Finite Group Databasebeta

← #27281433

Subgroups — #27281433

Order 13568.

532 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283134C211yesmaximal, characteristic
2#20273317C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20272608C212yesmaximal, complemented
5C53×#2611C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810680C2212yes—
8#16810579—24no—
9C4×#11719917—21no—
10C4×#11719917C2212yes—
11#16810365—24no—
12#16810349—21no—
13#16810347—21no—
14C2×#125×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#400C411yescharacteristic
17C53×#399C411yescharacteristic
18C53×#398C411yescharacteristic
19C53×#397C411yescharacteristic
20C53×#389—21no—
21C53×#388—21no—
22C53×#387—21no—
23C53×#385—21no—
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—24no—
27C2×#11719917—24no—
28C2×#11719917—22no—
29C22×#88322—22no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133128—22no—
34#13133128D414yes—
35C4×#88322—22no—
36C4×#88322—22no—
37C23×C4×C53D411yescharacteristic
38#125×C53—22no—
39#125×C53—22no—
40#117×C53—22no—
41#117×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53—21no—
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53C2×C411yescharacteristic
63#95×C53C2×C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C22×C4×C53—22no—
66C22×C4×C53—22no—
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×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—42no—
95C42×C53C2×D412yes—
96C42×C53#4112yes—
97#11719917—28no—
98#11719917—24no—
99#11719917—24no—
100C2×#88322—28no—
101C2×#88322—24no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
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#10412yes—
135C2×C4×C53C4×D412yes—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12112yes—
138C2×C4×C53#12311yescharacteristic
139#88322—28no—
140#88322—82no—
141#9388—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53#38611yescharacteristic
150C22×C53C2×#10411yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C4×C53—22no—
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—42no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—81no—
161C4×C53—81no—
162C4×C53—81no—
163#2873—531no—
164#2611D5311yescharacteristic
165#1528—532no—
166#1464—531no—
167#1300—532no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—42no—
171C2×C53#188411yescharacteristic
172C2×C53#189611yescharacteristic
173C2×C53#191511yescharacteristic
174C22×M16—1061no—
175C2×#125C2×D5311yescharacteristic
176C2×#117C2×D5311yescharacteristic
177#432—532no—
178#405—1064no—
179C4×M16—532no—
180C4×M16—1061no—
181#400Dic5311yescharacteristic
182#399Dic5311yescharacteristic
183#398Dic5311yescharacteristic
184#397Dic5311yescharacteristic
185#389—21no—
186#388—21no—
187#387—21no—
188#385—21no—
189#376C2×D5311yescharacteristic
190#330—1064no—
191#324—1061no—
192#323—1061no—
193C53#938811yescharacteristic, complemented, Sylow
194C23×C4C22⋊D5311yescharacteristic
195C2×M16—1062no—
196C2×M16—1064no—
197C2×M16—1064no—
198C2×M16—1062no—
199C22×C8—1062no—
200#125—22no—
201#125—22no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic5311yescharacteristic
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C42—21no—
217C2×C42C22⋊D5311yescharacteristic
218C2×C42C22×D5311yescharacteristic
219C4⋊C8—1062no—
220C22⋊C8—1062no—
221#97—534no—
222#97—1062no—
223#97—1062no—
224C4×C8—1062no—
225C4×C8—1062no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×Dic5311yescharacteristic
232#95C2×Dic5311yescharacteristic
233C24C22⋊Dic5311yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22×Dic5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—1068no—
249M16—1064no—
250M16—1064no—
251C2×C8—1068no—
252C2×C8—1064no—
253C2×C8—2122no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—21no—
268C42—21no—
269C42—42no—
270C42D4×D5312yes—
271C42#1171994712yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1313314412yes—
277C23C2×C22⋊Dic5311yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4#1313314412yes—
304C2×C4D4×Dic5312yes—
305C2×C4#1313318511yescharacteristic
306C2×C4#1313318912yes—
307C2×C4#1313319811yescharacteristic
308C8—1068no—
309C8—4242no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1681050311yescharacteristic
318C22C2×#1313314411yescharacteristic
319C22C2×#1313314411yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2027509911yescharacteristic
335C2#2027515911yescharacteristic
336C2#2027526811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2611C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2611D53
hypocenterC53#9388
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #20273317C2semidirect
4. #20272608C2semidirect
193. C53#9388semidirect

Maximal subgroups

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

subgroupindex
#202831342
#202733172
#202730492
#202726082
C53×#26112
#938853