Group #27142021 — subgroups

Subgroups — #27142021

Order 12416.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20261502	C₂	1	1	yes	maximal, characteristic
2	#20260761	C₂	1	2	yes	maximal, complemented
3	#20260709	C₂	1	1	yes	maximal, characteristic
4	#20260507	C₂	1	2	yes	maximal, complemented
5	#408×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690817	—	2	4	no	—
7	C₂×#13095495	—	2	2	no	—
8	C₂×#13095495	C₂²	1	2	yes	—
9	#15690769	C₂²	1	2	yes	—
10	#15690739	—	2	4	no	—
11	#15690725	C₄	1	4	yes	—
12	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₉₇	—	2	2	no	—
16	#13095495	—	2	4	no	—
17	#13095495	—	2	2	no	—
18	#13095495	—	2	2	no	—
19	C₂×#11715895	—	2	4	no	—
20	C₂×#11715895	D₄	1	4	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
23	C₄×Dic₉₇	—	2	2	no	—
24	#41×C₉₇	—	2	2	no	—
25	#41×C₉₇	—	2	2	no	—
26	C₂×Q₈×C₉₇	Q₈	1	1	yes	characteristic
27	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₉₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	—	2	1	no	—
33	C₂×C₈×C₉₇	—	2	1	no	—
34	C₂×C₈×C₉₇	—	2	2	no	—
35	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₉₇	—	2	2	no	—
38	D₄×C₉₇	—	2	4	no	—
39	D₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂×Dic₉₇	—	2	4	no	—
48	C₈×C₉₇	—	4	1	no	—
49	C₈×C₉₇	—	4	1	no	—
50	#11715895	—	2	4	no	—
51	C₂²×C₉₇	—	2	2	no	—
52	C₂²×C₉₇	—	4	1	no	—
53	C₂²×C₉₇	—	4	2	no	—
54	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₉₇	C₄×D₄	1	2	yes	—
56	C₄×C₉₇	—	2	2	no	—
57	C₄×C₉₇	#120	1	1	yes	characteristic
58	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₉₇	#123	1	1	yes	characteristic
60	Dic₉₇	—	4	4	no	—
61	C₂×C₉₇	—	2	2	no	—
62	C₂×C₉₇	—	2	1	no	—
63	C₂×C₉₇	—	4	2	no	—
64	C₂×C₉₇	#385	1	1	yes	characteristic
65	#1868	—	97	1	no	maximal, Sylow
66	C₉₇	#1868	1	1	yes	characteristic, complemented, Sylow
67	#568	—	97	1	no	—
68	#424	—	97	2	no	—
69	#410	—	97	1	no	—
70	#408	D₉₇	1	1	yes	characteristic
71	#344	—	97	2	no	—
72	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
73	#131	—	194	4	no	—
74	C₂×M₁₆	—	97	2	no	—
75	C₂×M₁₆	—	194	2	no	—
76	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
78	#117	—	97	2	no	—
79	C₄⋊C₈	—	194	4	no	—
80	#99	—	97	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₁₉₄	1	1	yes	characteristic
85	C₂×D₄	Dic₁₉₄	1	1	yes	characteristic
86	C₂×D₄	C₄×D₉₇	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	194	4	no	—
91	M₁₆	—	194	2	no	—
92	M₁₆	—	194	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	97	4	no	—
97	C₂×C₈	—	194	4	no	—
98	C₄⋊C₄	—	194	2	no	—
99	C₂²⋊C₄	—	97	4	no	—
100	C₄²	—	194	2	no	—
101	C₂³	#13095490	1	2	yes	—
102	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	194	4	no	—
109	C₂×C₄	#13095490	1	2	yes	—
110	C₂×C₄	D₄×D₉₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
113	C₂×C₄	#13095510	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	194	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#13095490	1	1	yes	characteristic
121	C₂²	#15690770	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	388	4	no	—
124	C₄	#15690778	1	1	yes	characteristic
125	C₄	#15690787	1	2	yes	—
126	C₄	#15690791	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#20260607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261502

C₂

yes

maximal, characteristic

#20260761

C₂

yes

maximal, complemented

#20260709

C₂

yes

maximal, characteristic

#20260507

C₂

yes

maximal, complemented

#408×C₉₇

C₂

yes

maximal, characteristic

#15690817

—

C₂×#13095495

—

C₂×#13095495

C₂²

yes

—

#15690769

C₂²

yes

—

#15690739

—

#15690725

C₄

yes

—

C₂×#41×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²⋊C₈×C₉₇

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

Q₈

yes

characteristic

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂³×C₉₇

C₄⋊C₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

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

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#123

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic

#344

—

C₂×#41

C₂×D₉₇

yes

characteristic

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

194

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

C₄⋊C₈

—

194

—

#99

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₁₉₄

yes

characteristic

C₂×D₄

Dic₁₉₄

yes

characteristic

C₂×D₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

100

C₄²

—

194

—

101

C₂³

#13095490

yes

—

102

C₂³

C₂×C₄×D₉₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

194

—

109

C₂×C₄

#13095490

yes

—

110

C₂×C₄

D₄×D₉₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

112

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

113

C₂×C₄

#13095510

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

194

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#13095490

yes

characteristic

121

C₂²

#15690770

yes

—

122

C₄

—

123

C₄

—

388

—

124

C₄

#15690778

yes

characteristic

125

C₄

#15690787

yes

—

126

C₄

#15690791

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20260607

yes

characteristic

	subgroup	G/H
center	C₄	#15690778
commutator	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#408×C₉₇	C₂
socle	C₂×C₉₇	#385
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#408	D₉₇
hypocenter	C₉₇	#1868
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#15690778

commutator

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#408×C₉₇

C₂

socle

C₂×C₉₇

#385

prime radical

C₂×#41×C₉₇

C₂²

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #20260761	C₂	semidirect
4. #20260507	C₂	semidirect
66. C₉₇	#1868	semidirect

H ≅ G/N

type

2. #20260761

C₂

semidirect

4. #20260507

C₂

semidirect

66. C₉₇

#1868

semidirect

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

subgroup	index
#20261502	2
#20260761	2
#20260709	2
#20260507	2
#408×C₉₇	2
#1868	97

subgroup

index

#408×C₉₇

Subgroups — #27142021

Conjugacy classes of subgroups

Complements

Maximal subgroups