Finite Group Databasebeta

← #27300066

Subgroups — #27300066

Order 13568.

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

normal subgroups only

This group has 600 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#20277369C211yesmaximal, characteristic
2C2×#16810411C212yesmaximal
3#20274703C211yesmaximal, characteristic, complemented
4#20272834C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic, complemented
6C2×#13133185C2211yescharacteristic
7C22×#11719921C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810523—21no—
10#16810514—21no—
11#16810513—21no—
12#16810503C2211yescharacteristic
13#16810502—21no—
14#16810495C411yescharacteristic, complemented
15#16810483—21no—
16#16810478C411yescharacteristic, complemented
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810411—24no—
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#423—21no—
22C2×#100×C53C2212yes—
23C53×#343—21no—
24C24×D53C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C23×Dic53C2×C411yescharacteristic
27#13133185—21no—
28#13133185—21no—
29#13133185D412yes—
30#13133185Q812yes—
31C2×C4⋊Dic53—41no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—41no—
43C2×#11719921—41no—
44C2×C4×Dic53—21no—
45#13133132—22no—
46#13133132—22no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—41no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C2×M16×C53—22no—
55C4×D4×C53—41no—
56C4×D4×C53—41no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C4⋊C8×C53—41no—
62C4⋊C8×C53—41no—
63#100×C53—22no—
64#100×C53—22no—
65C24×C53C22×C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C23×D53—21no—
68C23×D53—21no—
69C23×D53—41no—
70C23×D53—41no—
71C23×D53—81no—
72C23×D53—81no—
73C23×D53C4⋊C412yescomplemented
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53C2×D411yescharacteristic
77C2×D4×C53C2×Q811yescharacteristic
78C2×D4×C53#4111yescharacteristic
79C2×D4×C53#4111yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—81no—
85C22×C4×C53—81no—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C2×C22⋊D53—22no—
92C2×C22⋊D53—22no—
93C2×C22⋊D53—81no—
94C2×C22⋊D53—81no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C22×Dic53—42no—
100C22×Dic53—81no—
101M16×C53—42no—
102M16×C53—42no—
103C2×C8×C53—42no—
104C4⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—21no—
108C42×C53—21no—
109C22⋊Dic53—21no—
110C22⋊Dic53—41no—
111C22⋊Dic53—42no—
112C22⋊Dic53—42no—
113C22⋊Dic53—81no—
114C22⋊Dic53C4⋊C412yescomplemented
115#11719921—22no—
116#11719921—22no—
117#11719921—42no—
118#11719921—41no—
119#11719921—41no—
120#11719921—81no—
121#11719921—81no—
122C4⋊Dic53—82no—
123#11719919—81no—
124C4×Dic53—81no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C22⋊C411yescharacteristic
134C23×C53C2×C4⋊C411yescharacteristic
135C23×C53#11711yescharacteristic
136C23×C53#12111yescharacteristic
137C23×C53C22⋊Q811yescharacteristic
138C22×D53—22no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—42no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—41no—
146C22×D53—42no—
147C22×D53—82no—
148C22×D53—82no—
149C22×D53—84no—
150C22×D53—82no—
151C22×D53—82no—
152D4×C53—41no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53#9911yescharacteristic
170C2×C4×C53#9911yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53C22⋊Q811yescharacteristic
173C22⋊D53—82no—
174C22⋊D53—82no—
175C2×Dic53—22no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—82no—
185C2×Dic53—82no—
186C2×Dic53—82no—
187C2×Dic53—82no—
188C8×C53—82no—
189#12795—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—42no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—81no—
206C22×C53#38011yescharacteristic
207C22×C53C2×#9911yescharacteristic
208C22×C53#41011yescharacteristic
209C2×D53—42no—
210C2×D53—42no—
211C2×D53—42no—
212C2×D53—81no—
213C2×D53—81no—
214C2×D53—84no—
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—84no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225Dic53—42no—
226Dic53—81no—
227Dic53—81no—
228Dic53—162no—
229#2321—531no—
230C2×AΣL(1,𝔽16)—532no—
231#1800D5311yescharacteristic, complemented
232#1799—531no—
233#1390—532no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—41no—
239C2×C53—41no—
240C2×C53#179911yescharacteristic
241C2×C53#212111yescharacteristic
242C2×C53#212211yescharacteristic
243D53—82no—
