Finite Group Databasebeta

← #27288489

Subgroups — #27288489

Order 13568.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285487C211yesmaximal, characteristic
2#20275700C211yesmaximal, characteristic
3#20275139C211yesmaximal, characteristic
4#20275136C211yesmaximal, characteristic
5#20274895C211yesmaximal, characteristic
6#20272582C211yesmaximal, characteristic
7C53×#1996C211yesmaximal, characteristic
8C23×#88322C2211yescharacteristic
9C2×#13133196C2211yescharacteristic
10#16810958—21no—
11#16810956—21no—
12#16810862—21no—
13#16810862—21no—
14#16810779—21no—
15#16810778—21no—
16C2×C4⋊#88322—21no—
17C2×#13133141—21no—
18C2×#13133140—21no—
19C2×C22⋊#88322C2211yescharacteristic
20#16810483C2211yescharacteristic
21#16810389—21no—
22#16810389—21no—
23#16810385—21no—
24#16810385—21no—
25#16810378—21no—
26C2×C22⋊Q8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C4×C22⋊C4×C53C2211yescharacteristic
29C53×#342—21no—
30C53×#342—21no—
31C22×Dic106D411yescharacteristic
32C2×#11719925—21no—
33C2×#11719925—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—42no—
38C22×#88322D411yescharacteristic
39C22×#88322D411yescharacteristic
40#13133196—21no—
41#13133196—21no—
42#13133196—21no—
43#13133196—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×#11719919—21no—
47C2×C22⋊Dic53—21no—
48C2×C22⋊Dic53—21no—
49#13133150—21no—
50#13133150—21no—
51C4⋊#88322—42no—
52#13133141—21no—
53#13133141—21no—
54#13133141—42no—
55#13133140—21no—
56#13133140—21no—
57#13133140—42no—
58C22⋊#88322—21no—
59C22⋊#88322—21no—
60#13133124—21no—
61C22×Q8×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53D411yescharacteristic
73C2×C22⋊C4×C53D411yescharacteristic
74C2×C42×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—41no—
98C2×Dic106—21no—
99C2×Dic106—21no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53Q1612yescomplemented
116C42×C53—41no—
117C42×C53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120C22⋊Dic53—41no—
121C22⋊Dic53—41no—
122#11719925—81no—
123#11719925—81no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128#11719919—41no—
129#11719919—41no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53#12111yescharacteristic
144C23×C53#12111yescharacteristic
145C23×C53#12111yescharacteristic
146C23×C53C2×Q1611yescharacteristic
147C23×C53#13511yescharacteristic
148C23×C53#13711yescharacteristic
149C23×C53#13711yescharacteristic
150Q8×C53—42no—
151Q8×C53—41no—
152Q8×C53—41no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—81no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×C4×C53#12411yescharacteristic
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186Dic106—42no—
187Dic106—41no—
188Dic106—41no—
189#88322—42no—
190#88322—41no—
191#88322—41no—
192#10566—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#39411yescharacteristic
204C22×C53#46211yescharacteristic
205C22×C53#46211yescharacteristic
206C22×C53#46411yescharacteristic
207C22×C53#46411yescharacteristic
208C22×C53#46711yescharacteristic
209C22×C53#47011yescharacteristic
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53—161no—
218C4×C53B(4,𝔽2)11yescharacteristic
219C4×C53#45811yescharacteristic
220Dic53—81no—
221Dic53—81no—
222Dic53—161no—
223#3189—531no—
224#1996—531no—
225#1996D5311yescharacteristic
226#1892—531no—
227#1892—531no—
228#1850—531no—
229#1289—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53#204411yescharacteristic
237C2×C53#205311yescharacteristic
238C2×C53#205611yescharacteristic
239C23×C8—531no—
240C2×C22⋊Q8—531no—
241C2×C22⋊Q8C2×D5311yescharacteristic
242#484—1061no—
243#484—1061no—
244#467—1061no—
245#467—1061no—
246#451—1061no—
247#451—1061no—
248C2×C4⋊C8—1061no—
249C2×#103—1061no—
250C2×#103—1061no—
251C2×C22⋊C8—531no—
252#380—531no—
253#380C2×D5311yescharacteristic
254C4×C22⋊C4C2×D5311yescharacteristic
255#342—21no—
256#342—21no—
257#342—1061no—
258#342—1061no—
259#340—1061no—
260#340—1061no—
261#336—1061no—
262C53#1056611yescharacteristic, complemented, Sylow
263C22×Q8—531no—
264C22×Q8D21211yescharacteristic
265C23×C4C22×D5311yescharacteristic
266C2×Q16—1061no—
267C2×Q16—1061no—
268C22×C8—531no—
269C22×C8—531no—
270C22×C8—1061no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8—2122no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—1061no—
279C22⋊Q8—1061no—
280C22⋊Q8—1061no—
281C22⋊Q8—1061no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—531no—
285C2×C4⋊C4—1061no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4D21211yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4C22⋊D5311yescharacteristic
293C2×C22⋊C4C22⋊D5311yescharacteristic
294C2×C42—21no—
295#107—1061no—
296#107—1061no—
297C4⋊C8—2122no—
298#103—1061no—
299#103—1061no—
300#103—1061no—
301#103—1061no—
302#103—2122no—
303#103—2122no—
304C22⋊C8—1061no—
305C22⋊C8—1061no—
306#95—21no—
307#95—21no—
308#95—1061no—
309C24C2×C22⋊D5311yescharacteristic
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—1061no—
315C2×Q8—1061no—
316C2×Q8—2121no—
317C2×Q8—2121no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4—2121no—
328C22×C4C2×D21211yescharacteristic
329C22×C4#1171994511yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4#1171994711yescharacteristic
334Q16—4241no—
335Q16—4241no—
336C2×C8—1062no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—1061no—
340C2×C8—1061no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4—2121no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C22⋊C4—21no—
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—2121no—
367C22⋊C4—2121no—
368C22⋊C4—2121no—
369C22⋊C4—2121no—
370C22⋊C4#1171992512yescomplemented
371C42—41no—
372C42—41no—
373C23—41no—
374C23—41no—
375C23#1313318811yescharacteristic
376C23#1313318911yescharacteristic
377C23#1313319111yescharacteristic
378C23C2×#1171992511yescharacteristic
379C23#1313323711yescharacteristic
380C23#1313324711yescharacteristic
381C23#1313324811yescharacteristic
382Q8—42no—
383Q8—41no—
384Q8—41no—
385Q8—2122no—
386Q8—2121no—
387Q8—2121no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4#1313318711yescharacteristic
418C2×C4#1313319911yescharacteristic
419C2×C4#1313320111yescharacteristic
420C2×C4#1313320311yescharacteristic
421C8—2122no—
422C8—2121no—
423C8—2121no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#1681053911yescharacteristic
435C22#1681083011yescharacteristic
436C22#1681083311yescharacteristic
437C22#1681084211yescharacteristic
438C22#1681084511yescharacteristic
439C22#1681086211yescharacteristic
440C22#1681087911yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C4—8481no—
450C4—4241no—
451C4—4241no—
452C4#1681081111yescharacteristic
453C4#1681081411yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#2027596511yescharacteristic
461C2#2027602011yescharacteristic
462C2#2027603711yescharacteristic
subgroupG/H
centerC22#16810539
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1996C2
socleC22×C53#394
prime radicalC24×C53C2×D4
hypercenter#1996D53
hypocenterC53#10566
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
115. C22⋊C4×C53Q16semidirect
262. C53#10566semidirect
370. C22⋊C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202757002
#202751392
#202751362
#202748952
#202725822
C53×#19962
#1056653