₄	—	97	4	no	—
74	C₂×C₂²⋊C₄	C₂×D₉₇	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
76	#95	—	97	4	no	—
77	C₂⁴	—	4	4	no	—
78	C₂⁴	—	4	2	no	—
79	C₂⁴	Dic₁₉₄	1	4	yes	—
80	C₂⁴	C₂²×D₉₇	1	2	yes	—
81	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
82	C₂²×C₄	—	97	8	no	—
83	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
84	C₄⋊C₄	—	388	8	no	—
85	C₄⋊C₄	—	388	4	no	—
86	C₂²⋊C₄	—	4	4	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	388	4	no	—
89	C₂²⋊C₄	—	388	4	no	—
90	C₂²⋊C₄	—	388	4	no	—
91	C₂³	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	4	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	—	4	8	no	—
97	C₂³	—	4	4	no	—
98	C₂³	—	4	16	no	—
99	C₂³	C₂×Dic₁₉₄	1	4	yes	—
100	C₂³	C₂×Dic₁₉₄	1	2	yes	—
101	C₂³	C₂³×D₉₇	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	194	8	no	—
105	C₂×C₄	—	194	8	no	—
106	C₂×C₄	—	194	8	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	16	no	—
113	C₂²	—	4	8	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
116	C₂²	#15690855	1	4	yes	—
117	C₂²	#15690855	1	2	yes	—
118	C₄	—	4	4	no	—
119	C₄	—	388	8	no	—
120	C₂	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#20261361	1	2	yes	—
123	C₂	#20261361	1	1	yes	characteristic
124	C₂	#20261474	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260671

C₂

yes

maximal, complemented

#20260658

C₂

yes

maximal, complemented

#20260591

C₂

yes

maximal

#379×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

complemented

C₂×#13095490

C₂²

yes

complemented

C₂×C₂²⋊Dic₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₉₇

C₂²

yes

—

#15690717

C₂²

yes

complemented

C₂⁵×C₉₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095490

—

C₂²⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₂⁴×C₉₇

—

C₂⁴×C₉₇

—

C₂⁴×C₉₇

Q₈

yes

—

C₂⁴×C₉₇

C₂³

yes

—

C₂⁴×C₉₇

C₂³

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×Q₈

yes

—

C₂³×C₉₇

C₂×Q₈

yes

—

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²×Q₈

yes

characteristic

C₂²×C₉₇

H(5,𝔽₂)

yes

—

C₂²×C₉₇

H(5,𝔽₂)

yes

—

C₄×C₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#543

yes

—

C₂×C₉₇

#543

yes

characteristic

C₂×C₉₇

#560

yes

—

#2842

—

maximal, Sylow

C₉₇

#2842

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic

C₂⁵

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₉₇

yes

—

C₂×C₂²⋊C₄

C₂×D₉₇

yes

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₁₉₄

yes

—

C₂⁴

C₂²×D₉₇

yes

—

C₂⁴

C₂²×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

388

—

C₂²⋊C₄

—

388

—

C₂²⋊C₄

—

388

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₉₄

yes

—

100

C₂³

C₂×Dic₁₉₄

yes

—

101

C₂³

C₂³×D₉₇

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

194

—

105

C₂×C₄

—

194

—

106

C₂×C₄

—

194

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²×Dic₁₉₄

yes

characteristic

116

C₂²

#15690855

yes

—

117

C₂²

#15690855

yes

—

118

C₄

—

119

C₄

—

388

—

120

C₂

—

121

C₂

—

122

C₂

#20261361

yes

—

123

C₂

#20261361

yes

characteristic

124

C₂

#20261474

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#379×C₉₇	C₂
prime radical	C₂⁵×C₉₇	C₂²
hypercenter	#379	D₉₇
hypocenter	C₉₇	#2842
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, cosocle

C₂³×C₉₇

C₂⁴

Fitting

#379×C₉₇

C₂

prime radical

C₂⁵×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. #20260671	C₂	semidirect
2. #20260658	C₂	semidirect
5. C₂×C₄⋊Dic₉₇	C₂²	semidirect
6. C₂×#13095490	C₂²	semidirect
10. #15690717	C₂²	semidirect
63. C₉₇	#2842	semidirect

H ≅ G/N

type

1. #20260671

C₂

semidirect

2. #20260658

C₂

semidirect

5. C₂×C₄⋊Dic₉₇

C₂²

semidirect

6. C₂×#13095490

C₂²

semidirect

10. #15690717

C₂²

semidirect

63. C₉₇

#2842

semidirect

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

subgroup	index
#20260671	2
#20260658	2
#20260591	2
#379×C₉₇	2
#2842	97

subgroup

index

#379×C₉₇

Subgroups — #27150000

Conjugacy classes of subgroups

Complements

Maximal subgroups