Finite Group Databasebeta

← #27281265

Subgroups — #27281265

Order 13568.

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

normal subgroups only

This group has 500 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#20284663C211yesmaximal, characteristic
2#20277600C211yesmaximal, characteristic
3#20275700C211yesmaximal, characteristic
4#20273236C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic
6#20272601C211yesmaximal, characteristic
7C53×#1826C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810964—22no—
10#16810958—21no—
11#16810953—21no—
12#16810666C412yescomplemented
13#16810631C412yescomplemented
14#16810587C2211yescharacteristic
15#16810522—21no—
16#16810516—22no—
17#16810489—21no—
18#16810483C2211yescharacteristic
19C4⋊C4×Dic53—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810469—21no—
22#16810389—22no—
23#16810361—21no—
24#16810361—21no—
25C2×#117×C53C2211yescharacteristic
26C53×#407C2211yescharacteristic
27C4×C22⋊C4×C53C2211yescharacteristic
28C53×#339—22no—
29C22×Dic106C2×C411yescharacteristic
30C2×#11719917C2×C412yes—
31#13133196—21no—
32#13133196—21no—
33#13133196—21no—
34#13133196—21no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53D412yes—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—21no—
44#13133150—22no—
45#13133148—22no—
46#13133140—22no—
47#13133140—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133124—22no—
51#13133124—22no—
52#13133124—21no—
53#13133124D411yescharacteristic
54#13133124D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×M16×C53—21no—
57#117×C53—21no—
58#117×C53—21no—
59#117×C53C2×C412yes—
60#117×C53D412yes—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53—21no—
67C22⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69#95×C53—21no—
70C24×C53C2×D411yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—41no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—22no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—42no—
89M16×C53—42no—
90C2×C8×C53—41no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53C22⋊C412yes—
94C4⋊C4×C53C22⋊C412yes—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—41no—
99C42×C53—22no—
100C42×C53—22no—
101C42×C53—42no—
102C22⋊Dic53—42no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105C22⋊Dic53—42no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—42no—
109C4⋊Dic53—82no—
110#11719919—41no—
111#11719919—41no—
112C4×Dic53—44no—
113C4×Dic53—42no—
114#11719917—42no—
115C2×#88322—22no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53C4×D411yescharacteristic
121C23×C53#12011yescharacteristic
122C23×C53#12111yescharacteristic
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×Dic53—22no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154Dic106—42no—
155Dic106—81no—
156C8×C53—81no—
157#88322—42no—
158#9363—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53#38611yescharacteristic
170C22×C53#45511yescharacteristic
171C22×C53#45611yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—82no—
178C4×C53#41012yes—
179Dic53—42no—
180Dic53—82no—
181Dic53—81no—
182Dic53—81no—
183Dic53—81no—
184#3095—531no—
185#2346—531no—
186#1996—531no—
187#1826D5311yescharacteristic
188#1508—531no—
189#1305—531no—
190#1297—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#188211yescharacteristic
196C2×C53#189911yescharacteristic
197C2×C53#190611yescharacteristic
198C2×C22⋊Q8—531no—
199C2×#117C2×D5311yescharacteristic
200#485—1062no—
201#484—1061no—
202#483—1061no—
203#428—532no—
204#419—532no—
205#407—531no—
206#407C2×D5311yescharacteristic
207#389—1061no—
208#388—1062no—
209#382—1061no—
210#380—531no—
211C4×C4⋊C4—1061no—
212C4×C22⋊C4—531no—
213C4×C22⋊C4C2×D5311yescharacteristic
214#376—1061no—
215#342—1062no—
216#339—22no—
217#328—1061no—
218#328—1061no—
219C53#936311yescharacteristic, complemented, Sylow
220C22×Q8—531no—
221C23×C4C22×D5311yescharacteristic
222C2×M16—21no—
223C2×M16—532no—
224C22⋊Q8—1061no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8—1061no—
228#117—21no—
229#117—21no—
230#117C4×D5312yes—
231#117C22⋊D5312yes—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4D21211yescharacteristic
237C2×C4⋊C4D21211yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—532no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—1062no—
246C2×C42—1061no—
247#107—1062no—
248#106—1062no—
249#103—1062no—
250#103—1062no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1061no—
254C22⋊C8—1061no—
255#95—21no—
256#95—531no—
257#95—531no—
258#95—1062no—
259#95—1062no—
260#95—1061no—
261C24D4×D5311yescharacteristic
262C2×Q8—1061no—
263C2×Q8—1061no—
264C2×Q8—2122no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1062no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4C2×D21211yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C22×C4#1171994711yescharacteristic
278C22×C4#1171994711yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280M16—42no—
281M16—2122no—
282C2×C8—41no—
283C2×C8—1062no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—4242no—
289C4⋊C4—2122no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4#1171992112yes—
293C4⋊C4#1171992112yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—2122no—
299C22⋊C4—2121no—
300C22⋊C4—2121no—
301C22⋊C4—2122no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—2122no—
306C42—2124no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#1313317711yescharacteristic
312C23#1313318711yescharacteristic
313C23#1313318911yescharacteristic
314Q8—4241no—
315Q8—2122no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—1062no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—2122no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2122no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4C2×#1171992111yescharacteristic
344C2×C4#1313317711yescharacteristic
345C2×C4#1313318911yescharacteristic
346C2×C4#1313320111yescharacteristic
347C8—81no—
348C8—2122no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1681050011yescharacteristic
360C22#1681080211yescharacteristic
361C22#1681080711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—4241no—
369C4—4242no—
370C4—4241no—
371C4—4241no—
372C4—2122no—
373C4#1681059612yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2027508211yescharacteristic
379C2#2027517111yescharacteristic
380C2#2027521411yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1826C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1826D53
hypocenterC53#9363
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
12. #16810666C4semidirect
13. #16810631C4semidirect
219. C53#9363semidirect

Maximal subgroups

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

subgroupindex
#202846632
#202776002
#202757002
#202732362
#202726222
#202726012
C53×#18262
#936353