Finite Group Databasebeta

← #27300080

Subgroups — #27300080

Order 13568.

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

normal subgroups only

This group has 588 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#20277437C211yesmaximal, characteristic
2C2×#16810415C212yesmaximal
3#20274703C211yesmaximal, characteristic, complemented
4#20272839C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic, complemented
6C2×#13133213C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810526C2211yescharacteristic
10#16810524C411yescharacteristic, complemented
11#16810502—21no—
12#16810502—21no—
13#16810499—21no—
14#16810499—21no—
15#16810488C411yescharacteristic, complemented
16#16810483—21no—
17#16810469C411yescharacteristic, complemented
18#16810415—24no—
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C53×#380—21no—
23C53×#342—21no—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27C22×C4×D53—21no—
28#13133213—21no—
29#13133213—21no—
30#13133213D412yes—
31#13133213Q812yes—
32C4×D212—41no—
33C4×D212—41no—
34C2×C4⋊Dic53—41no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—22no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44#13133132—22no—
45#13133132—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—41no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C4×D4×C53—41no—
55C4×D4×C53—41no—
56C2×C4⋊C4×C53—41no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C42×C53C2×C411yescharacteristic
62#99×C53—22no—
63#99×C53—22no—
64#95×C53—41no—
65C24×C53C2×D411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C23×D53—21no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53C2×D411yescharacteristic
71C2×D4×C53C2×Q811yescharacteristic
72C2×D4×C53#4111yescharacteristic
73C2×D4×C53#4111yescharacteristic
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—42no—
79C22×C4×C53—81no—
80C22×C4×C53—81no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22×C411yescharacteristic
86C2×C22⋊D53—22no—
87C2×C22⋊D53—22no—
88C2×C22⋊D53—22no—
89C2×C22⋊D53—22no—
90C2×C22⋊D53—81no—
91C2×C22⋊D53—81no—
92C2×C22⋊D53—81no—
93C2×C22⋊D53—81no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—42no—
98C2×D212—21no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212C4⋊C412yescomplemented
102C2×C4×D53—42no—
103C2×C4×D53—81no—
104C2×C4×D53—81no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—82no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C42×C53—21no—
112C42×C53—21no—
113C22⋊Dic53—42no—
114C22⋊Dic53—42no—
115C22⋊Dic53—81no—
116C22⋊Dic53—81no—
117#11719921—41no—
118#11719921—41no—
119#11719921—81no—
120#11719921—81no—
121C4⋊Dic53—41no—
122C4⋊Dic53—82no—
123C4×Dic53—21no—
124C4×Dic53—41no—
125C4×Dic53C4⋊C412yescomplemented
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53#12111yescharacteristic
137C23×C53C22⋊Q811yescharacteristic
138C23×C53C22⋊Q811yescharacteristic
139C22×D53—22no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149D4×C53—41no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—82no—
164C2×C4×C53—82no—
165C2×C4×C53—82no—
166C2×C4×C53#9911yescharacteristic
167C2×C4×C53#9911yescharacteristic
168C2×C4×C53C2×C4⋊C411yescharacteristic
169C2×C4×C53#11711yescharacteristic
170C22⋊D53—82no—
171C22⋊D53—82no—
172C22⋊D53—82no—
173C22⋊D53—82no—
174C2×Dic53—22no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—82no—
184C2×Dic53—82no—
185C2×Dic53—82no—
186D212—42no—
187D212—42no—
188C4×D53—84no—
189#12799—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—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—81no—
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—81no—
215C2×D53—81no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—162no—
224Dic53—42no—
225Dic53—81no—
226Dic53—81no—
227Dic53—162no—
228#2328—531no—
229C2×#353—532no—
230#1799—531no—
231#1799D5311yescharacteristic, complemented
232#1393—532no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—41no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—41no—
239C2×C53#179911yescharacteristic
240C2×C53#212711yescharacteristic
