Finite Group Databasebeta

← #27293546

Subgroups — #27293546

Order 13568.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C4×#13133198C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272662C212yesmaximal, complemented
4#20272644C212yesmaximal, complemented
5C53×#1466C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8#16810587—21no—
9#16810511—21no—
10#16810504—21no—
11#16810503C411yescharacteristic
12#16810495C2211yescharacteristic
13#16810484C2211yescharacteristic
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C2×#11719917—41no—
26#13133198—22no—
27#13133198—22no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—41no—
38C2×C4×Dic53—41no—
39C22⋊#88322—22no—
40C22⋊#88322—41no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57#97×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22×C411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—42no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53M1612yescomplemented
85C2×C8×C53—42no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C42×C53—21no—
90C42×C53—21no—
91C22⋊Dic53—24no—
92C22⋊Dic53—21no—
93C22⋊Dic53—42no—
94C22⋊Dic53—41no—
95C22⋊Dic53M1612yescomplemented
96#11719919—24no—
97#11719919—42no—
98C4×Dic53—42no—
99C4×Dic53—42no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102#11719917—82no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×M1611yescharacteristic
115C23×C53C2×M1611yescharacteristic
116D4×C53—42no—
117D4×C53—42no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—81no—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—41no—
138C2×Dic53—42no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C8×C53—82no—
151#88322—82no—
152#11526—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—81no—
167C22×C53#40711yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41311yescharacteristic
170C4×C53—21no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53#35212yes—
176Dic53—42no—
177Dic53—42no—
178Dic53—42no—
179Dic53—81no—
180C4×#123—531no—
181#1466D5311yescharacteristic
182#1465—531no—
183#1326—532no—
184#1317—532no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—41no—
188C2×C53—41no—
189C2×C53—81no—
190C2×C53#146511yescharacteristic
191C2×C53C2×#35211yescharacteristic
192C2×C53#212011yescharacteristic
193C2×#123—531no—
194C2×C4×D4C2×D5311yescharacteristic
195C22×C42—531no—
196#432—21no—
197#407—1061no—
198#388—1061no—
199#387—1061no—
200#386—531no—
201#385—531no—
202#381—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—531no—
205C4×C22⋊C4—1061no—
206#324—1061no—
207#323—21no—
208#323—532no—
209#323C2×D5312yes—
210C53#1152611yescharacteristic, complemented, Sylow
211C22×D4C4×D5311yescharacteristic
212C23×C4—531no—
213C23×C4—531no—
214C23×C4C4×D5311yescharacteristic
215C23×C4D21211yescharacteristic
216C2×M16—2121no—
217#123—1062no—
218#123—1062no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4D21211yescharacteristic
224C2×C22⋊C4—531no—
225C2×C22⋊C4—531no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42—1061no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—2121no—
234C2×C42—2121no—
235C2×C42C22⋊D5311yescharacteristic
236C4⋊C8—41no—
237C22⋊C8—22no—
238C22⋊C8—1062no—
239C22⋊C8—2121no—
240#97—41no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24#1171992111yescharacteristic
247C24C2×C4×D5311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—532no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—2122no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4#1171992111yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273M16—4242no—
274C2×C8—42no—
275C2×C8—2122no—
276C4⋊C4—41no—
277C4⋊C4—1064no—
278C4⋊C4—2122no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—532no—
282C22⋊C4—1064no—
283C22⋊C4—1061no—
284C22⋊C4—2122no—
285C22⋊C4—2121no—
286C42—21no—
287C42—21no—
288C42—2122no—
289C42—2122no—
290C42—2121no—
291C42—2121no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×#1171992111yescharacteristic
302C23M16×D5311yescharacteristic
303C23M16×D5311yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4#1313313111yescharacteristic
337C2×C4#1313313111yescharacteristic
338C8—82no—
339C8—4242no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1681058611yescharacteristic
355C22C2×#1313313111yescharacteristic
356C22#1681060511yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—4241no—
363C4—2122no—
364C4—2122no—
365C4—2122no—
366C4#1681041412yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#2027305111yescharacteristic
373C2C2×#1681041411yescharacteristic
374C2#2027635411yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1466C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1466D53
hypocenterC53#11526
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #20272662C2semidirect
4. #20272644C2semidirect
84. C22×Dic53M16semidirect
95. C22⋊Dic53M16semidirect
210. C53#11526semidirect

Maximal subgroups

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

subgroupindex
C4×#131331982
#202730532
#202726622
#202726442
C53×#14662
#1152653