Finite Group Databasebeta

← #32006905

Subgroups — #32006905

Order 18176.

1130 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590938C212yesmaximal
2#22589326C218yesmaximal, complemented
3#22588414C214yesmaximal
4C2×#543×C71C211yesmaximal, characteristic
5#17985892—28no—
6#17985859C2214yes—
7#17985855C2214yes—
8C4×C22⋊Dic71C2214yes—
9#17985766C22116yes—
10#543×C71—28no—
11C2×C22⋊Q8×C71C418yescomplemented
12C2×C22⋊Q8×C71C2214yes—
13C22×C22⋊C4×C71C2212yes—
14C22×C22⋊C4×C71C2211yescharacteristic
15C2×C4⋊Dic71—24no—
16C2×C22⋊Dic71—24no—
17C2×C22⋊Dic71—24no—
18C2×C22⋊Dic71C2318yes—
19C2×C4×Dic71—24no—
20C71⋊#95—24no—
21C71⋊#95—24no—
22C25×C71C2311yescharacteristic
23C22×Q8×C71C2×C414yes—
24C23×C4×C71C2×C414yes—
25C23×C4×C71C2312yes—
26C22⋊Q8×C71—28no—
27C22⋊Q8×C71—28no—
28C22⋊Q8×C71—232no—
29C2×C4⋊C4×C71C2×C4112yes—
30C2×C22⋊C4×C71—28no—
31C2×C22⋊C4×C71C2×C414yes—
32C2×C22⋊C4×C71C2×C414yes—
33C2×C22⋊C4×C71D418yes—
34C2×C22⋊C4×C71C2314yes—
35C22×Dic71—28no—
36C4⋊Dic71—84no—
37C22⋊Dic71—216no—
38C22⋊Dic71—84no—
39C22⋊Dic71—84no—
40C4×Dic71—84no—
41C24×C71—24no—
42C24×C71—82no—
43C24×C71C22×C412yes—
44C24×C71C2×D414yes—
45C24×C71C2411yescharacteristic
46C2×Q8×C71—28no—
47C2×Q8×C71—84no—
48C22×C4×C71—28no—
49C22×C4×C71—82no—
50C22×C4×C71—84no—
51C22×C4×C71C22×C418yes—
52C22×C4×C71C22×C414yes—
53C22×C4×C71C2×D414yes—
54C22×C4×C71C2×D414yes—
55C4⋊C4×C71—224no—
56C22⋊C4×C71—28no—
57C22⋊C4×C71—28no—
58C22⋊C4×C71C22⋊C4116yes—
59C23×C71—24no—
60C23×C71—28no—
61C23×C71—44no—
62C23×C71—42no—
63C23×C71—44no—
64C23×C71—44no—
65C23×C71—88no—
66C23×C71C2×C22⋊C414yes—
67C23×C71C23×C411yescharacteristic
68C23×C71C22×D412yes—
69C23×C712−1+414yes—
70Q8×C71—88no—
71C2×C4×C71—28no—
72C2×C4×C71—44no—
73C2×C4×C71—44no—
74C2×C4×C71—44no—
75C2×C4×C71—48no—
76C2×C4×C71—88no—
77C2×C4×C71C2×C22⋊C418yes—
78C2×Dic71—48no—
79C2×Dic71—48no—
80C2×Dic71—48no—
81C22×C71—22no—
82C22×C71—22no—
83C22×C71—22no—
84C22×C71—24no—
85C22×C71—28no—
86C22×C71—44no—
87C22×C71—48no—
88C22×C71—88no—
89C22×C71C22×C22⋊C411yescharacteristic
90C22×C71#51912yes—
91C22×C71#53614yes—
92C4×C71—28no—
93C4×C71—48no—
94Dic71—88no—
95#21827—711nomaximal, Sylow
96C2×C71—24no—
97C2×C71—22no—
98C2×C71—44no—
99C2×C71#231011yescharacteristic
100C2×C71#288312yes—
101C2×#543D7111yescharacteristic
102#2312—712no—
103#1996—718no—
104#1819—714no—
105C71#2182711yescharacteristic, complemented, Sylow
106#543—28no—
107C2×C22⋊Q8Dic7118yescomplemented
108C2×C22⋊Q8C2×D7114yes—
109C22×C22⋊C4C2×D7112yes—
110C22×C22⋊C4C2×D7111yescharacteristic
111#388—1428no—
112#380—714no—
113#379—714no—
114C4×C22⋊C4—714no—
115#342—7116no—
116C25C22×D7111yescharacteristic
117C22×Q8C2×Dic7114yes—
118C23×C4C2×Dic7114yes—
119C23×C4C22×D7112yes—
120C22⋊Q8—28no—
121C22⋊Q8—28no—
122C22⋊Q8—232no—
123C2×C4⋊C4—1424no—
124C2×C4⋊C4C2×Dic71112yes—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—718no—
127C2×C22⋊C4—1424no—
128C2×C22⋊C4—1424no—
129C2×C22⋊C4C2×Dic7114yes—
130C2×C22⋊C4C2×Dic7114yes—
131C2×C22⋊C4C22⋊D7118yes—
132C2×C22⋊C4C22×D7114yes—
133C2×C42—1424no—
134#95—1424no—
135#95—1424no—
136C24—24no—
137C24—82no—
138C24C22×Dic7112yes—
139C24C2×C22⋊D7114yes—
140C24C23×D7111yescharacteristic
141C2×Q8—28no—
142C2×Q8—84no—
143C22×C4—28no—
144C22×C4—82no—
145C22×C4—84no—
146C22×C4—1428no—
147C22×C4C22×Dic7118yes—
148C22×C4C22×Dic7114yes—
149C22×C4C2×C22⋊D7114yes—
150C22×C4C2×C22⋊D7114yes—
151C4⋊C4—224no—
152C4⋊C4—5684no—
153C22⋊C4—28no—
154C22⋊C4—28no—
155C22⋊C4—5684no—
156C22⋊C4—5684no—
157C22⋊C4—14216no—
158C22⋊C4C22⋊Dic71116yes—
159C42—5684no—
160C23—24no—
161C23—28no—
162C23—44no—
163C23—42no—
164C23—44no—
165C23—44no—
166C23—88no—
167C23C2×C22⋊Dic7114yes—
168C23C23×Dic7111yescharacteristic
169C23C22×C22⋊D7112yes—
170C23#1455617614yes—
171Q8—88no—
172C2×C4—28no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—88no—
178C2×C4—2848no—
179C2×C4—2848no—
180C2×C4—2848no—
181C2×C4C2×C22⋊Dic7118yes—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—28no—
186C22—24no—
187C22—44no—
188C22—48no—
189C22—88no—
190C22C22×C22⋊Dic7111yescharacteristic
191C22#1798647712yes—
192C22#1798657314yes—
193C4—28no—
194C4—48no—
195C4—5688no—
196C2—24no—
197C2—22no—
198C2—44no—
199C2#2259092611yescharacteristic
200C2#2259680112yes—
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC2×#543×C71C2
socleC22×C71C22×C22⋊C4
prime radicalC25×C71C23
cosocleC24×C71C24
hypercenterC2×#543D71
hypocenterC71#21827
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ C2×#543

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22589326C2semidirect
11. C2×C22⋊Q8×C71C4semidirect
105. C71#21827semidirect
107. C2×C22⋊Q8Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909382
#225893262
#225884142
C2×#543×C712
#2182771