Finite Group Databasebeta

← #28302203

Subgroups — #28302203

Order 13568.

1158 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 160 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#20290606C212yesmaximal, complemented
2C2×#16811468C212yesmaximal, complemented
3C2×#16811467C212yesmaximal
4C22×#13133175C211yesmaximal, characteristic
5C2×C22⋊#88324C2214yes—
6#16811489—24no—
7#16811468—24no—
8#16811467—24no—
9C23×C4×D53C411yescharacteristic
10C2×#13133175—24no—
11C2×#13133175—24no—
12C2×#13133175—24no—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719921C411yescharacteristic
15C22×#11719921C2211yescharacteristic
16C22×C4×Dic53C411yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C22×#88324—24no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C24×D53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26C22×C4×D53—22no—
27C22×C4×D53—22no—
28C22×C4×D53—22no—
29C22×C4×D53—22no—
30C22×C4×D53—22no—
31C22×C4×D53—22no—
32C22×C4×D53C814yescomplemented
33#13133175—232no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53D414yes—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921D414yes—
43C2×C4×Dic53—22no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53D414yes—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53—22no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—22no—
55C24×C53C22×C411yescharacteristic
56C23×D53—21no—
57C23×D53—21no—
58C23×D53—21no—
59C23×D53—21no—
60C23×D53—42no—
61C23×D53—42no—
62C23×D53C22⋊C412yes—
63C23×D53C2×C812yes—
64C23×D53M1612yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
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×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53C22⋊C412yes—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C2×C812yes—
79C22×C4×C53M1612yes—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—42no—
87C22×Dic53—42no—
88C22×Dic53C22⋊C412yes—
89C22×Dic53C22⋊C412yes—
90C22×Dic53C2×C812yes—
91C22×Dic53M1612yes—
92C22×Dic53C2×D412yes—
93C22×Dic53C2×D412yes—
94C22×Dic53C2×D412yes—
95C2×C4×D53—28no—
96C2×C4×D53—28no—
97C2×C4×D53—28no—
98C2×C4×D53—28no—
99C2×C4×D53—28no—
100C2×C4×D53—28no—
101C2×C4×D53—28no—
102C2×#88324—48no—
103C2×#88324—44no—
104C4⋊C4×C53—28no—
105#11719921—28no—
106#11719921—28no—
107C4⋊Dic53—28no—
108C4×Dic53—28no—
109C4×Dic53—28no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C22×C811yescharacteristic
116C23×C53C2×M1611yescharacteristic
117C23×C53C2×M1611yescharacteristic
118C23×C53C2×M1611yescharacteristic
119C22×D53—22no—
120C22×D53—22no—
121C22×D53—22no—
122C22×D53—22no—
123C22×D53—22no—
124C22×D53—22no—
125C22×D53—44no—
126C22×D53—44no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—44no—
134C22×D53—44no—
135C22×D53C22⋊C814yes—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—24no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53C22⋊C814yes—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—24no—
158C2×Dic53—22no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53C22⋊C814yes—
165C2×Dic53#12014yes—
166C4×D53—232no—
167#88324—84no—
168#8300—531nomaximal, Sylow
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53#37911yescharacteristic
178C22×C53C2×C22⋊C811yescharacteristic
179C22×C53C2×C22⋊C811yescharacteristic
180C22×C53C2×C22⋊C811yescharacteristic
181C22×C53#40711yescharacteristic
182C22×C53#40711yescharacteristic
183C22×C53#40711yescharacteristic
184C22×C53#41012yes—
185C22×C53#41312yes—
186C2×D53—24no—
187C2×D53—44no—
188C2×D53—44no—
189C2×D53—44no—
190C2×D53—44no—
191C2×D53—44no—
192C2×D53—42no—
193C2×D53—42no—
194C2×D53—44no—
195C4×C53—24no—
196C4×C53—42no—
197C4×C53—42no—
198Dic53—24no—
199Dic53—24no—
200Dic53—42no—
201C22×#117—531no—
202#1817—532no—
203C2×#324—532no—
204C2×#323—532no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53#145812yes—
210C2×C53#146412yes—
211C2×C53#177911yescharacteristic
212C2×C53#178211yescharacteristic
