Finite Group Databasebeta

← #27294608

Subgroups — #27294608

Order 13568.

470 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274454C211yesmaximal, characteristic, complemented
2#20273049C211yesmaximal, characteristic, complemented
3#20272745C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2610C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810680—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×#124×C53C411yescharacteristic
13C2×#117×C53C2211yescharacteristic
14C53×#395C2211yescharacteristic
15C53×#394C2211yescharacteristic
16C53×#393C411yescharacteristic
17C53×#392C411yescharacteristic
18C53×#390—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#386—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C22×#88322—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30#13133128—41no—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35#124×C53—41no—
36#124×C53—41no—
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60C2×Q8×C53—41no—
61C2×Q8×C53—81no—
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—81no—
66C2×D4×C53—81no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—81no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—81no—
93C22⋊C4×C53—81no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—42no—
99#11719917—24no—
100#11719917—44no—
101#11719917—42no—
102C2×#88322—44no—
103C2×#88322—42no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53#10412yes—
117C23×C53C2×C22⋊C411yescharacteristic
118Q8×C53—82no—
119D4×C53—84no—
120D4×C53—82no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
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—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#9911yescharacteristic
147C2×C4×C53#9911yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150#88322—44no—
151#88322—82no—
152#11765—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—22no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53C2×#9911yescharacteristic
168C22×C53#41711yescharacteristic
169C22×C53C2×#10411yescharacteristic
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—42no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—81no—
180C4×C53—81no—
181#2610D5311yescharacteristic
182#1742—531no—
183#1464—531no—
184#1354—532no—
185#1345—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—42no—
189C2×C53—161no—
190C2×C53#150411yescharacteristic
191C2×C53#212211yescharacteristic
192C2×C53#213211yescharacteristic
193C22×M16—1061no—
194C2×#124Dic5311yescharacteristic
195C2×#117C2×D5311yescharacteristic
196#432—1061no—
197#395C2×D5311yescharacteristic
198#394C2×D5311yescharacteristic
199#393Dic5311yescharacteristic
200#392Dic5311yescharacteristic
201#390—21no—
202#388—21no—
203#388—21no—
204#386—21no—
205#376Dic5311yescharacteristic
206#343—532no—
207#343—1061no—
208#324—532no—
209#323—1061no—
210C53#1176511yescharacteristic, complemented, Sylow
211C22×Q8C2×Dic5311yescharacteristic
212C22×D4—21no—
213C22×D4C2×Dic5311yescharacteristic
214C23×C4C22⋊D5311yescharacteristic
215C2×M16—1064no—
216C2×M16—1062no—
217C2×M16—2121no—
218C2×M16—2122no—
219C22×C8—2121no—
220#124—41no—
221#124—41no—
222#117—41no—
223#117—41no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊D5311yescharacteristic
226C2×C4⋊C4C22×D5311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C2×Dic5311yescharacteristic
234C2×C22⋊C4C22⋊D5311yescharacteristic
235C2×C42—21no—
236C2×C42C2×Dic5311yescharacteristic
237C2×C42C22⋊D5311yescharacteristic
238C4⋊C8—2121no—
239C22⋊C8—1062no—
240#97—2121no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95C2×Dic5311yescharacteristic
245#95C2×Dic5311yescharacteristic
246C24—21no—
247C24C22⋊Dic5311yescharacteristic
248C2×Q8—41no—
249C2×Q8—81no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22×Dic5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269M16—1064no—
270M16—2124no—
271M16—2122no—
272C2×C8—2122no—
273C2×C8—2124no—
274C2×C8—2122no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—42no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1313314412yes—
305C23C2×C22⋊Dic5311yescharacteristic
306Q8—82no—
307D4—84no—
308D4—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
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—41no—
334C2×C4#1313313211yescharacteristic
335C2×C4#1313313211yescharacteristic
336C2×C4#1313324211yescharacteristic
337C2×C4#1313324711yescharacteristic
338C8—4242no—
339C8—2124no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22C2×#1313313211yescharacteristic
355C22#1681062411yescharacteristic
356C22C2×#1313314411yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—161no—
372C2#2027321911yescharacteristic
373C2#2027636511yescharacteristic
374C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2610C2
socleC22×C53C2×#99
prime radicalC53×#392C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2610D53
hypocenterC53#11765
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #20274454C2semidirect
2. #20273049C2semidirect
3. #20272745C2semidirect
4. #20272718C2semidirect
210. C53#11765semidirect

Maximal subgroups

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

subgroupindex
#202744542
#202730492
#202727452
#202727182
C53×#26102
#1176553