Finite Group Databasebeta

← #27302759

Subgroups — #27302759

Order 13568.

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

normal subgroups only

This group has 488 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#20274707C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20272833C212yesmaximal
4#20272833C212yesmaximal
5C53×#2579C211yesmaximal, characteristic
6#16810647—21no—
7C2×#13133134C2212yes—
8C2×#13133132C2212yes—
9C4×C22⋊Dic53—21no—
10#16810392—21no—
11#16810390—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic, complemented
14C53×#398—21no—
15C53×#396—21no—
16C53×#387—21no—
17C53×#386C2211yescharacteristic
18C53×#386C2211yescharacteristic
19C53×#385—21no—
20C53×#385C411yescharacteristic, complemented
21C53×#385C411yescharacteristic, complemented
22C53×#380C411yescharacteristic, complemented
23C2×#11719917—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—41no—
26C4⋊#88322—41no—
27C4⋊#88322—41no—
28#13133134—24no—
29#13133132—22no—
30#13133132Q814yes—
31#13133124—41no—
32C22×D4×C53C2311yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—41no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59C24×C53C22×C411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53C2×Q812yes—
64C2×D4×C53#4112yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53—81no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—81no—
91C4⋊C4×C53—81no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—81no—
98C42×C53—21no—
99C42×C53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C4×Dic53—82no—
103#11719917—42no—
104#11719917—42no—
105C2×#88322—42no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C4×Q812yes—
116C23×C53C22⋊Q812yes—
117D4×C53—42no—
118D4×C53—42no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—82no—
147C2×C4×C53#9911yescharacteristic
148C2×C4×C53#9911yescharacteristic
149C2×C4×C53#12411yescharacteristic
150C2×C4×C53C4⋊Q811yescharacteristic
151C2×Dic53—44no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154#88322—82no—
155#13297—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—81no—
168C22×C53#39111yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53#41011yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—42no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—81no—
180C4×C53—81no—
181C4×C53—81no—
182Dic53—84no—
183#2579D5311yescharacteristic
184#1800—531no—
185C4×#99—531no—
186#1390—532no—
187#1390—532no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—42no—
191C2×C53—42no—
192C2×C53#199611yescharacteristic
193C2×C53#212011yescharacteristic
194C2×C53#212111yescharacteristic
195C2×C4×D4C2×D5311yescharacteristic
196C22×C4⋊C4Dic5311yescharacteristic, complemented
197#423—1061no—
198C2×#100—532no—
199C2×#99—532no—
200#398—21no—
201#396—21no—
202#387—21no—
203#386C2×D5311yescharacteristic
204#386C2×D5311yescharacteristic
205#385—21no—
206#385Dic5311yescharacteristic, complemented
207#385Dic5311yescharacteristic, complemented
208#380Dic5311yescharacteristic, complemented
209C4×C22⋊C4—1061no—
210#343—1061no—
211#342—1061no—
212C53#1329711yescharacteristic, complemented, Sylow
213C22×D4C22×D5311yescharacteristic
214C23×C4C2×Dic5311yescharacteristic
215C23×C4C2×Dic5311yescharacteristic
216C23×C4C2×Dic5311yescharacteristic
217C23×C4C22⋊D5311yescharacteristic
218C2×M16—1062no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4C2×Dic5311yescharacteristic
233C2×C22⋊C4C2×Dic5311yescharacteristic
234C2×C22⋊C4C2×Dic5311yescharacteristic
235C2×C22⋊C4C22⋊D5311yescharacteristic
236C2×C42—2121no—
237C2×C42C22⋊D5311yescharacteristic
238C4⋊C8—2121no—
239C4⋊C8—2121no—
240#100—1064no—
241#99—534no—
242#99—1062no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—2121no—
249C24C22×Dic5311yescharacteristic
250C24C2×C22⋊D5311yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D5312yes—
254C2×D4#1171994912yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—2122no—
270C22×C4C22⋊Dic5311yescharacteristic
271C22×C4C22⋊Dic5311yescharacteristic
272C22×C4C22⋊Dic5311yescharacteristic
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275M16—2122no—
276M16—2122no—
277C2×C8—2122no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—1064no—
292C22⋊C4—2122no—
293C42—21no—
294C42—21no—
295C42—4242no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×C22⋊Dic5311yescharacteristic
305C23Q8×Dic5312yes—
306C23#1313319312yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—4242no—
338C2×C4—4242no—
339C2×C4—2124no—
340C2×C4#1313313211yescharacteristic
341C2×C4#1313313211yescharacteristic
342C2×C4#1313320611yescharacteristic
343C2×C4#1313321511yescharacteristic
344C8—4242no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1681053011yescharacteristic
358C22C2×#1313313211yescharacteristic
359C22#1681059711yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—4244no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2027570111yescharacteristic
377C2#2027635611yescharacteristic
378C2#2027636111yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2579C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2579D53
hypocenterC53#13297
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C53C4semidirect
20. C53×#385C4semidirect
21. C53×#385C4semidirect
22. C53×#380C4semidirect
196. C22×C4⋊C4Dic53semidirect
206. #385Dic53semidirect
207. #385Dic53semidirect
208. #380Dic53semidirect
212. C53#13297semidirect

Maximal subgroups

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

subgroupindex
#202747072
C4×#131331322
#202728332
C53×#25792
#1329753