Group #27141387 — subgroups

Subgroups — #27141387

Order 12416.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20261138	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261087	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261086	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261081	C₂	1	1	yes	maximal, characteristic
5	#20261081	C₂	1	1	yes	maximal, characteristic
6	#20260457	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₉₇	C₂	1	1	yes	maximal, characteristic
8	#15690810	C₂²	1	1	yes	characteristic
9	#15690810	C₂²	1	1	yes	characteristic
10	#15690743	—	2	2	no	—
11	#15690735	—	2	2	no	—
12	#15690733	—	2	2	no	—
13	#15690731	—	2	2	no	—
14	#15690731	—	2	2	no	—
15	C₄×#11715895	C₂²	1	1	yes	characteristic
16	C₄×#11715895	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₉₇	—	2	2	no	—
18	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
19	#127×C₉₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉₄	—	2	1	no	—
22	C₂×Dic₁₉₄	—	2	1	no	—
23	C₂×#11715895	D₄	1	2	yes	—
24	C₂×#11715895	D₄	1	2	yes	—
25	C₄⋊Dic₉₇	—	2	1	no	—
26	C₄⋊Dic₉₇	—	2	1	no	—
27	C₄⋊Dic₉₇	—	2	1	no	—
28	C₄⋊Dic₉₇	—	2	1	no	—
29	C₂×Q₈×C₉₇	—	2	1	no	—
30	C₂×D₄×C₉₇	—	2	1	no	—
31	C₂×D₄×C₉₇	—	2	1	no	—
32	C₂×D₄×C₉₇	—	2	1	no	—
33	SD₁₆×C₉₇	—	4	4	no	—
34	C₂×C₈×C₉₇	D₄	1	2	yes	—
35	C₄⋊C₄×C₉₇	—	2	1	no	—
36	C₄⋊C₄×C₉₇	—	2	1	no	—
37	C₄²×C₉₇	C₂³	1	1	yes	characteristic
38	C₂³×C₉₇	—	4	1	no	—
39	Q₈×C₉₇	—	4	2	no	—
40	D₄×C₉₇	—	4	2	no	—
41	D₄×C₉₇	—	4	2	no	—
42	D₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	—	4	1	no	—
44	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₉₇	—	4	1	no	—
48	C₂×Dic₉₇	—	4	1	no	—
49	Dic₁₉₄	—	4	2	no	—
50	Dic₁₉₄	—	4	2	no	—
51	C₈×C₉₇	—	2	2	no	—
52	#11715895	—	2	2	no	—
53	#11715895	—	2	2	no	—
54	C₂²×C₉₇	—	8	1	no	—
55	C₂²×C₉₇	—	8	1	no	—
56	C₂²×C₉₇	—	8	1	no	—
57	C₂²×C₉₇	#127	1	1	yes	characteristic
58	C₄×C₉₇	—	8	1	no	—
59	C₄×C₉₇	#135	1	2	yes	—
60	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₉₇	#137	1	2	yes	—
62	Dic₉₇	—	8	1	no	—
63	Dic₉₇	—	8	1	no	—
64	C₂×C₉₇	—	16	1	no	—
65	C₂×C₉₇	#495	1	1	yes	characteristic
66	C₂×C₉₇	#496	1	1	yes	characteristic
67	C₂×C₉₇	#497	1	1	yes	characteristic
68	#1719	—	97	1	no	maximal, Sylow
69	C₉₇	#1719	1	1	yes	characteristic, complemented, Sylow
70	#500	—	97	1	no	—
71	#492	D₉₇	1	1	yes	characteristic
72	#487	—	97	1	no	—
73	#487	—	97	1	no	—
74	#486	—	97	1	no	—
75	#486	—	97	1	no	—
76	#322	—	97	1	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	—	97	1	no	—
79	C₄⋊Q₈	—	97	1	no	—
80	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
81	#127	C₂×D₉₇	1	1	yes	characteristic
82	#107	—	194	2	no	—
83	#103	—	194	2	no	—
84	#103	—	194	2	no	—
85	#102	—	194	2	no	—
86	#102	—	194	2	no	—
87	C₄×C₈	—	97	1	no	—
88	C₄×C₈	—	97	1	no	—
89	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	194	1	no	—
92	C₂×Q₈	—	194	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	97	2	no	—
98	C₂×C₈	—	97	2	no	—
99	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	194	1	no	—
103	C₄⋊C₄	—	194	1	no	—
104	C₄⋊C₄	—	194	1	no	—
105	C₄⋊C₄	—	194	1	no	—
106	C₄²	C₂²×D₉₇	1	1	yes	characteristic
107	C₂³	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	388	2	no	—
110	Q₈	—	388	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	388	1	no	—
116	C₂×C₄	—	388	1	no	—
117	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	194	2	no	—
122	C₈	—	194	2	no	—
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	#15690806	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	776	1	no	—
129	C₄	—	776	1	no	—
130	C₄	#15690831	1	2	yes	—
131	C₄	#15690836	1	2	yes	—
132	C₄	#15690839	1	2	yes	—
133	C₂	—	16	1	no	—
134	C₂	#20261116	1	1	yes	characteristic
135	C₂	#20261122	1	1	yes	characteristic
136	C₂	#20261126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261138