213C2×C53#178511yescharacteristic
214D53—44no—
215C24×C4—531no—
216C2×#117—1064no—
217C2×#117—1064no—
218C2×#117—1064no—
219C22×C4⋊C4—531no—
220C22×C4⋊C4#615711yescharacteristic
221C22×C22⋊C4—531no—
222C22×C22⋊C4—531no—
223C22×C42—531no—
224C22×C42—531no—
225C2×C22⋊C8—534no—
226#336—1064no—
227#324—1064no—
228#323—1064no—
229C53#830011yescharacteristic, complemented, Sylow
230C25—531no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—534no—
235C23×C4—1062no—
236C23×C4—1062no—
237C23×C4—1062no—
238C23×C4—1062no—
239C23×C4—1062no—
240C23×C4—1062no—
241C23×C4C2×#615711yescharacteristic
242C23×C4C2×#615711yescharacteristic
243C23×C4C2×#615711yescharacteristic
244C22×C8—1064no—
245#117—10632no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—534no—
250C2×C4⋊C4—1062no—
251C2×C4⋊C4—1062no—
252C2×C22⋊C4—534no—
253C2×C22⋊C4—1062no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4—1062no—
257C2×C22⋊C4—1062no—
258C2×C42—534no—
259C2×C42—1062no—
260C2×C42—1062no—
261C2×C42—1062no—
262C2×C42—1062no—
263C2×C42—1062no—
264C22⋊C8—1064no—
265C22⋊C8—1064no—
266C24—532no—
267C24—532no—
268C24—532no—
269C24—1061no—
270C24—1061no—
271C24—1061no—
272C24—1061no—
273C24—2122no—
274C24—2122no—
275C24C22×#615711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—532no—
289C22×C4—532no—
290C22×C4—532no—
291C22×C4—532no—
292C22×C4—532no—
293C22×C4—1068no—
294C22×C4—1068no—
295C22×C4—1068no—
296C22×C4—1068no—
297C22×C4—1068no—
298C22×C4—1068no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1068no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—2122no—
307C22×C4—2122no—
308C22×C4D53⋊C812yes—
309C22×C4#1171993612yes—
310C22×C4C22⋊#615712yes—
311C22×C4C22⋊#615712yes—
312C22×C4C22⋊#615712yes—
313C2×C8—2124no—
314C2×C8—2128no—
315C4⋊C4—28no—
316C4⋊C4—1068no—
317C22⋊C4—1068no—
318C22⋊C4—1068no—
319C42—1068no—
320C42—1068no—
321C23—22no—
322C23—22no—
323C23—534no—
324C23—1062no—
325C23—1062no—
326C23—1062no—
327C23—1062no—
328C23—1062no—
329C23—1062no—
330C23—2124no—
331C23—2124no—
332C23—2122no—
333C23—2122no—
334C23—2124no—
335C23—2124no—
336C23—2122no—
337C23—2122no—
338C23—2122no—
339C23—2122no—
340C23C2×C22⋊#615711yescharacteristic
341C23C2×C22⋊#615711yescharacteristic
342C23C2×C22⋊#615711yescharacteristic
343C23C2×D53⋊C811yescharacteristic
344C23D53⋊M1611yescharacteristic
345C23D53⋊M1611yescharacteristic
346C23C2×#1171993611yescharacteristic
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—534no—
362C2×C4—534no—
363C2×C4—10632no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1062no—
367C2×C4—1062no—
368C2×C4—1062no—
369C2×C4—1062no—
370C2×C4—1064no—
371C2×C4—1062no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4—2122no—
376C2×C4—2122no—
377C2×C4#1313327414yes—
378C8—4244no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—1064no—
388C22—2124no—
389C22—2124no—
390C22—2124no—
391C22—2124no—
392C22—2124no—
393C22—2124no—
394C22—2122no—
395C22—2122no—
396C22#1681156411yescharacteristic
397C22D53⋊(C22⋊C8)11yescharacteristic
398C22D53⋊(C22⋊C8)11yescharacteristic
399C22C2×#1313327411yescharacteristic
400C22#1681159411yescharacteristic
401C22#1681159411yescharacteristic
402C22#1681159511yescharacteristic
403C22#1681160412yes—
404C22#1681161012yes—
405C4—24no—
406C4—42no—
407C4—42no—
408C4—1064no—
409C4—1064no—
410C4—2122no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—2124no—
416C2#2028994912yes—
417C2#2028996912yes—
418C2#2029053811yescharacteristic
419C2#2029054611yescharacteristic
420C2#2029055211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8300
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290606C2semidirect
2. C2×#16811468C2semidirect
32. C22×C4×D53C8semidirect
229. C53#8300semidirect

Maximal subgroups

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

subgroupindex
#202906062
C2×#168114682
C2×#168114672
C22×#131331752
#830053