241C2×C53#212811yescharacteristic
242D53—82no—
243C2×#127—531no—
244C2×C4×D4—531no—
245C2×C4×D4C2×D5311yescharacteristic
246C2×#99—532no—
247C2×#99C2×D5312yes—
248#390—531no—
249#390—531no—
250#386—1061no—
251#386—1061no—
252#386—1061no—
253#386—1061no—
254#382—531no—
255#380—21no—
256#380—1061no—
257#376—531no—
258#353—1064no—
259#342—21no—
260#342—1061no—
261C53#1279911yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4C22×D5311yescharacteristic
266C23×C4—1061no—
267C23×C4D21211yescharacteristic
268C23×C4C22⋊D5311yescharacteristic
269#127—532no—
270#127—532no—
271#127—1061no—
272#127—1061no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—2121no—
276C4×D4—2121no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—2121no—
280C2×C4⋊C4C4×D5311yescharacteristic
281C2×C22⋊C4—22no—
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⋊C4D21211yescharacteristic
289C2×C22⋊C4C22⋊D5311yescharacteristic
290C2×C42—531no—
291C2×C42—531no—
292C2×C42C4×D5311yescharacteristic
293#99—22no—
294#99—22no—
295#99—1062no—
296#99—1062no—
297#95—41no—
298#95—1061no—
299#95—1061no—
300#95—1061no—
301#95—1061no—
302#95—2121no—
303C24—1061no—
304C24C2×D21211yescharacteristic
305C24C2×C22⋊D5311yescharacteristic
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—532no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—1061no—
314C2×D4—4241no—
315C2×D4—4241no—
316C2×D4—4241no—
317C2×D4—4241no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4D4×D5311yescharacteristic
321C2×D4#1171994711yescharacteristic
322C2×D4Q8×D5311yescharacteristic
323C2×D4#1171994911yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4—4241no—
335C22×C4—4241no—
336C22×C4—2122no—
337C22×C4—2122no—
338C22×C4#1171992111yescharacteristic
339C22×C4#1171992111yescharacteristic
340C22×C4#1171992111yescharacteristic
341C22×C4#1171992111yescharacteristic
342C22×C4C2×C4×D5311yescharacteristic
343C4⋊C4—41no—
344C4⋊C4—82no—
345C4⋊C4—4242no—
346C4⋊C4—2121no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—4241no—
352C22⋊C4—4241no—
353C22⋊C4—4241no—
354C22⋊C4—4241no—
355C22⋊C4—2122no—
356C22⋊C4—2122no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C42—21no—
360C42—21no—
361C42—532no—
362C42—1061no—
363C42—2121no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—1062no—
373C23—2121no—
374C23—2121no—
375C23—2121no—
376C23—2121no—
377C23—2121no—
378C23—2121no—
379C23C2×#1171992111yescharacteristic
380C23#1313318811yescharacteristic
381C23#1313319211yescharacteristic
382C23#1313319311yescharacteristic
383C23#1313319711yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—4242no—
389D4—4242no—
390D4—4242no—
391D4—4242no—
392D4—2122no—
393D4—2122no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—1062no—
411C2×C4—4244no—
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×C4C4⋊C4×D5311yescharacteristic
426C2×C4#1313317511yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—81no—
442C22—81no—
443C22—4241no—
444C22—4241no—
445C22—4241no—
446C22—4241no—
447C22—2122no—
448C22—2122no—
449C22—2122no—
450C22#1681048011yescharacteristic
451C22C2×#1313313111yescharacteristic
452C22#1681059611yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—162no—
461C4—8482no—
462C4—4241no—
463C4—4241no—
464C4—2122no—
465C2—21no—
466C2—21no—
467C2—41no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—4242no—
472C2#2027470011yescharacteristic
473C2#2027638211yescharacteristic
474C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D4×C53C23
hypercenter#1799D53
hypocenterC53#12799
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20274703C2semidirect
4. #20272839C2semidirect
5. C53×#1799C2semidirect
7. C2×C4×D212C4semidirect
10. #16810524C4semidirect
15. #16810488C4semidirect
17. #16810469C4semidirect
101. C2×D212C4⋊C4semidirect
125. C4×Dic53C4⋊C4semidirect
231. #1799D53semidirect
261. C53#12799semidirect

Maximal subgroups

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

subgroupindex
#202774372
C2×#168104152
#202747032
#202728392
C53×#17992
#1279953