244C2×#120—531no—
245C2×C4×D4C2×D5311yescharacteristic
246C22×C22⋊C4—531no—
247#423—21no—
248C2×#100C2×D5312yes—
249C2×#99—532no—
250#390—1061no—
251#388—1061no—
252#388—1061no—
253#386—531no—
254#386—1061no—
255#385—531no—
256#380—1061no—
257#379—531no—
258C4×C22⋊C4—531no—
259AΣL(1,𝔽16)—1064no—
260#343—21no—
261#342—1061no—
262C53#1279511yescharacteristic, complemented, Sylow
263C25—531no—
264C22×D4—1061no—
265C22×D4C22×D5311yescharacteristic
266C23×C4—531no—
267C23×C4C4×D5311yescharacteristic
268C23×C4D21211yescharacteristic
269C2×M16—22no—
270#120—532no—
271#120—532no—
272#120—1061no—
273#120—1061no—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—2121no—
278C2×C4⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—1061no—
282C2×C22⋊C4—1061no—
283C2×C22⋊C4—1061no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4—1061no—
286C2×C22⋊C4—1061no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—2121no—
289C2×C22⋊C4—2121no—
290C2×C22⋊C4C4×D5311yescharacteristic
291C2×C22⋊C4D21211yescharacteristic
292C2×C42—1061no—
293C2×C42C22⋊D5311yescharacteristic
294C4⋊C8—41no—
295C4⋊C8—41no—
296#100—22no—
297#100—22no—
298#99—1062no—
299#99—1062no—
300#95—1061no—
301#95—1061no—
302#95—1061no—
303#95—2121no—
304C24—532no—
305C24—1061no—
306C24—1061no—
307C24—4241no—
308C24—4241no—
309C24—2121no—
310C24—2121no—
311C24C2×C4×D5311yescharacteristic
312C24C2×D21211yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—4241no—
318C2×D4—4241no—
319C2×D4D4×D5311yescharacteristic
320C2×D4#1171994711yescharacteristic
321C2×D4Q8×D5311yescharacteristic
322C2×D4#1171994911yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—4241no—
333C22×C4—2122no—
334C22×C4—2121no—
335C22×C4#1171992111yescharacteristic
336C22×C4#1171992111yescharacteristic
337C22×C4#1171992111yescharacteristic
338C22×C4#1171992111yescharacteristic
339C22×C4C2×C22⋊D5311yescharacteristic
340M16—42no—
341M16—42no—
342C2×C8—42no—
343C4⋊C4—41no—
344C4⋊C4—4242no—
345C4⋊C4—4241no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—532no—
349C22⋊C4—1061no—
350C22⋊C4—1062no—
351C22⋊C4—1062no—
352C22⋊C4—4241no—
353C22⋊C4—4241no—
354C22⋊C4—4241no—
355C22⋊C4—2122no—
356C22⋊C4—2122no—
357C22⋊C4—2122no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C42—21no—
362C42—21no—
363C42—4241no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—1062no—
373C23—4242no—
374C23—4242no—
375C23—4244no—
376C23—4242no—
377C23—4242no—
378C23—2122no—
379C23—2121no—
380C23—2121no—
381C23—2121no—
382C23—2121no—
383C23—2122no—
384C23—2121no—
385C23—2121no—
386C23C2×#1171992111yescharacteristic
387C23C4⋊C4×D5311yescharacteristic
388C23#1313317511yescharacteristic
389C23#1313319211yescharacteristic
390C23#1313319711yescharacteristic
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—4242no—
396D4—4242no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—1062no—
411C2×C4—4242no—
412C2×C4—4242no—
413C2×C4—4242no—
414C2×C4—4242no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4#1313313111yescharacteristic
424C2×C4#1313313111yescharacteristic
425C2×C4#1313318811yescharacteristic
426C2×C4#1313319311yescharacteristic
427C8—82no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22—4241no—
445C22—4241no—
446C22—4244no—
447C22—4244no—
448C22—4241no—
449C22—4241no—
450C22—2122no—
451C22—2122no—
452C22—2122no—
453C22#1681048011yescharacteristic
454C22C2×#1313313111yescharacteristic
455C22#1681059611yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—8482no—
464C4—4241no—
465C4—4241no—
466C4—2122no—
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—4242no—
474C2#2027470011yescharacteristic
475C2#2027636211yescharacteristic
476C2#2027636611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radicalC2×#13133185C22
cosocleC22×D4×C53C23
hypercenter#1800D53
hypocenterC53#12795
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20274703C2semidirect
4. #20272834C2semidirect
5. C53×#1800C2semidirect
7. C22×#11719921C4semidirect
14. #16810495C4semidirect
16. #16810478C4semidirect
17. C4×C22⋊Dic53C4semidirect
73. C23×D53C4⋊C4semidirect
114. C22⋊Dic53C4⋊C4semidirect
231. #1800D53semidirect
262. C53#12795semidirect

Maximal subgroups

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

subgroupindex
#202773692
C2×#168104112
#202747032
#202728342
C53×#18002
#1279553