Finite Group Databasebeta

← #27288505

Subgroups — #27288505

Order 13568.

560 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 119 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#20280829C211yesmaximal, characteristic
2#20274903C212yesmaximal
3#20272594C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#1442C211yesmaximal, characteristic
6C2×#13133196—21no—
7C2×#13133150—22no—
8C2×C22⋊#88322C2212yes—
9#16810555—22no—
10#16810544—22no—
11#16810483—21no—
12#16810483C2212yes—
13#16810385—22no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C2×#95×C53—21no—
16C2×#95×C53—21no—
17C2×#95×C53C2211yescharacteristic
18C2×#95×C53C2211yescharacteristic
19C22×Dic106—21no—
20C22×#88322—22no—
21#13133196—42no—
22#13133196—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30#13133150—44no—
31C22⋊#88322—22no—
32C22⋊#88322Q814yescomplemented
33#13133124—22no—
34#13133124—22no—
35C23×C4×C53—21no—
36C23×C4×C53—21no—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—42no—
42#95×C53—22no—
43#95×C53—21no—
44#95×C53—42no—
45#95×C53—42no—
46#95×C53Q812yescomplemented
47C24×C53C2×D411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53D812yes—
73C22×C4×C53Q1612yes—
74C22×C4×C53C2×D412yes—
75C22×C4×C53C2×Q812yes—
76C22×C4×C53C2×Q811yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—22no—
79C22×Dic53—22no—
80C2×Dic106—42no—
81C2×Dic106—81no—
82C2×Dic106—81no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4⋊Dic53—42no—
90C4⋊Dic53—81no—
91C4⋊Dic53—81no—
92#11719919—44no—
93C2×#88322—24no—
94C2×#88322—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53#12011yescharacteristic
101C23×C53C22⋊Q811yescharacteristic
102C23×C53C4⋊Q811yescharacteristic
103C23×C53C2×D811yescharacteristic
104C23×C53C2×Q1611yescharacteristic
105C23×C53AGL(1,ℤ/8)11yescharacteristic
106C23×C53#13711yescharacteristic
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C22⋊Q812yes—
137C2×C4×C53C4⋊Q812yes—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144Dic106—84no—
145#88322—44no—
146#10569—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#39511yescharacteristic
160C22×C53#44711yescharacteristic
161C22×C53#45111yescharacteristic
162C22×C53#45411yescharacteristic
163C22×C53#45511yescharacteristic
164C22×C53B(4,𝔽2)11yescharacteristic
165C22×C53#45811yescharacteristic
166C22×C53#47411yescharacteristic
167C22×C53#47711yescharacteristic
168C22×C53#50011yescharacteristic
169C22×C53#50111yescharacteristic
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—42no—
173C4×C53—42no—
174C4×C53—82no—
175C4×C53—82no—
176Dic53—82no—
177Dic53—82no—
178#2659—531no—
179#1852—532no—
180#1442D5311yescharacteristic
181#1294—532no—
182#1280—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#159811yescharacteristic
188C2×C53#160211yescharacteristic
189C2×C53#160711yescharacteristic
190C2×C53#161711yescharacteristic
191C2×C53#205711yescharacteristic
192C2×C53#206111yescharacteristic
193C2×C53#206211yescharacteristic
194C2×C22⋊Q8—1061no—
195C22×C4⋊C4C2×D5311yescharacteristic
196C2×#107—1062no—
197C2×C22⋊C8—532no—
198#398—1062no—
199#395—1062no—
200#380—532no—
201#380—1061no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95C2×D5311yescharacteristic
205C2×#95C2×D5311yescharacteristic
206#340—1062no—
207C53#1056911yescharacteristic, complemented, Sylow
208C22×Q8—1061no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22⋊D5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C22×C8—1062no—
215C22⋊Q8—2122no—
216C22⋊Q8—2122no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—532no—
220C2×C4⋊C4—532no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1061no—
225C2×C22⋊C4—1062no—
226#107—2124no—
227C22⋊C8—534no—
228C22⋊C8—1062no—
229#95—21no—
230#95—22no—
231#95—42no—
232#95—42no—
233#95—1062no—
234#95—1062no—
235#95Dic10612yescomplemented
236C24C2×C22⋊D5311yescharacteristic
237C2×Q8—4241no—
238C2×Q8—4241no—
239C2×Q8—2122no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—1062no—
265C22×C4—1062no—
266C22×C4#1171992212yes—
267C22×C4#1171992512yes—
268C22×C4C2×Dic10611yescharacteristic
269C22×C4#1171994511yescharacteristic
270C22×C4D4×D5312yes—
271C22×C4Q8×D5312yes—
272C2×C8—1064no—
273C2×C8—2122no—
274C2×C8—2122no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—4241no—
278C4⋊C4—4241no—
279C4⋊C4—2122no—
280C4⋊C4—2122no—
281C4⋊C4—2124no—
282C4⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1313318511yescharacteristic
290C23#1313319611yescharacteristic
291C23#1313321511yescharacteristic
292C23C2×#1171992211yescharacteristic
293C23C2×#1171992511yescharacteristic
294C23#1313324211yescharacteristic
295C23#1313324711yescharacteristic
296Q8—4244no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4#1313319512yes—
333C2×C4#1313321612yes—
334C8—2124no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1681054311yescharacteristic
348C22#1681075611yescharacteristic
349C22#1681077811yescharacteristic
350C22#1681079511yescharacteristic
351C22#1681080011yescharacteristic
352C22#1681080911yescharacteristic
353C22#1681081211yescharacteristic
354C22#1681090311yescharacteristic
355C22#1681091811yescharacteristic
356C22#1681102611yescharacteristic
357C22#1681103311yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—4242no—
365C4—4242no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2027364911yescharacteristic
371C2#2027367111yescharacteristic
372C2#2027370011yescharacteristic
373C2#2027375611yescharacteristic
374C2#2027604311yescharacteristic
375C2#2027606511yescharacteristic
376C2#2027606811yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1442C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1442D53
hypocenterC53#10569
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1442

Complements

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

NH ≅ G/Ntype
32. C22⋊#88322Q8semidirect
46. #95×C53Q8semidirect
207. C53#10569semidirect
235. #95Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202749032
#202725942
#202725582
C53×#14422
#1056953