Finite Group Databasebeta

← #27281757

Subgroups — #27281757

Order 13568.

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

normal subgroups only

This group has 486 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#20283800C211yesmaximal, characteristic, complemented
2#20273339C212yesmaximal
3#20273059C211yesmaximal, characteristic, complemented
4#20272571C212yesmaximal, complemented
5C53×#2545C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—21no—
8#16810680C2212yes—
9#16810680C2212yes—
10#16810676—21no—
11#16810363—24no—
12#16810349—21no—
13#16810349—21no—
14C2×C4⋊Q8×C53C411yescharacteristic
15C2×#124×C53C411yescharacteristic
16C2×#124×C53C2211yescharacteristic
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#396—21no—
19C53×#396—21no—
20C53×#391C411yescharacteristic
21C53×#390C411yescharacteristic
22C53×#388—21no—
23C53×#388—21no—
24C4×C4⋊C4×C53C2211yescharacteristic
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33#13133128D414yes—
34C4×#88322—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53C2×C411yescharacteristic
38C4⋊Q8×C53—42no—
39#124×C53—22no—
40#124×C53—22no—
41#124×C53—42no—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64C24×C53—21no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×D4×C53—22no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—41no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—21no—
100C42×C53—21no—
101C42×C53—41no—
102C42×C53—41no—
103C42×C53C2×D412yes—
104C42×C53#4112yes—
105C2×#88322—24no—
106C2×#88322—42no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115Q8×C53—44no—
116Q8×C53—82no—
117D4×C53—82no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
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#10012yes—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12112yes—
144C2×C4×C53#12311yescharacteristic
145#88322—44no—
146#88322—82no—
147#9448—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53#38611yescharacteristic
158C22×C53C2×#10011yescharacteristic
159C22×C53#41311yescharacteristic
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—42no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170C4×C53—81no—
171#2983—531no—
172#2545D5311yescharacteristic
173#1533—532no—
174#1467—531no—
175#1285—532no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—161no—
179C2×C53#188411yescharacteristic
180C2×C53#190011yescharacteristic
181C2×C53#191211yescharacteristic
182C2×C4⋊Q8Dic5311yescharacteristic
183C2×#124Dic5311yescharacteristic
184C2×#124C2×D5311yescharacteristic
185C2×C4×Q8C2×D5311yescharacteristic
186#446—1064no—
187#432—532no—
188#432—532no—
189#432—1061no—
190#431—1061no—
191#396—21no—
192#396—21no—
193#391Dic5311yescharacteristic
194#390Dic5311yescharacteristic
195#388—21no—
196#388—21no—
197C4×C4⋊C4C2×D5311yescharacteristic
198#329—1064no—
199#324—1061no—
200#324—1061no—
201C53#944811yescharacteristic, complemented, Sylow
202C22×Q8C2×Dic5311yescharacteristic
203C22×Q8C2×Dic5311yescharacteristic
204C22×D4C2×Dic5311yescharacteristic
205C4⋊Q8—42no—
206#124—22no—
207#124—22no—
208#124—42no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic5311yescharacteristic
224C2×C42C2×Dic5311yescharacteristic
225C2×C42C22⋊D5311yescharacteristic
226C2×C42C22⋊D5311yescharacteristic
227C2×C42C22×D5311yescharacteristic
228C4⋊C8—1062no—
229C4⋊C8—1062no—
230C4⋊C8—1062no—
231C4⋊C8—1062no—
232C22⋊C8—1062no—
233C22⋊C8—1062no—
234#97—534no—
235#97—1062no—
236#97—1062no—
237C4×C8—1062no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22⋊Dic5311yescharacteristic
255C22×C4C22⋊Dic5311yescharacteristic
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22×Dic5311yescharacteristic
259C22×C4C2×C22⋊D5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C2×C8—1064no—
262C2×C8—2122no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—41no—
282C42D4×D5312yes—
283C42#1171994712yes—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic5311yescharacteristic
292Q8—44no—
293Q8—82no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#1313313412yes—
318C2×C4D4×Dic5312yes—
319C2×C4#1313318511yescharacteristic
320C2×C4#1313318912yes—
321C2×C4#1313319811yescharacteristic
322C8—4242no—
323C8—2124no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#1681050311yescharacteristic
334C22C2×#1313313411yescharacteristic
335C22#1681060711yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2#2027509911yescharacteristic
351C2#2027517911yescharacteristic
352C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2545C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2545D53
hypocenterC53#9448
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
3. #20273059C2semidirect
4. #20272571C2semidirect
201. C53#9448semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202733392
#202730592
#202725712
C53×#25452
#944853