Finite Group Databasebeta

← #27279077

Subgroups — #27279077

Order 13568.

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

normal subgroups only

This group has 558 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#20283756C211yesmaximal, characteristic
2#20274814C212yesmaximal, complemented
3C4×#13133132C211yesmaximal, characteristic
4#20272601C212yesmaximal, complemented
5C53×#2395C211yesmaximal, characteristic
6#16810701—24no—
7#16810676—21no—
8C2×#13133132—21no—
9#16810587—21no—
10#16810587C2212yes—
11C4×C22⋊Dic53C2212yes—
12#16810390—21no—
13#16810382—24no—
14C2×C4×D4×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#400C411yescharacteristic, complemented
17C53×#399C411yescharacteristic, complemented
18C53×#398C411yescharacteristic, complemented
19C53×#397—21no—
20C53×#397C411yescharacteristic, complemented
21C53×#394—21no—
22C4×C4⋊C4×C53—21no—
23C4×C22⋊C4×C53—21no—
24C4×C22⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917D414yes—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29C4⋊#88322—22no—
30#13133132—24no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34C4×#88322—22no—
35#13133124—22no—
36C22×D4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41#117×C53—22no—
42#117×C53—22no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
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×C53—21no—
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—22no—
67C2×D4×C53—41no—
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—42no—
75C22×C4×C53—42no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4112yes—
83C22×Dic53—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—42no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—42no—
98C42×C53—41no—
99C42×C53—41no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C4×Dic53—44no—
103#11719917—44no—
104#11719917—82no—
105C2×#88322—24no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#12112yes—
117D4×C53—44no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
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—41no—
141C2×C4×C53—81no—
142C2×C4×C53#10412yes—
143C2×C4×C53C4×D412yes—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×Dic53—44no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149#88322—44no—
150#9012—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53#38611yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53—81no—
175C4×C53#45812yes—
176Dic53—84no—
177#2975—531no—
178#2395D5311yescharacteristic
179#1826—532no—
180C4×#99—531no—
181#1297—532no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—81no—
186C2×C53#187111yescharacteristic
187C2×C53#188411yescharacteristic
188C2×C53#189011yescharacteristic
189C2×C4×D4C2×D5311yescharacteristic
190C2×#117C2×D5311yescharacteristic
191#436—1064no—
192#431—1061no—
193C2×#99—1061no—
194#407—532no—
195#407—1061no—
196#400Dic5311yescharacteristic, complemented
197#399Dic5311yescharacteristic, complemented
198#398Dic5311yescharacteristic, complemented
199#397—21no—
200#397Dic5311yescharacteristic, complemented
201#394—21no—
202C4×C4⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—532no—
205C4×C22⋊C4C2×D5311yescharacteristic
206#342—1061no—
207#339—1064no—
208C53#901211yescharacteristic, complemented, Sylow
209C22×D4C22⋊D5311yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C23×C4C22×D5311yescharacteristic
212C2×M16—534no—
213C2×M16—1062no—
214C4×D4—22no—
215C4×D4—22no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C4⋊C4C2×Dic5311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C2×Dic5311yescharacteristic
229C2×C22⋊C4C22⋊D5311yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—1062no—
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—1062no—
235#99—1064no—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C4×C8—1062no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—1062no—
245#95C2×Dic5311yescharacteristic
246#95C2×Dic5311yescharacteristic
247C24C22⋊Dic5311yescharacteristic
248C24C2×C22⋊D5311yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—2122no—
261C22×C4C22⋊Dic5311yescharacteristic
262C22×C4C22⋊Dic5311yescharacteristic
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4D4×D5312yes—
265C22×C4#1171994712yes—
266C22×C4C22×Dic5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268M16—4242no—
269M16—2124no—
270C2×C8—1064no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—1064no—
283C22⋊C4—2122no—
284C42—21no—
285C42—21no—
286C42—42no—
287C42—41no—
288C42—41no—
289C42—2124no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic5311yescharacteristic
300C23#1313318912yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4—2124no—
329C2×C4#1313314412yes—
330C2×C4D4×Dic5312yes—
331C2×C4#1313318511yescharacteristic
332C2×C4#1313319811yescharacteristic
333C8—2124no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1681050311yescharacteristic
347C22C2×#1313314411yescharacteristic
348C22#1681064011yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—4244no—
359C4#1681081212yes—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2027501811yescharacteristic
365C2#2027509911yescharacteristic
366C2#2027512911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2395C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2395D53
hypocenterC53#9012
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2395

Complements

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

NH ≅ G/Ntype
2. #20274814C2semidirect
4. #20272601C2semidirect
16. C53×#400C4semidirect
17. C53×#399C4semidirect
18. C53×#398C4semidirect
20. C53×#397C4semidirect
196. #400Dic53semidirect
197. #399Dic53semidirect
198. #398Dic53semidirect
200. #397Dic53semidirect
208. C53#9012semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202748142
C4×#131331322
#202726012
C53×#23952
#901253