Finite Group Databasebeta

← #27299649

Subgroups — #27299649

Order 13568.

986 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 135 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#20276410C211yesmaximal, characteristic, complemented
2#20276390C211yesmaximal, characteristic, complemented
3#20276356C212yesmaximal, complemented
4#20274685C212yesmaximal, complemented
5C53×#3485C211yesmaximal, characteristic
6#16810640—22no—
7#16810628—22no—
8#16810607C2212yes—
9#16810597C2212yes—
10#16810422—22no—
11#16810418—22no—
12#16810413—22no—
13#16810410—22no—
14C2×H(5,𝔽2)×C53C2211yescharacteristic
15C22×#41×C53C411yescharacteristic, complemented
16C53×#547—22no—
17C53×#547—22no—
18C53×#546—22no—
19C53×#546—22no—
20C53×#536C411yescharacteristic, complemented
21C53×#536C2211yescharacteristic
22C53×#534—22no—
23C53×#525C411yescharacteristic, complemented
24C53×#525C2211yescharacteristic
25C2×#123×C53—22no—
26C53×#518C411yescharacteristic, complemented
27C2×#11719917—22no—
28#13133176—22no—
29#13133144—44no—
30#13133138—44no—
31#13133136—22no—
32#13133134—22no—
33#13133132—22no—
34#13133132—22no—
35H(5,𝔽2)×C53—44no—
36C2×#41×C53—22no—
37C2×#41×C53—21no—
38C2×#41×C53—22no—
39C2×#41×C53—44no—
40C2×#41×C53C2×C411yescharacteristic
41C2×#41×C53D412yes—
42C2×#41×C53C2311yescharacteristic
43C22×Q8×C53C2×C411yescharacteristic
44C22×D4×C53—21no—
45C22×D4×C53—22no—
46C22×D4×C53—21no—
47C22×D4×C53C2×C411yescharacteristic
48C22×D4×C53D412yes—
49C23×C4×C53—21no—
50C23×C4×C53C2×C411yescharacteristic
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53D412yes—
53#124×C53—21no—
54#124×C53D412yes—
55#123×C53—24no—
56#123×C53—24no—
57#123×C53—24no—
58#123×C53—22no—
59#123×C53—22no—
60#123×C53—21no—
61#123×C53D412yes—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—22no—
66#121×C53—24no—
67#121×C53—24no—
68#121×C53—21no—
69#121×C53—21no—
70#120×C53—24no—
71#120×C53—21no—
72#120×C53D412yes—
73C4×D4×C53—22no—
74C4×D4×C53—22no—
75C4×D4×C53—22no—
76C4×D4×C53—22no—
77#117×C53C2×C411yescharacteristic
78#117×C53C2×C411yescharacteristic
79C2×C4⋊C4×C53—24no—
80C2×C22⋊C4×C53—24no—
81C2×C22⋊C4×C53—22no—
82C24×C53—21no—
83C24×C53—22no—
84C24×C53C22⋊C411yescharacteristic
85#41×C53—24no—
86#41×C53—42no—
87#41×C53—42no—
88#41×C53—42no—
89#41×C53—48no—
90#41×C53—44no—
91#41×C53—44no—
92#41×C53—44no—
93C2×Q8×C53—21no—
94C2×Q8×C53—42no—
95C2×Q8×C53C22⋊C411yescharacteristic
96C2×Q8×C53C22⋊C412yes—
97C2×Q8×C53C2×D411yescharacteristic
98C2×D4×C53—24no—
99C2×D4×C53—24no—
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53—22no—
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—42no—
108C2×D4×C53—44no—
109C2×D4×C53—44no—
110C2×D4×C53—48no—
111C2×D4×C53C22⋊C411yescharacteristic
112C2×D4×C53C22⋊C412yes—
113C2×D4×C53C2×D411yescharacteristic
114C2×D4×C53C2×D412yes—
115C22×C4×C53—24no—
116C22×C4×C53—21no—
117C22×C4×C53—24no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—42no—
123C22×C4×C53—44no—
124C22×C4×C53C22⋊C411yescharacteristic
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C2×D412yes—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—24no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—24no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—24no—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—21no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—24no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53C22⋊C412yes—
142C42×C53—21no—
143C42×C53C22⋊C412yes—
144C22⋊Dic53—42no—
145C4⋊Dic53—42no—
146C4×Dic53—42no—
147#11719917—42no—
148#11719917—42no—
149C2×#88322—42no—
150C23×C53—24no—
151C23×C53—22no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—44no—
155C23×C53—41no—
156C23×C53—81no—
157C23×C53—82no—
158C23×C53—84no—
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53#12012yes—
161Q8×C53—22no—
162Q8×C53—41no—
163Q8×C53—42no—
164Q8×C53—42no—
165D4×C53—24no—
166D4×C53—22no—
167D4×C53—22no—
168D4×C53—42no—
169D4×C53—41no—
170D4×C53—44no—
171D4×C53—44no—
172D4×C53—42no—
173D4×C53—48no—
174D4×C53—82no—
175D4×C53—82no—
176D4×C53#9914yes—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—44no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—48no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—84no—
193C2×C4×C53C2×C22⋊C411yescharacteristic
194C2×C4×C53C2×C22⋊C411yescharacteristic
195C2×C4×C53#12012yes—
196C2×Dic53—82no—
197#88322—82no—
198#12713—531nomaximal, Sylow
199C22×C53—22no—
200C22×C53—21no—
201C22×C53—24no—
202C22×C53—42no—
203C22×C53—41no—
204C22×C53—44no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—42no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—84no—
