Finite Group Databasebeta

← #28307196

Subgroups — #28307196

Order 13568.

1434 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C22×#13133278C212yesmaximal, complemented
2C2×#16811593C212yesmaximal
3C2×#16811503C212yesmaximal, complemented
4C22×#13133194C211yesmaximal, characteristic
5C22×#11719940C2214yes—
6C2×#13133278—28no—
7C2×#13133278—24no—
8#16811593—28no—
9C2×C22⋊#88324—24no—
10#16811503—24no—
11C23×Dic106C2211yescharacteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133194—24no—
14C2×#13133194—24no—
15C2×#13133194—24no—
16C22×C4⋊Dic53C411yescharacteristic
17C22×#11719919C411yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×C22⋊Dic53C411yescharacteristic
20C22×C22⋊C4×C53C411yescharacteristic
21C2×#11719940—24no—
22C2×#11719940—24no—
23C2×#11719940—24no—
24C2×#11719940—24no—
25C2×#11719940—24no—
26C2×#11719940D418yes—
27C22×#88324—24no—
28#13133278—216no—
29C22⋊#88324—216no—
30C22×Dic106—24no—
31C22×Dic106—22no—
32C22×Dic106—22no—
33C22×Dic106—24no—
34C22×Dic106D414yes—
35C23×Dic53—24no—
36C23×Dic53—22no—
37C23×Dic53—22no—
38C23×Dic53—24no—
39C23×Dic53C2×C411yescharacteristic
40C23×Dic53C2×C411yescharacteristic
41C23×Dic53C2×C411yescharacteristic
42C23×Dic53D414yes—
43C23×Dic53C2311yescharacteristic
44#13133194—232no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—22no—
48C2×#11719919—22no—
49C2×#11719919—22no—
50C2×#11719919—22no—
51C2×#11719919—22no—
52C2×#11719919—22no—
53C2×#11719919D414yes—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C25×C53C2×C411yescharacteristic
58C23×C4×C53C2×C411yescharacteristic
59C23×C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53—22no—
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—22no—
63C24×C53—24no—
64C24×C53—21no—
65C24×C53—21no—
66C24×C53—24no—
67C24×C53—42no—
68C24×C53C22⋊C412yes—
69C24×C53C22×C411yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—24no—
75C22×C4×C53—42no—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22⋊C412yes—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—24no—
85C22×Dic53—24no—
86C22×Dic53—28no—
87C22×Dic53—24no—
88C22×Dic53—28no—
89C22×Dic53—24no—
90C22×Dic53—42no—
91C22×Dic53—44no—
92C22×Dic53—44no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C22×Dic53C22⋊C412yes—
98C22×Dic53C22⋊C412yes—
99C22×Dic53C22⋊C412yes—
100C22×Dic53C2×D412yes—
101C22×Dic53C2×D412yes—
102C22×Dic53C2×D414yes—
103C22×Dic53C2×D412yes—
104C22×Dic53#4114yes—
105C2×Dic106—28no—
106C2×Dic106—24no—
107C2×Dic106—28no—
108C2×Dic106—24no—
109C2×Dic106—42no—
110C2×Dic106—44no—
111C2×Dic106—44no—
112C2×Dic106—42no—
113C2×Dic106—42no—
114C2×Dic106—42no—
115#11719940—28no—
116#11719940—28no—
117#11719940—44no—
118#11719940—44no—
119C2×#88324—28no—
120C2×#88324—44no—
121C2×#88324—44no—
122C22⋊C4×C53—28no—
123C22⋊Dic53—28no—
124C4⋊Dic53—28no—
125#11719919—28no—
126#11719919—28no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—24no—
130C23×C53—24no—
131C23×C53—22no—
132C23×C53—24no—
133C23×C53—22no—
134C23×C53—24no—
135C23×C53—416no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53#10114yescomplemented
141C23×C53C2×C22⋊C411yescharacteristic
142C23×C53C2×C22⋊C411yescharacteristic
143C23×C53C2×C22⋊C411yescharacteristic
144C23×C53C4×D414yes—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—24no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—24no—
157C2×Dic53—24no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—48no—
165C2×Dic53—48no—
166C2×Dic53—42no—
167C2×Dic53#12012yes—
168C2×Dic53#12012yes—
169C2×Dic53#12014yes—
170C2×Dic53#12114yes—
171C2×Dic53#12114yes—
172C2×Dic53#12114yes—
173C2×Dic53#12312yes—
174C2×Dic53#12312yes—
175C2×Dic53#12412yes—
176C2×Dic53#12712yes—
177Dic106—48no—
178Dic106—48no—
179#88324—48no—
180#10968—531nomaximal, Sylow
181C22×C53—28no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—28no—
187C22×C53—24no—
188C22×C53—48no—
189C22×C53—48no—
190C22×C53—44no—
191C22×C53—42no—
192C22×C53—42no—
193C22×C53#37911yescharacteristic
194C22×C53#38612yes—
195C22×C53#38612yes—
196C22×C53#39012yes—
197C22×C53C2×#10112yes—
198C22×C53#41312yes—
199C4×C53—42no—
200C4×C53—42no—
201Dic53—42no—
202Dic53—42no—
203Dic53—42no—
