Finite Group Databasebeta

← #27301642

Subgroups — #27301642

Order 13568.

676 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 115 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×#13133136C211yesmaximal, characteristic
2C2×#16810422C212yesmaximal, complemented
3#20274639C211yesmaximal, characteristic
4#20272842C212yesmaximal, complemented
5C53×#2434C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133136—24no—
8C2×#13133136—21no—
9C2×#13133136—21no—
10C2×#13133136C2212yes—
11C2×#13133136C2212yes—
12#16810587—21no—
13#16810422—24no—
14C23×Q8×C53C2211yescharacteristic
15C2×C4⋊Q8×C53C411yescharacteristic
16C2×C4⋊Q8×C53C2211yescharacteristic
17C2×C22⋊Q8×C53C411yescharacteristic
18C53×#395—21no—
19C53×#395—21no—
20C53×#391C411yescharacteristic
21C53×#391C2211yescharacteristic
22C53×#385—21no—
23C53×#385—21no—
24C4×C22⋊C4×C53C411yescharacteristic
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—22no—
28C2×#11719917—41no—
29C2×#11719917—42no—
30C22×#88322—41no—
31#13133136—24no—
32#13133136—22no—
33#13133136—22no—
34#13133136—22no—
35#13133136—42no—
36#13133136D414yes—
37C22⋊#88322—41no—
38C22⋊#88322—41no—
39C22×Q8×C53—22no—
40C22×Q8×C53—22no—
41C22×Q8×C53—41no—
42C22×Q8×C53—41no—
43C22×Q8×C53—42no—
44C22×Q8×C53C2×C411yescharacteristic
45C22×Q8×C53C2×C411yescharacteristic
46C22×Q8×C53D411yescharacteristic
47C22×Q8×C53D411yescharacteristic
48C22×Q8×C53C2311yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4⋊Q8×C53—22no—
53C4⋊Q8×C53—22no—
54C4⋊Q8×C53—42no—
55C22⋊Q8×C53—41no—
56C22⋊Q8×C53—41no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C42×C53C2×C411yescharacteristic
66C2×C42×C53C2×C411yescharacteristic
67#95×C53—21no—
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53C22⋊C411yescharacteristic
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—22no—
78C2×Q8×C53—21no—
79C2×Q8×C53—22no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C2×Q8×C53—44no—
87C2×Q8×C53—44no—
88C2×Q8×C53—44no—
89C2×Q8×C53—44no—
90C2×Q8×C53—41no—
91C2×Q8×C53—41no—
92C2×Q8×C53C2×D412yes—
93C2×Q8×C53#4112yes—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C22⋊C411yescharacteristic
113C22×C4×C53C22⋊C411yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C4⋊C4×C53—42no—
118C4⋊C4×C53—42no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—42no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—42no—
123C4⋊C4×C53—41no—
124C22⋊C4×C53—24no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53—22no—
128C42×C53—41no—
129#11719917—24no—
130#11719917—44no—
131#11719917—42no—
132#11719917—42no—
133#11719917—42no—
134C2×#88322—44no—
135C2×#88322—42no—
136C2×#88322—42no—
137C23×C53—22no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—41no—
141C23×C53#9911yescharacteristic
142C23×C53#9911yescharacteristic
143C23×C53C2×C22⋊C411yescharacteristic
144Q8×C53—42no—
145Q8×C53—42no—
146Q8×C53—44no—
147Q8×C53—44no—
148Q8×C53—42no—
149Q8×C53—42no—
150Q8×C53—82no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53#9912yes—
185C2×C4×C53C4×D412yes—
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12112yes—
188C2×C4×C53#12311yescharacteristic
189#88322—44no—
190#88322—82no—
191#13094—531nomaximal, Sylow
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53#35612yescomplemented
202C22×C53#38611yescharacteristic
203C22×C53C2×#9911yescharacteristic
204C22×C53C2×#9911yescharacteristic
205C4×C53—21no—
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—21no—
209C4×C53—22no—
210C4×C53—42no—
211C4×C53—42no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—81no—
215C4×C53—81no—
216C22×#101—531no—
