Finite Group Databasebeta

← #27287084

Subgroups — #27287084

Order 13568.

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

normal subgroups only

This group has 640 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.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274649C212yesmaximal
2#20272985C211yesmaximal, characteristic
3#20272593C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#2646C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133140—22no—
8C2×#13133140—22no—
9C2×C22⋊#88322C2212yes—
10C2×#13133124—22no—
11C2×#13133124C2211yescharacteristic
12C53×#400—22no—
13C53×#398—22no—
14C53×#385—22no—
15C53×#380C2212yes—
16C2×#95×C53C2211yescharacteristic
17C22×Dic106—22no—
18C22×Dic106—22no—
19C22×Dic106—21no—
20C22×Dic106—44no—
21C22×Dic106Q812yes—
22C23×Dic53—21no—
23C23×Dic53—21no—
24C22×#88322—22no—
25#13133140—44no—
26#13133140—44no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30#13133124—44no—
31C23×C4×C53D412yes—
32C23×C4×C53C2311yescharacteristic
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53D412yes—
37C2×C22⋊C4×C53—22no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42C24×C53C2×D411yescharacteristic
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—42no—
49C22×C4×C53—42no—
50C22×C4×C53—42no—
51C22×C4×C53C2×D412yes—
52C22×C4×C53C2×Q811yescharacteristic
53C22×C4×C53#4112yes—
54C22×C4×C53#4111yescharacteristic
55C22×Dic53—24no—
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53—42no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53—42no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—42no—
65C2×Dic106—24no—
66C2×Dic106—24no—
67C2×Dic106—24no—
68C2×Dic106—44no—
69C2×Dic106—48no—
70C2×Dic106—44no—
71C2×Dic106—42no—
72C2×Dic106—42no—
73C2×Dic106—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—82no—
77C4⋊C4×C53—82no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—22no—
86C23×C53#12011yescharacteristic
87C23×C53C22⋊Q811yescharacteristic
88C23×C53#12411yescharacteristic
89C23×C53#13512yes—
90C23×C53#13712yes—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—44no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—82no—
107C2×C4×C53C22⋊Q812yes—
108C2×C4×C53#12412yes—
109C2×Dic53—24no—
110C2×Dic53—42no—
111C2×Dic53—44no—
112C2×Dic53—42no—
113C2×Dic53—44no—
114C2×Dic53—44no—
115C2×Dic53—42no—
116C2×Dic53—82no—
117C2×Dic53—81no—
118C2×Dic53—81no—
119Dic106—28no—
120Dic106—44no—
121Dic106—44no—
122Dic106—44no—
123#88322—82no—
124#10345—531nomaximal, Sylow
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#39311yescharacteristic
134C22×C53#45212yes—
135C22×C53#45511yescharacteristic
136C22×C53#45611yescharacteristic
137C22×C53B(4,𝔽2)12yes—
138C22×C53#47912yes—
139C22×C53#48811yescharacteristic
140C22×C53#49111yescharacteristic
141C4×C53—22no—
142C4×C53—22no—
143C4×C53—82no—
144C4×C53—82no—
145C4×C53—82no—
146Dic53—44no—
147Dic53—82no—
148#2646D5311yescharacteristic
149#1787—532no—
150#1446—531no—
151#1294—532no—
152#1280—531no—
153C2×C53—21no—
154C2×C53—22no—
155C2×C53—21no—
156C2×C53#160312yes—
157C2×C53#160612yes—
158C2×C53#202611yescharacteristic
159C2×C53#202811yescharacteristic
160C2×C53#202911yescharacteristic
161C23×Q8—531no—
162C2×#103—1062no—
163C2×#103—1062no—
164C2×C22⋊C8—532no—
165#400—22no—
166#398—22no—
167#385—22no—
168#380C2×D5312yes—
169C2×#95—531no—
170C2×#95—1062no—
171C2×#95C2×D5311yescharacteristic
172C53#1034511yescharacteristic, complemented, Sylow
173C22×Q8—532no—
174C22×Q8—1061no—
175C22×Q8—1062no—
176C22×Q8—1062no—
177C22×Q8—2124no—
178C23×C4—1061no—
179C23×C4—1061no—
180C23×C4C22⋊D5312yes—
181C23×C4C22×D5311yescharacteristic
182C22×C8—1062no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊D5312yes—
186C2×C4⋊C4C22⋊D5312yes—
187C2×C22⋊C4—22no—
188#103—2124no—
189#103—2124no—
190C22⋊C8—1062no—
191C22⋊C8—1062no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—1062no—
197#95—2124no—
198C24C2×C22⋊D5311yescharacteristic
199C2×Q8—1064no—
200C2×Q8—1064no—
201C2×Q8—1064no—
202C2×Q8—2122no—
203C2×Q8—2122no—
204C2×Q8—2124no—
205C2×Q8—2124no—
206C2×Q8—2128no—
207C2×Q8—2122no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—1064no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—2122no—
220C22×C4—2121no—
221C22×C4—2121no—
222C22×C4—2122no—
223C22×C4—2122no—
224C22×C4—2121no—
225C22×C4—2121no—
226C22×C4Q8×D5311yescharacteristic
227C22×C4#1171994912yes—
228C22×C4#1171994911yescharacteristic
229C22×C4C2×C22⋊D5312yes—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1313318611yescharacteristic
243C23#1313319311yescharacteristic
244C23#1313320611yescharacteristic
245C23#1313323712yes—
246C23#1313324712yes—
247Q8—1068no—
248Q8—2124no—
249Q8—2124no—
250Q8—2124no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—1064no—
268C2×C4—4242no—
269C2×C4—4241no—
270C2×C4—4241no—
271C2×C4—2124no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2124no—
275C2×C4—2122no—
276C2×C4—2124no—
277C2×C4#1313319312yes—
278C2×C4#1313320612yes—
279C8—4242no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22#1681053411yescharacteristic
289C22#1681078512yes—
290C22#1681080111yescharacteristic
291C22#1681080611yescharacteristic
292C22#1681081012yes—
293C22#1681092712yes—
294C22#1681098011yescharacteristic
295C22#1681099311yescharacteristic
296C4—22no—
297C4—22no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—4242no—
302C4—2124no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#2027367812yes—
307C2#2027369512yes—
308C2#2027586311yescharacteristic
309C2#2027587311yescharacteristic
310C2#2027587811yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2646C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#2646D53
hypocenterC53#10345
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
172. C53#10345semidirect

Maximal subgroups

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

subgroupindex
#202746492
#202729852
#202725932
#202725582
C53×#26462
#1034553