204Dic53#39214yes—
205Dic53#39414yes—
206C22×C22⋊Q8—531no—
207C22×#101—532no—
208C2×#407—532no—
209C2×#343—532no—
210C2×C53—22no—
211C2×C53—22no—
212C2×C53—44no—
213C2×C53#143811yescharacteristic
214C2×C53#178411yescharacteristic
215C2×C53#178511yescharacteristic
216C2×C53#190114yes—
217C23×Q8—531no—
218C24×C4—531no—
219C22×M16—534no—
220C2×C22⋊Q8—1064no—
221C2×C22⋊Q8—1064no—
222C2×C22⋊Q8—1064no—
223C22×C4⋊C4—531no—
224C22×C4⋊C4—531no—
225C22×C4⋊C4—531no—
226C22×C22⋊C4—531no—
227C22×C22⋊C4#615711yescharacteristic
228C2×#101—1068no—
229C2×#101—1064no—
230#407—1068no—
231C2×C22⋊C8—1064no—
232#343—1064no—
233C53#1096811yescharacteristic, complemented, Sylow
234C25C2×#615711yescharacteristic
235C22×Q8—534no—
236C22×Q8—1062no—
237C22×Q8—1062no—
238C22×Q8—1064no—
239C22×Q8—1064no—
240C23×C4—531no—
241C23×C4—534no—
242C23×C4—531no—
243C23×C4—531no—
244C23×C4—531no—
245C23×C4—1062no—
246C23×C4—1062no—
247C23×C4—1064no—
248C23×C4—1064no—
249C23×C4C2×#615711yescharacteristic
250C23×C4C2×#615711yescharacteristic
251C2×M16—538no—
252C2×M16—1064no—
253C2×M16—1064no—
254C2×M16—1064no—
255C2×M16—1064no—
256C2×M16—1064no—
257C22×C8—1064no—
258C22⋊Q8—10632no—
259C2×C4⋊C4—534no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1062no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4—1062no—
265C2×C4⋊C4—1062no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1062no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—1062no—
272C2×C22⋊C4—1062no—
273C2×C22⋊C4—1062no—
274#101—10616no—
275C22⋊C8—10616no—
276C24—24no—
277C24—21no—
278C24—21no—
279C24—24no—
280C24—42no—
281C24C22⋊#615712yes—
282C24C22×#615711yescharacteristic
283C2×Q8—1064no—
284C2×Q8—1068no—
285C2×Q8—1068no—
286C2×Q8—1064no—
287C2×Q8—2122no—
288C2×Q8—2124no—
289C2×Q8—2122no—
290C2×Q8—2124no—
291C2×Q8—2122no—
292C2×Q8—2122no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—42no—
299C22×C4—532no—
300C22×C4—532no—
301C22×C4—532no—
302C22×C4—534no—
303C22×C4—534no—
304C22×C4—532no—
305C22×C4—532no—
306C22×C4—532no—
307C22×C4—1064no—
308C22×C4—1068no—
309C22×C4—1068no—
310C22×C4—1064no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1064no—
318C22×C4—1064no—
319C22×C4—2122no—
320C22×C4—2124no—
321C22×C4—2122no—
322C22×C4—2124no—
323C22×C4—2122no—
324C22×C4—2122no—
325C22×C4—2122no—
326C22×C4C22⋊#615712yes—
327C22×C4C22⋊#615712yes—
328M16—1068no—
329M16—1068no—
330M16—2124no—
331M16—2124no—
332C2×C8—1068no—
333C2×C8—2124no—
334C2×C8—2124no—
335C4⋊C4—1068no—
336C4⋊C4—1068no—
337C4⋊C4—1068no—
338C22⋊C4—28no—
339C22⋊C4—1068no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—416no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23#1313327814yescomplemented
354C23C2×C22⋊#615711yescharacteristic
355C23C2×C22⋊#615711yescharacteristic
356C23C2×C22⋊#615711yescharacteristic
357C23D4×#615714yes—
358Q8—2128no—
359Q8—2128no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—534no—
369C2×C4—532no—
370C2×C4—532no—
371C2×C4—534no—
372C2×C4—534no—
373C2×C4—532no—
374C2×C4—532no—
375C2×C4—534no—
376C2×C4—532no—
377C2×C4—532no—
378C2×C4—1062no—
379C2×C4—1062no—
380C2×C4—1062no—
381C2×C4—1064no—
382C2×C4—1064no—
383C2×C4—2128no—
384C2×C4—2128no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C8—2128no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—28no—
399C22—24no—
400C22—48no—
401C22—48no—
402C22—42no—
403C22—44no—
404C22—42no—
405C22#1681156411yescharacteristic
406C22#1681157112yes—
407C22#1681157112yes—
408C22#1681157512yes—
409C22C2×#1313327812yes—
410C22#1681161012yes—
411C4—42no—
412C4—42no—
413C4—534no—
414C4—534no—
415C4—2122no—
416C4—2122no—
417C4—2122no—
418C2—22no—
419C2—22no—
420C2—44no—
421C2#2028989711yescharacteristic
422C2#2029054911yescharacteristic
423C2#2029055211yescharacteristic
424C2#2029074214yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#10968
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13133278C2semidirect
3. C2×#16811503C2semidirect
140. C23×C53#101semidirect
233. C53#10968semidirect
353. C23#13133278semidirect

Maximal subgroups

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

subgroupindex
C22×#131332782
C2×#168115932
C2×#168115032
C22×#131331942
#1096853