Finite Group Databasebeta

← #27300349

Subgroups — #27300349

Order 13568.

670 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274633C212yesmaximal
2#20272840C214yesmaximal, complemented
3C53×#2297C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810486—22no—
7C22×#41×C53C2211yescharacteristic
8C2×#117×C53C411yescharacteristic, complemented
9C22×C42×C53C411yescharacteristic, complemented
10C53×#391C412yescomplemented
11C53×#390C2212yes—
12C53×#386—22no—
13C4×C22⋊C4×C53—22no—
14C2×C22⋊Dic53—24no—
15#13133132—28no—
16#13133132—44no—
17#13133124—44no—
18C2×#41×C53—24no—
19C2×#41×C53—42no—
20C2×#41×C53—42no—
21C22×Q8×C53C2×C411yescharacteristic
22C22×D4×C53D412yes—
23C22×D4×C53C2311yescharacteristic
24C23×C4×C53—21no—
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53D412yes—
27#117×C53—42no—
28C2×C4⋊C4×C53C2×C412yes—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53—24no—
31C2×C42×C53—21no—
32C2×C42×C53—21no—
33C2×C42×C53—42no—
34C2×C42×C53C2×C412yes—
35#95×C53—24no—
36C24×C53C22⋊C411yescharacteristic
37C24×C53C2×D412yes—
38#41×C53—416no—
39C2×Q8×C53—22no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—42no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53#4114yes—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—24no—
53C22×C4×C53—24no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—44no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—44no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—48no—
64C22⋊C4×C53—42no—
65C42×C53—42no—
66C42×C53—42no—
67C42×C53—42no—
68C22⋊Dic53—44no—
69C22⋊Dic53—44no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—42no—
75C23×C53—44no—
76C23×C53#9912yes—
77C23×C53C2×C22⋊C411yescharacteristic
78C23×C53#12314yes—
79Q8×C53—44no—
80D4×C53—44no—
81D4×C53—48no—
82C2×C4×C53—28no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—48no—
92C2×C4×C53—42no—
93C2×C4×C53—48no—
94C2×C4×C53—42no—
95C2×C4×C53—44no—
96C2×C4×C53—42no—
97C2×C4×C53#9912yes—
98C2×C4×C53#11712yes—
99C2×Dic53—84no—
100C2×Dic53—88no—
101#12857—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53—24no—
104C22×C53—22no—
105C22×C53—42no—
106C22×C53—44no—
107C22×C53—42no—
108C22×C53—44no—
109C22×C53—42no—
110C22×C53—82no—
111C22×C53#38111yescharacteristic
112C22×C53C2×#9911yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—44no—
117C4×C53—42no—
118C4×C53—82no—
119Dic53—164no—
120#2297D5311yescharacteristic
121#1782—532no—
122#1393—534no—
123C2×C53—21no—
124C2×C53—21no—
125C2×C53—44no—
126C2×C53—42no—
127C2×C53#181911yescharacteristic
128C2×C53#212612yes—
129C22×#41C2×D5311yescharacteristic
130C2×#117Dic5311yescharacteristic, complemented
131C22×C42Dic5311yescharacteristic, complemented
132C2×#99—534no—
133C2×#99—1062no—
134#391Dic5312yescomplemented
135#390C2×D5312yes—
136#386—22no—
137#381—1062no—
138C4×C22⋊C4—22no—
139C53#1285711yescharacteristic, complemented, Sylow
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8C2×Dic5311yescharacteristic
144C22×D4C22⋊D5312yes—
145C22×D4C22×D5311yescharacteristic
146C23×C4—21no—
147C23×C4C2×Dic5311yescharacteristic
148C23×C4C22⋊D5312yes—
149#117—42no—
150C2×C4⋊C4C2×Dic5312yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—1064no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C2×Dic5312yes—
158#99—1068no—
159#99—2124no—
160#95—24no—
161#95—2124no—
162C24C22⋊Dic5311yescharacteristic
163C24C2×C22⋊D5312yes—
164#41—416no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4#1171994514yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—2124no—
185C22×C4C22⋊Dic5312yes—
186C22×C4C22⋊Dic5311yescharacteristic
187C22×C4C22×Dic5311yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—2124no—
192C22⋊C4—2124no—
193C42—42no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—42no—
201C23—44no—
202C23#1313313212yes—
203C23C2×C22⋊Dic5311yescharacteristic
204C23#1313320014yes—
205Q8—44no—
206D4—44no—
207D4—48no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—4244no—
224C2×C4—4248no—
225C2×C4#1313313212yes—
226C2×C4#1313317612yes—
227C22—22no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22#1681048611yescharacteristic
237C22C2×#1313313211yescharacteristic
238C22C2×#1313313211yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—42no—
243C4—82no—
244C4—8484no—
245C2—21no—
246C2—21no—
247C2—44no—
248C2—42no—
249C2#2027478911yescharacteristic
250C2#2027638012yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2297C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#2297D53
hypocenterC53#12857
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #20272840C2semidirect
8. C2×#117×C53C4semidirect
9. C22×C42×C53C4semidirect
10. C53×#391C4semidirect
130. C2×#117Dic53semidirect
131. C22×C42Dic53semidirect
134. #391Dic53semidirect
139. C53#12857semidirect

Maximal subgroups

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

subgroupindex
#202746332
#202728402
C53×#22972
#1285753