211C22×C53—82no—
212C22×C53—164no—
213C22×C53#37911yescharacteristic
214C4×C53—22no—
215C4×C53—21no—
216C4×C53—42no—
217C4×C53—42no—
218C4×C53—41no—
219C4×C53—41no—
220C4×C53—42no—
221C4×C53C2×#9912yes—
222Dic53—162no—
223#3485D5311yescharacteristic
224#2135—531no—
225#2129—531no—
226#2120—532no—
227#1795—532no—
228C2×C53—21no—
229C2×C53—42no—
230C2×C53—44no—
231C2×C53—41no—
232C2×C53—81no—
233C2×C53—81no—
234C2×C53#178111yescharacteristic
235C2×H(5,𝔽2)C2×D5311yescharacteristic
236C22×#41Dic5311yescharacteristic, complemented
237#547—22no—
238#547—22no—
239#546—22no—
240#546—22no—
241#536Dic5311yescharacteristic, complemented
242#536C2×D5311yescharacteristic
243#534—22no—
244#525Dic5311yescharacteristic, complemented
245#525C2×D5311yescharacteristic
246C2×#123—22no—
247#518Dic5311yescharacteristic, complemented
248#421—1062no—
249#418—1062no—
250#413—532no—
251#410—532no—
252#356—1062no—
253#354—1062no—
254#352—1062no—
255AΣL(1,𝔽16)—1062no—
256C53#1271311yescharacteristic, complemented, Sylow
257H(5,𝔽2)—44no—
258C2×#41—21no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—44no—
262C2×#41C2×Dic5311yescharacteristic
263C2×#41C22⋊D5312yes—
264C2×#41C22×D5311yescharacteristic
265C22×Q8C2×Dic5311yescharacteristic
266C22×D4—21no—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4C2×Dic5311yescharacteristic
270C22×D4C22⋊D5312yes—
271C23×C4—21no—
272C23×C4C2×Dic5311yescharacteristic
273C2×M16—1062no—
274C4⋊Q8—21no—
275C4⋊Q8C22⋊D5312yes—
276#124—21no—
277#124C22⋊D5312yes—
278#123—24no—
279#123—24no—
280#123—24no—
281#123—22no—
282#123—21no—
283#123—22no—
284#123C22⋊D5312yes—
285C22⋊Q8—21no—
286C22⋊Q8—21no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289#121—24no—
290#121—24no—
291#121—21no—
292#121—21no—
293#120—24no—
294#120—21no—
295#120C22⋊D5312yes—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300#117—1062no—
301#117C2×Dic5311yescharacteristic
302#117C2×Dic5311yescharacteristic
303C2×C4⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306#104—2124no—
307#102—2124no—
308#101—1062no—
309#100—1062no—
310#99—1062no—
311#99—1062no—
312C24—21no—
313C24—22no—
314C24C22⋊Dic5311yescharacteristic
315#41—24no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—48no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8C22⋊Dic5312yes—
326C2×Q8C22⋊Dic5311yescharacteristic
327C2×Q8C2×C22⋊D5311yescharacteristic
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4C22⋊Dic5311yescharacteristic
342C2×D4C22⋊Dic5312yes—
343C2×D4C2×C22⋊D5311yescharacteristic
344C2×D4C2×C22⋊D5312yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4C22⋊Dic5311yescharacteristic
355C22×C4C22×Dic5311yescharacteristic
356C22×C4C2×C22⋊D5312yes—
357M16—2122no—
358M16—2122no—
359C2×C8—2122no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—2122no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—21no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—2122no—
376C22⋊C4C22⋊Dic5312yes—
377C42—21no—
378C42—2122no—
379C42C22⋊Dic5312yes—
380C23—24no—
381C23—22no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—41no—
386C23—81no—
387C23—82no—
388C23—84no—
389C23C2×C22⋊Dic5311yescharacteristic
390C23#1313318612yes—
391Q8—22no—
392Q8—41no—
393Q8—42no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—22no—
398D4—42no—
399D4—41no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—48no—
404D4—82no—
405D4—82no—
406D4#1313313214yes—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—84no—
423C2×C4—4242no—
424C2×C4C2×C22⋊Dic5311yescharacteristic
425C2×C4C2×C22⋊Dic5311yescharacteristic
426C2×C4#1313318612yes—
427C8—4242no—
428C22—21no—
429C22—22no—
430C22—24no—
431C22—42no—
432C22—41no—
433C22—44no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—84no—
440C22—82no—
441C22—164no—
442C22#1681047911yescharacteristic
443C4—22no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—41no—
448C4—41no—
449C4—42no—
450C4—8482no—
451C4C2×#1313313212yes—
452C2—21no—
453C2—42no—
454C2—44no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3485C2
socleC2×C53#1781
cosocleC2×#41×C53C23
hypercenter#3485D53
hypocenterC53#12713
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #20276410C2semidirect
2. #20276390C2semidirect
3. #20276356C2semidirect
4. #20274685C2semidirect
15. C22×#41×C53C4semidirect
20. C53×#536C4semidirect
23. C53×#525C4semidirect
26. C53×#518C4semidirect
236. C22×#41Dic53semidirect
241. #536Dic53semidirect
244. #525Dic53semidirect
247. #518Dic53semidirect
256. C53#12713semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202763902
#202763562
#202746852
C53×#34852
#1271353