Finite Group Databasebeta

← #27279137

Subgroups — #27279137

Order 13568.

740 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277313C211yesmaximal, characteristic
2#20274805C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272594C212yesmaximal, complemented
5C2×C53×#399C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810516—24no—
8#16810483C2212yes—
9#16810479—21no—
10C4×C22⋊Dic53—21no—
11#16810385—24no—
12#16810347—22no—
13C22×C4⋊C4×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#399—22no—
18C53×#399C414yescomplemented
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21C2×C4⋊Dic53D414yes—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26C22⋊#88322—22no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30C25×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C2×C4⋊C4×C53—24no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—44no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C22⋊C4×C53—24no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—44no—
45C2×C22⋊C4×C53C2×C412yes—
46C2×C22⋊C4×C53D412yes—
47#95×C53—21no—
48#95×C53C2×C412yes—
49C24×C53—21no—
50C24×C53—22no—
51C24×C53—21no—
52C24×C53—22no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53C22⋊C412yes—
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—24no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—48no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—48no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C22⋊Dic53—82no—
85C22⋊Dic53—82no—
86C4⋊Dic53—44no—
87C4×Dic53—82no—
88C2×#88322—44no—
89C2×#88322—42no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—48no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—81no—
103C23×C53—81no—
104C23×C53—84no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53#12011yescharacteristic
107C23×C53#12311yescharacteristic
108C23×C53#13512yes—
109C23×C53#13712yes—
110C2×C4×C53—24no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—48no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—81no—
125C2×C4×C53—82no—
126C2×C4×C53—82no—
127C2×C4×C53C4×D412yes—
128C2×C4×C53#12112yes—
129C2×Dic53—44no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132#88322—82no—
133#9024—531nomaximal, Sylow
134C22×C53—24no—
135C22×C53—24no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—44no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—82no—
145C22×C53—84no—
146C22×C53—84no—
147C22×C53#38611yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41911yescharacteristic
150C22×C53#45212yes—
151C22×C53#45312yes—
152C22×C53#45812yes—
153C22×C53#48512yes—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—44no—
157C4×C53—42no—
158C4×C53—82no—
159Dic53—84no—
160C2×#399D5311yescharacteristic
161#2312—531no—
162#1824—532no—
163C2×#323—531no—
164#1294—532no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—22no—
168C2×C53—82no—
169C2×C53#161814yes—
170C2×C53#187311yescharacteristic
171C2×C53#188711yescharacteristic
172C2×C53#189011yescharacteristic
173C22×C4⋊C4—21no—
174C22×C4⋊C4C2×D5311yescharacteristic
175C22×C22⋊C4—21no—
176C22×C22⋊C4C2×D5311yescharacteristic
177C2×C22⋊C8—532no—
178#399—22no—
179#399Dic5314yescomplemented
180#388—1064no—
181#380—532no—
182#379—1061no—
183C4×C22⋊C4—1061no—
184C2×#95C2×D5311yescharacteristic
185#340—1064no—
186#323—1062no—
187C53#902411yescharacteristic, complemented, Sylow
188C25C22⋊D5311yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22⋊D5311yescharacteristic
192C23×C4C22×D5311yescharacteristic
193C22×C8—1062no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—534no—
200C2×C4⋊C4C2×Dic5312yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4C2×Dic5312yes—
209C2×C22⋊C4C22⋊D5312yes—
210C2×C42—1062no—
211C22⋊C8—1062no—
212C22⋊C8—1062no—
213#95—21no—
214#95—1062no—
215#95—1062no—
216#95C2×Dic5312yes—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24C22⋊Dic5312yes—
224C24C2×C22⋊D5311yescharacteristic
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—1064no—
239C22×C4C22⋊Dic5312yes—
240C22×C4D4×D5312yes—
241C22×C4#1171994712yes—
242C22×C4C22×Dic5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244C2×C8—2124no—
245C2×C8—2122no—
246C4⋊C4—24no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—2124no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—48no—
253C22⋊C4—4242no—
254C22⋊C4—4242no—
255C22⋊C4—2122no—
256C22⋊C4—2122no—
257C42—4242no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×C22⋊Dic5311yescharacteristic
274C23#1313318511yescharacteristic
275C23#1313319811yescharacteristic
276C23#1313323712yes—
277C23#1313324812yes—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—2124no—
296C2×C4—2124no—
297C2×C4—2124no—
298C2×C4D4×Dic5312yes—
299C2×C4#1313318912yes—
300C8—4242no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1681050311yescharacteristic
315C22#1681061911yescharacteristic
316C22#1681063211yescharacteristic
317C22#1681078312yes—
318C22#1681079012yes—
319C22#1681081212yes—
320C22#1681096112yes—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—82no—
326C4—4244no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#2027376114yes—
332C2#2027502811yescharacteristic
333C2#2027511411yescharacteristic
334C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#399C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#399D53
hypocenterC53#9024
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. #20274805C2semidirect
4. #20272594C2semidirect
18. C53×#399C4semidirect
179. #399Dic53semidirect
187. C53#9024semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202748052
C2×#168103472
#202725942
C2×C53×#3992
#902453