Group #27142693 — subgroups

Subgroups — #27142693

Order 12416.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#15690802	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	2	yes	maximal
4	#20260497	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690802	—	2	2	no	—
8	#15690802	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×#13095490	—	2	1	no	—
12	C₂×#13095490	—	2	1	no	—
13	C₄⋊#11715895	—	2	2	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	—	2	1	no	—
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	#95×C₉₇	—	2	1	no	—
18	C₂²×Dic₉₇	—	2	1	no	—
19	C₂²×Dic₉₇	—	2	1	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₂×#11715895	—	2	4	no	—
22	C₄⋊Dic₉₇	—	2	1	no	—
23	C₄⋊Dic₉₇	—	2	1	no	—
24	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
26	#13095490	—	2	4	no	—
27	#13095490	—	2	4	no	—
28	C₂²×C₄×C₉₇	—	2	2	no	—
29	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₉₇	—	2	2	no	—
33	C₄⋊C₄×C₉₇	Q₈	1	4	yes	—
34	C₄²×C₉₇	—	4	2	no	—
35	C₄²×C₉₇	C₂×C₄	1	2	yes	—
36	C₄²×C₉₇	D₄	1	2	yes	—
37	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	—	4	2	no	—
44	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₉₇	#41	1	2	yes	—
50	C₂×Dic₉₇	—	2	2	no	—
51	C₂×Dic₉₇	—	2	2	no	—
52	C₂×Dic₉₇	—	4	2	no	—
53	C₂×Dic₉₇	—	4	1	no	—
54	C₂×Dic₉₇	—	4	1	no	—
55	#11715895	—	4	2	no	—
56	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₉₇	#135	1	2	yes	—
59	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₉₇	#137	1	1	yes	characteristic
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	2	2	no	—
63	C₄×C₉₇	—	4	4	no	—
64	C₄×C₉₇	C₄×Q₈	1	2	yes	—
65	C₄×C₉₇	#124	1	1	yes	characteristic
66	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₉₇	—	4	2	no	—
68	Dic₉₇	—	4	2	no	—
69	C₂×C₉₇	#391	1	1	yes	characteristic
70	C₂×C₉₇	#416	1	1	yes	characteristic
71	C₂×C₉₇	#417	1	1	yes	characteristic
72	C₂×C₉₇	#478	1	2	yes	—
73	C₂×C₉₇	#479	1	2	yes	—
74	#1989	—	97	1	no	maximal, Sylow
75	C₉₇	#1989	1	1	yes	characteristic, complemented, Sylow
76	C₂×#125	—	97	1	no	—
77	C₂×C₄⋊C₈	—	97	1	no	—
78	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
79	#340	—	97	2	no	—
80	#340	—	97	2	no	—
81	C₂²×C₈	—	97	2	no	—
82	#125	—	97	4	no	—
83	#125	—	194	2	no	—
84	C₂×C₄⋊C₄	—	97	1	no	—
85	C₂×C₄⋊C₄	—	97	1	no	—
86	C₂×C₄⋊C₄	—	194	1	no	—
87	C₂×C₄⋊C₄	—	194	1	no	—
88	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
91	C₄⋊C₈	—	194	2	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	194	1	no	—
95	C₂²×C₄	—	194	1	no	—
96	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	C₂×C₈	—	194	2	no	—
100	C₂×C₈	—	194	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	97	2	no	—
103	C₄⋊C₄	—	97	2	no	—
104	C₄⋊C₄	—	194	4	no	—
105	C₄⋊C₄	—	194	4	no	—
106	C₄⋊C₄	—	194	1	no	—
107	C₄⋊C₄	—	194	1	no	—
108	C₄⋊C₄	Dic₁₉₄	1	4	yes	—
109	C₄²	—	4	2	no	—
110	C₄²	C₄×D₉₇	1	2	yes	—
111	C₄²	D₃₈₈	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	388	2	no	—
120	C₂×C₄	—	388	1	no	—
121	C₂×C₄	—	388	1	no	—
122	C₂×C₄	—	194	2	no	—
123	C₂×C₄	—	194	2	no	—
124	C₂×C₄	#13095488	1	2	yes	—
125	C₂×C₄	#13095488	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
127	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
129	C₂×C₄	#13095511	1	2	yes	—
130	C₈	—	388	2	no	—
131	C₂²	C₂×#13095488	1	1	yes	characteristic
132	C₂²	#15690789	1	2	yes	—
133	C₂²	#15690830	1	2	yes	—
134	C₂²	#15690836	1	1	yes	characteristic
135	C₂²	#15690841	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	388	2	no	—
140	C₄	—	388	2	no	—
141	C₄	C₄×Dic₁₉₄	1	2	yes	—
142	C₄	#15690798	1	1	yes	characteristic
143	C₄	#15690810	1	1	yes	characteristic
144	C₂	#20260641	1	1	yes	characteristic
145	C₂	#20260729	1	1	yes	characteristic
146	C₂	#20260734	1	1	yes	characteristic
147	C₂	#20261035	1	2	yes	—
148	C₂	#20261042	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#20260497

C₂

yes

maximal

C₄×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690802

—

#15690802

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

#13095490

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

Q₈

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂²×C₉₇

#135

yes

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#479

yes

—

#1989

—

maximal, Sylow

C₉₇

#1989

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₉₇

yes

characteristic

#340

—

#340

—

C₂²×C₈

—

#125

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

100

C₂×C₈

—

194

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

194

—

105

C₄⋊C₄

—

194

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

—

194

—

108

C₄⋊C₄

Dic₁₉₄

yes

—

109

C₄²

—

110

C₄²

C₄×D₉₇

yes

—

111

C₄²

D₃₈₈

yes

—

112

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

388

—

120

C₂×C₄

—

388

—

121

C₂×C₄

—

388

—

122

C₂×C₄

—

194

—

123

C₂×C₄

—

194

—

124

C₂×C₄

#13095488

yes

—

125

C₂×C₄

#13095488

yes

—

126

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

127

C₂×C₄

C₂×D₃₈₈

yes

characteristic

128

C₂×C₄

C₂×Dic₁₉₄

yes

—

129

C₂×C₄

#13095511

yes

—

130

C₈

—

388

—

131

C₂²

C₂×#13095488

yes

characteristic

132

C₂²

#15690789

yes

—

133

C₂²

#15690830

yes

—

134

C₂²

#15690836

yes

characteristic

135

C₂²

#15690841

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

388

—

140

C₄

—

388

—

141

C₄

C₄×Dic₁₉₄

yes

—

142

C₄

#15690798

yes

characteristic

143

C₄

#15690810

yes

characteristic

144

C₂

#20260641

yes

characteristic

145

C₂

#20260729

yes

characteristic

146

C₂

#20260734

yes

characteristic

147

C₂

#20261035

yes

—

148

C₂

#20261042

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#1989
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₄×C₄⋊C₄×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₉₇

hypocenter

C₉₇

#1989

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #15690802	C₄	semidirect
75. C₉₇	#1989	semidirect

H ≅ G/N

type

8. #15690802

C₄

semidirect

75. C₉₇

#1989

semidirect

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

subgroup	index
C₂×#15690802	2
C₂×C₄⋊#11715895	2
#20260497	2
C₄×C₄⋊C₄×C₉₇	2
#1989	97

subgroup

index

C₄×C₄⋊C₄×C₉₇

Subgroups — #27142693

Conjugacy classes of subgroups

Complements

Maximal subgroups