Finite Group Databasebeta

← #27280085

Subgroups — #27280085

Order 13568.

502 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280583C211yesmaximal, characteristic
2C4×#13133134C211yesmaximal, characteristic
3#20273318C212yesmaximal, complemented
4#20272581C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7C2×#13133134—21no—
8C4×#11719917C2212yes—
9#16810559C411yescharacteristic
10#16810559C411yescharacteristic
11#16810554—21no—
12#16810548C411yescharacteristic
13#16810548C411yescharacteristic
14#16810537—21no—
15#16810496—21no—
16#16810496—21no—
17#16810491C2211yescharacteristic
18#16810392—21no—
19#16810364—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432C2212yes—
23C53×#407—21no—
24C2×C22⋊C8×C53—21no—
25C2×#11719917—22no—
26#13133209—22no—
27#13133209—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38#13133134—24no—
39#13133128—22no—
40C4×#88322—22no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C2×M16×C53—22no—
51C22×C8×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—22no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62#97×C53D414yes—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
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—81no—
73C22×C4×C53—81no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83M16×C53—82no—
84C2×C8×C53—24no—
85C2×C8×C53—82no—
86C4⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—22no—
89C42×C53C2×D412yes—
90C42×C53#4112yes—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—81no—
94C22⋊Dic53—81no—
95C4⋊Dic53—22no—
96C4⋊Dic53—22no—
97#11719919—41no—
98#11719919—42no—
99#11719919—41no—
100#11719919—81no—
101#11719919—81no—
102#11719917—24no—
103#11719917—42no—
104C2×#88322—42no—
105C23×C53—21no—
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×C53#10412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116D4×C53—44no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C4×D412yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12112yes—
132C2×C4×C53#12311yescharacteristic
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C8×C53—44no—
146#88322—44no—
147#9189—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#38611yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—22no—
167C4×C53—41no—
168C4×C53—41no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172Dic53—81no—
173#2973D5311yescharacteristic
174#2633—531no—
175C4×#100—531no—
176#1528—532no—
177#1288—532no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—42no—
181C2×C53—81no—
182C2×C53#187111yescharacteristic
183C2×C53#191111yescharacteristic
184C2×C53#191511yescharacteristic
185C2×#125—531no—
186C2×C4×D4C2×D5311yescharacteristic
187#435—24no—
188#432C2×D5312yes—
189C2×#100—1061no—
190#407—21no—
191C2×C22⋊C8—21no—
192C4×M16—532no—
193#399—531no—
194#399—531no—
195#398—1061no—
196#397—531no—
197#397—531no—
198#394—1061no—
199#385—1061no—
200#385—1061no—
201#383—531no—
202#343—1061no—
203#330—1064no—
204C53#918911yescharacteristic, complemented, Sylow
205C22×D4D21211yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—22no—
209C2×M16—1062no—
210C22×C8—22no—
211#125—1062no—
212#125—1062no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4D21211yescharacteristic
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4C4×D5311yescharacteristic
227C2×C22⋊C4C4×D5311yescharacteristic
228C2×C42C22×D5311yescharacteristic
229C4⋊C8—22no—
230#100—1064no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234#97—1062no—
235#97C22⋊D5314yes—
236C4×C8—1062no—
237#95—531no—
238#95—531no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243C24#1171992111yescharacteristic
244C24#1171992111yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×D21211yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—82no—
264M16—1064no—
265M16—2122no—
266C2×C8—24no—
267C2×C8—82no—
268C2×C8—2122no—
269C4⋊C4—22no—
270C4⋊C4—1062no—
271C4⋊C4—1062no—
272C4⋊C4—4241no—
273C4⋊C4—4241no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C4⋊C4—2121no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—4241no—
280C22⋊C4—4241no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42#1171994512yes—
284C42C2×C22⋊D5312yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1313314312yes—
295C23C2×#1171992111yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4C4×C22⋊D5312yes—
322C2×C4#1313318611yescharacteristic
323C2×C4#1313319112yes—
324C2×C4#1313320011yescharacteristic
325C8—44no—
326C8—2124no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1681050211yescharacteristic
340C22C2×#1313314311yescharacteristic
341C22#1681063911yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—4241no—
349C4—4241no—
350C4—4241no—
351C4—4241no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2027501711yescharacteristic
357C2#2027524711yescharacteristic
358C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9189
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. #20273318C2semidirect
4. #20272581C2semidirect
204. C53#9189semidirect

Maximal subgroups

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

subgroupindex
#202805832
C4×#131331342
#202733182
#202725812
C53×#29732
#918953