217#2434D5311yescharacteristic
218C2×#356—532no—
219#1784—531no—
220#1394—532no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53—41no—
225C2×C53#189811yescharacteristic
226C2×C53C2×#35611yescharacteristic
227C2×C53#213311yescharacteristic
228C23×Q8C2×D5311yescharacteristic
229C22×M16—1061no—
230C2×C4⋊Q8Dic5311yescharacteristic
231C2×C4⋊Q8C2×D5311yescharacteristic
232C2×C22⋊Q8Dic5311yescharacteristic
233C2×#101—532no—
234C2×#101—532no—
235C2×#101—1064no—
236C2×#101—1061no—
237C2×#101—1061no—
238#407—1061no—
239#395—21no—
240#395—21no—
241#391Dic5311yescharacteristic
242#391C2×D5311yescharacteristic
243#385—21no—
244#385—21no—
245C4×C22⋊C4Dic5311yescharacteristic
246#356—1064no—
247C53#1309411yescharacteristic, complemented, Sylow
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—41no—
251C22×Q8—41no—
252C22×Q8—42no—
253C22×Q8C2×Dic5311yescharacteristic
254C22×Q8C2×Dic5311yescharacteristic
255C22×Q8C22⋊D5311yescharacteristic
256C22×Q8C22⋊D5311yescharacteristic
257C22×Q8C22×D5311yescharacteristic
258C23×C4C2×Dic5311yescharacteristic
259C23×C4C22⋊D5311yescharacteristic
260C23×C4C22⋊D5311yescharacteristic
261C2×M16—1064no—
262C2×M16—1062no—
263C2×M16—1062no—
264C2×M16—2121no—
265C2×M16—2122no—
266C22×C8—2121no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—42no—
270C22⋊Q8—41no—
271C22⋊Q8—41no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C2×Dic5311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42C2×Dic5311yescharacteristic
281C2×C42C2×Dic5311yescharacteristic
282#101—534no—
283#101—1064no—
284#101—1062no—
285#101—1062no—
286#101—1062no—
287#101—2122no—
288C22⋊C8—2121no—
289C22⋊C8—2121no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295C24C22⋊Dic5311yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8D4×D5312yes—
316C2×Q8#1171994712yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C22⋊Dic5311yescharacteristic
335C22×C4C22⋊Dic5311yescharacteristic
336C22×C4C22⋊Dic5311yescharacteristic
337C22×C4C22×Dic5311yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339C22×C4C2×C22⋊D5311yescharacteristic
340M16—1064no—
341M16—2122no—
342M16—2122no—
343M16—2124no—
344M16—2122no—
345C2×C8—2122no—
346C2×C8—2124no—
347C2×C8—2122no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42—22no—
359C42—41no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—41no—
364C23#1313313211yescharacteristic
365C23#1313313211yescharacteristic
366C23C2×C22⋊Dic5311yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—44no—
370Q8—44no—
371Q8—42no—
372Q8—42no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4#1313313212yes—
408C2×C4D4×Dic5312yes—
409C2×C4#1313318511yescharacteristic
410C2×C4#1313318912yes—
411C2×C4#1313319811yescharacteristic
412C8—4242no—
413C8—2124no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#1681042212yescomplemented
424C22#1681050311yescharacteristic
425C22C2×#1313313211yescharacteristic
426C22C2×#1313313211yescharacteristic
427C4—22no—
428C4—21no—
429C4—22no—
430C4—21no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#2027516911yescharacteristic
443C2C2×#1681042211yescharacteristic
444C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2434C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2434D53
hypocenterC53#13094
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#16810422C2semidirect
4. #20272842C2semidirect
201. C22×C53#356semidirect
247. C53#13094semidirect
423. C22#16810422semidirect

Maximal subgroups

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

subgroupindex
C22×#131331362
C2×#168104222
#202746392
#202728422
C53×#24342
#1309453