C₂

yes

maximal, characteristic, complemented

#20261087

C₂

yes

maximal, characteristic, complemented

#20261086

C₂

yes

maximal, characteristic, complemented

#20261081

C₂

yes

maximal, characteristic

#20261081

C₂

yes

maximal, characteristic

#20260457

C₂

yes

maximal, characteristic, complemented

#492×C₉₇

C₂

yes

maximal, characteristic

#15690810

C₂²

yes

characteristic

#15690810

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#11715895

C₂²

yes

characteristic

C₂×SD₁₆×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×#11715895

D₄

yes

—

C₂×#11715895

D₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

SD₁₆×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#127

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#135

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

#137

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#500

—

#492

D₉₇

yes

—

—

—

—

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

#127

C₂×D₉₇

yes

characteristic

#107

—

194

—

#103

—

194

—

#103

—

194

—

#102

—

194

—

#102

—

194

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₉₇

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

194

—

103

C₄⋊C₄

—

194

—

104

C₄⋊C₄

—

194

—

105

C₄⋊C₄

—

194

—

106

C₄²

C₂²×D₉₇

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

388

—

110

Q₈

—

388

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

388

—

116

C₂×C₄

—

388

—

117

C₂×C₄

D₄×D₉₇

yes

characteristic

118

C₂×C₄

D₄×D₉₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

120

C₈

—

121

C₈

—

194

—

122

C₈

—

194

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#15690806

yes

characteristic

127

C₄

—

128

C₄

—

776

—

129

C₄

—

776

—

130

C₄

#15690831

yes

—

131

C₄

#15690836

yes

—

132

C₄

#15690839

yes

—

133

C₂

—

134

C₂

#20261116

yes

characteristic

135

C₂

#20261122

yes

characteristic

136

C₂

#20261126

yes

characteristic

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#492×C₉₇	C₂
socle	C₂²×C₉₇	#127
prime radical	#127×C₉₇	C₂²
hypercenter	#492	D₉₇
hypocenter	C₉₇	#1719
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15690806

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#492×C₉₇

C₂

socle

C₂²×C₉₇

#127

prime radical

#127×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20261138	C₂	semidirect
2. #20261087	C₂	semidirect
3. #20261086	C₂	semidirect
6. #20260457	C₂	semidirect
69. C₉₇	#1719	semidirect

H ≅ G/N

type

1. #20261138

C₂

semidirect

2. #20261087

C₂

semidirect

3. #20261086

C₂

semidirect

6. #20260457

C₂

semidirect

69. C₉₇

#1719

semidirect

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

subgroup	index
#20261138	2
#20261087	2
#20261086	2
#20261081	2
#20260457	2
#492×C₉₇	2
#1719	97

subgroup

index

#492×C₉₇

Subgroups — #27141387

Conjugacy classes of subgroups

Complements

Maximal subgroups