Finite Group Databasebeta

← #27287300

Subgroups — #27287300

Order 13568.

404 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 77 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#20275699C211yesmaximal, characteristic
2#20273247C211yesmaximal, characteristic
3#20273234C211yesmaximal, characteristic
4#20272621C211yesmaximal, characteristic
5#20272601C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#2665C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810587C2211yescharacteristic
11C4×C22⋊Dic53C2211yescharacteristic
12#16810389—21no—
13#16810389—21no—
14#16810359—21no—
15#16810359—21no—
16#16810359—21no—
17#16810359—21no—
18C53×#400—21no—
19C53×#396—21no—
20C53×#395—21no—
21C53×#393—21no—
22C53×#391—21no—
23C53×#387—21no—
24C53×#381C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C22×Dic106Q811yescharacteristic, complemented
28C2×#11719917—21no—
29C2×#11719917—21no—
30#13133196—21no—
31#13133196—21no—
32#13133196—21no—
33#13133196—21no—
34C2×C4⋊Dic53Q811yescharacteristic, complemented
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43#13133124—21no—
44C22×Q8×C53—21no—
45C23×C4×C53C2311yescharacteristic
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53D411yescharacteristic
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C2×Q8×C53—42no—
67C2×Q8×C53—81no—
68C2×Q8×C53—81no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—81no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×Q811yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—41no—
85C2×Dic106—21no—
86C2×Dic106—21no—
87C2×Dic106—81no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—81no—
90C4⋊C4×C53—81no—
91C4⋊C4×C53—81no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—81no—
96C42×C53—81no—
97C22⋊Dic53—22no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C22⋊Dic53—41no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—81no—
106#11719917—81no—
107#11719917—81no—
108C2×#88322—41no—
109C2×#88322—41no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C23×C53C22⋊Q811yescharacteristic
116C23×C53#12411yescharacteristic
117Q8×C53—84no—
118C2×C4×C53—22no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53C22⋊Q811yescharacteristic
143C2×C4×C53C22⋊Q811yescharacteristic
144C2×C4×C53#12411yescharacteristic
145C2×C4×C53#12411yescharacteristic
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153Dic106—81no—
154Dic106—81no—
155#88322—81no—
156#88322—81no—
157#10383—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#39311yescharacteristic
169C22×C53#45511yescharacteristic
170C22×C53#45611yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53—81no—
180C4×C53—81no—
181Dic53—82no—
182Dic53—81no—
183Dic53—81no—
184#2665D5311yescharacteristic
185#1996—531no—
186#1511—531no—
187#1508—531no—
188#1305—531no—
189#1297—531no—
190#1284—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#202811yescharacteristic
196C2×C53#203111yescharacteristic
197C2×C53#203211yescharacteristic
198C2×C22⋊Q8—531no—
199#407—531no—
200#407—531no—
201#400—21no—
202#396—21no—
203#395—21no—
204#393—21no—
205#391—21no—
206#387—21no—
207#381C2×D5311yescharacteristic
208#380C2×D5311yescharacteristic
209C4×C22⋊C4—531no—
210C4×C22⋊C4C2×D5311yescharacteristic
211#342—1061no—
212#342—1061no—
213#328—1061no—
214#328—1061no—
215#328—1061no—
216#328—1061no—
217C53#1038311yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C22×Q8—531no—
220C23×C4C22×D5311yescharacteristic
221C2×M16—1061no—
222C2×M16—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8—1061no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—1061no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4C22⋊D5311yescharacteristic
238C2×C22⋊C4C22⋊D5311yescharacteristic
239C2×C42—21no—
240C2×C42—1061no—
241C22⋊C8—1061no—
242C22⋊C8—1061no—
243C22⋊C8—1061no—
244C22⋊C8—1061no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—1061no—
252#95C22⋊D5311yescharacteristic
253#95C22⋊D5311yescharacteristic
254#95C22⋊D5311yescharacteristic
255#95C22⋊D5311yescharacteristic
256C24C2×C22⋊D5311yescharacteristic
257C2×Q8—42no—
258C2×Q8—81no—
259C2×Q8—81no—
260C2×Q8—1061no—
261C2×Q8—1061no—
262C2×Q8—4241no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—81no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—2121no—
273C22×C4Q8×D5311yescharacteristic
274C22×C4#1171994911yescharacteristic
275C22×C4#1171994911yescharacteristic
276C22×C4#1171994911yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279M16—4241no—
280M16—4241no—
281C2×C8—2121no—
282C2×C8—2121no—
283C4⋊C4—42no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—81no—
295C22⋊C4—1062no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C42—81no—
300C42—4241no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23#1313318611yescharacteristic
306C23#1313319311yescharacteristic
307C23#1313320611yescharacteristic
308Q8—84no—
309Q8—4241no—
310Q8—4241no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4#1313319311yescharacteristic
343C2×C4#1313319311yescharacteristic
344C2×C4#1313320611yescharacteristic
345C2×C4#1313320611yescharacteristic
346C8—4241no—
347C8—4241no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#1681053411yescharacteristic
359C22#1681080111yescharacteristic
360C22#1681080611yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—4242no—
372C4—4241no—
373C4—4241no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2027587311yescharacteristic
379C2#2027588411yescharacteristic
380C2#2027589111yescharacteristic
subgroupG/H
centerC22#16810534
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2665C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#2665D53
hypocenterC53#10383
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2665

Complements

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

NH ≅ G/Ntype
27. C22×Dic106Q8semidirect
34. C2×C4⋊Dic53Q8semidirect
217. C53#10383semidirect

Maximal subgroups

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

subgroupindex
#202756992
#202732472
#202732342
#202726212
#202726012
#202725672
C53×#26652
#1038353