Group #27143290 — subgroups

Subgroups — #27143290

Order 12416.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20260784	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260762	C₂	1	2	yes	maximal, complemented
3	#20260753	C₂	1	2	yes	maximal
4	#20260703	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095511	C₂²	1	1	yes	characteristic
7	C₂×#13095495	C₂²	1	2	yes	—
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	#15690746	—	2	2	no	—
10	C₄⋊#11715895	—	2	4	no	—
11	#15690738	—	2	4	no	—
12	#15690722	—	2	2	no	—
13	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
14	#117×C₉₇	C₂²	1	2	yes	—
15	#13095511	—	2	2	no	—
16	#13095511	—	2	2	no	—
17	C₂×Dic₁₉₄	Q₈	1	1	yes	characteristic
18	C₂×D₃₈₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₉₇	—	2	1	no	—
20	C₂×C₄×D₉₇	—	2	1	no	—
21	#13095495	—	2	2	no	—
22	#13095495	—	2	2	no	—
23	C₂×#11715895	—	2	1	no	—
24	C₂×#11715895	—	2	2	no	—
25	C₂×#11715895	—	2	2	no	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₂²⋊C₄×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	D₄	1	4	yes	—
32	C₂³×C₉₇	#41	1	1	yes	characteristic
33	C₂²×D₉₇	—	2	1	no	—
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₉₇	#41	1	2	yes	—
39	C₂²⋊D₉₇	—	4	2	no	—
40	C₂×Dic₉₇	—	2	1	no	—
41	D₃₈₈	—	2	2	no	—
42	C₄×D₉₇	—	4	1	no	—
43	C₄×D₉₇	—	4	1	no	—
44	Dic₁₉₄	—	2	2	no	—
45	C₈×C₉₇	—	2	2	no	—
46	#11715895	—	4	1	no	—
47	#11715895	—	4	2	no	—
48	C₂²×C₉₇	—	4	1	no	—
49	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₉₇	#123	1	2	yes	—
51	C₂×D₉₇	—	4	2	no	—
52	C₂×D₉₇	—	4	1	no	—
53	C₄×C₉₇	—	4	4	no	—
54	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₉₇	#123	1	1	yes	characteristic
56	C₄×C₉₇	#127	1	1	yes	characteristic
57	Dic₉₇	—	4	1	no	—
58	C₂×C₉₇	—	2	2	no	—
59	C₂×C₉₇	—	2	1	no	—
60	C₂×C₉₇	#399	1	1	yes	characteristic
61	D₉₇	—	8	1	no	—
62	#2098	—	97	1	no	maximal, Sylow
63	C₉₇	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	97	1	no	—
65	#424	—	97	2	no	—
66	#421	—	97	2	no	—
67	#408	—	97	1	no	—
68	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic, complemented
69	C₂×#41	—	97	1	no	—
70	C₂×M₁₆	—	97	2	no	—
71	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
72	C₂²×C₈	—	97	1	no	—
73	#117	C₂×D₉₇	1	2	yes	—
74	#108	—	194	2	no	—
75	C₄⋊C₈	—	194	4	no	—
76	#104	—	194	4	no	—
77	C₂²⋊C₈	—	194	2	no	—
78	#41	—	194	2	no	—
79	#41	—	194	2	no	—
80	C₂×Q₈	—	97	1	no	—
81	C₂×D₄	—	97	1	no	—
82	C₂×D₄	—	194	1	no	—
83	C₂²×C₄	—	194	1	no	—
84	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	194	2	no	—
87	M₁₆	—	194	2	no	—
88	C₂×C₈	—	194	1	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₂×C₈	D₃₈₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₃₈₈	1	4	yes	—
95	C₂³	—	194	1	no	—
96	C₂³	#13095510	1	1	yes	characteristic
97	Q₈	—	194	2	no	—
98	D₄	—	388	2	no	—
99	D₄	—	194	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	388	1	no	—
102	C₂×C₄	—	388	1	no	—
103	C₂×C₄	—	194	1	no	—
104	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
105	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
107	C₂×C₄	#13095510	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	388	1	no	—
110	C₈	—	388	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	388	2	no	—
113	C₂²	—	388	1	no	—
114	C₂²	#15690790	1	1	yes	characteristic
115	C₂²	#15690794	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	388	1	no	—
118	C₄	#15690790	1	2	yes	—
119	C₄	#15690794	1	1	yes	characteristic
120	C₄	#15690807	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	776	1	no	—
124	C₂	#20260672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260784

C₂

yes

maximal, characteristic, complemented

#20260762

C₂

yes

maximal, complemented

#20260753

C₂

yes

maximal

#20260703

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

—

—

—

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

#13095511

—

#13095511

—

C₂×Dic₁₉₄

Q₈

yes

characteristic

C₂×D₃₈₈

Q₈

yes

characteristic

C₂×C₂²⋊D₉₇

—

C₂×C₄×D₉₇

—

—

—

—

—

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

#41

yes

characteristic

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#123

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#399

yes

characteristic

D₉₇

—

#2098

—

maximal, Sylow

C₉₇

#2098

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₉₇

yes

—

#108

—

194

—

C₄⋊C₈

—

194

—

#104

—

194

—

C₂²⋊C₈

—

194

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

D₃₈₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₃₈₈

yes

—

C₂³

—

194

—

C₂³

#13095510

yes

characteristic

Q₈

—

194

—

D₄

—

388

—

D₄

—

194

—

100

C₂×C₄

—

101

C₂×C₄

—

388

—

102

C₂×C₄

—

388

—

103

C₂×C₄

—

194

—

104

C₂×C₄

C₂×D₃₈₈

yes

—

105

C₂×C₄

C₂×D₃₈₈

yes

characteristic

106

C₂×C₄

Q₈×D₉₇

yes

characteristic

107

C₂×C₄

#13095510

yes

—

108

C₈

—

109

C₈

—

388

—

110

C₈

—

388

—

111

C₂²

—

112

C₂²

—

388

—

113

C₂²

—

388

—

114

C₂²

#15690790

yes

characteristic

115

C₂²

#15690794

yes

—

116

C₄

—

117

C₄

—

388

—

118

C₄

#15690790

yes

—

119

C₄

#15690794

yes

characteristic

120

C₄

#15690807

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

776

—

124

C₂

#20260672

yes

characteristic

	subgroup	G/H
center	C₄	#15690807
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#399
prime radical	C₂×#13095511	C₂²
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#2098
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15690807

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

H(3,ℤ/4)×C₉₇

C₂

C₂×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #20260784	C₂	semidirect
2. #20260762	C₂	semidirect
5. H(3,ℤ/4)×C₉₇	C₂	semidirect
63. C₉₇	#2098	semidirect
68. H(3,ℤ/4)	D₉₇	semidirect

H ≅ G/N

type

1. #20260784

C₂

semidirect

2. #20260762

C₂

semidirect

5. H(3,ℤ/4)×C₉₇

C₂

semidirect

63. C₉₇

#2098

semidirect

68. H(3,ℤ/4)

D₉₇

semidirect

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

subgroup	index
#20260784	2
#20260762	2
#20260753	2
#20260703	2
H(3,ℤ/4)×C₉₇	2
#2098	97

subgroup

index

H(3,ℤ/4)×C₉₇

Subgroups — #27143290

Conjugacy classes of subgroups

Complements

Maximal subgroups