Finite Group Databasebeta

← #27358286

Subgroups — #27358286

Order 13568.

950 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#16811254C211yesmaximal, characteristic
2#20285536C212yesmaximal
3#20273722C214yesmaximal, complemented
4#20273688C214yesmaximal, complemented
5#20273234C212yesmaximal
6#20273045C211yesmaximal, characteristic
7C53×#2583C211yesmaximal, characteristic
8#16811254—24no—
9#16811254—22no—
10#16811254C2214yes—
11C2×#13133216—22no—
12C2×#13133208—22no—
13C2×#13133196—21no—
14C2×#13133196C2212yes—
15C2×#13133176—21no—
16#16810953C2214yes—
17#16810951—22no—
18#16810879—24no—
19#16810779C2214yes—
20#16810778—22no—
21#16810587C2212yes—
22#16810359C2218yes—
23#16810349C2214yes—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#398C2212yes—
27C53×#395C2212yes—
28C53×#389—22no—
29C53×#385—21no—
30C53×#385—21no—
31C4×C22⋊C4×C53C2211yescharacteristic
32C22×Dic106—21no—
33C22×Dic106D412yes—
34C2×#11719925—24no—
35C2×#11719917—22no—
36#13133216—28no—
37#13133216—28no—
38#13133208—28no—
39#13133208—28no—
40#13133196—24no—
41#13133196—24no—
42#13133196—22no—
43#13133196—22no—
44#13133196D414yes—
45#13133196C2314yes—
46#13133176—24no—
47#13133176—24no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53D412yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×C22⋊Dic53—22no—
55C2×C4×Dic53—22no—
56#13133148—24no—
57#13133141—24no—
58#13133140—24no—
59C22⋊#88322—22no—
60C22⋊#88322C2314yes—
61C22×Q8×C53C2311yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53D412yes—
73C2×C4⋊C4×C53C2311yescharacteristic
74C2×C4⋊C4×C53C2312yes—
75C2×C22⋊C4×C53—21no—
76C2×C22⋊C4×C53—21no—
77C2×C42×C53—21no—
78#95×C53—22no—
79#95×C53—22no—
80#95×C53C2312yes—
81C24×C53C2×D411yescharacteristic
82C2×Q8×C53—22no—
83C2×Q8×C53—81no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—24no—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—81no—
93C22×C4×C53—81no—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2411yescharacteristic
100C22×Dic53—22no—
101C22×Dic53—22no—
102C2×Dic106—24no—
103C2×Dic106—22no—
104C2×Dic106—82no—
105C2×Dic106—82no—
106C2×Dic106C2×D414yes—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53—44no—
112C4⋊C4×C53—82no—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53—81no—
117C22⋊Dic53—24no—
118C22⋊Dic53—24no—
119#11719925—84no—
120C4⋊Dic53—24no—
121C4⋊Dic53—24no—
122C4⋊Dic53—24no—
123C4⋊Dic53—22no—
124C4⋊Dic53C2×D414yes—
125#11719919—28no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53—28no—
129#11719917—82no—
130C2×#88322—24no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—41no—
134C23×C53#12011yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53C22×D411yescharacteristic
137Q8×C53—44no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—24no—
143C2×C4×C53—22no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—81no—
157C2×C4×C53#12012yes—
158C2×C4×C53#13714yes—
159C2×C4×C53C22×D412yes—
160C2×Dic53—24no—
161C2×Dic53—24no—
162C2×Dic53—44no—
163C2×Dic53—44no—
164Dic106—44no—
165Dic106—82no—
166Dic106—84no—
167#88322—44no—
168#18327—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—44no—
177C22×C53#45612yes—
178C22×C53C2×#12011yescharacteristic
179C22×C53C2×#13712yes—
180C4×C53—24no—
181C4×C53—22no—
182C4×C53—22no—
183C4×C53—44no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53—81no—
187Dic53—44no—
188Dic53—44no—
189C2×#544—531no—
190#3195—532no—
191#2583D5311yescharacteristic
192#1611—534no—
193#1605—534no—
194#1508—532no—
195#1463—531no—
196C2×C53—22no—
197C2×C53—22no—
198C2×C53—41no—
199C2×C53#300411yescharacteristic
200C2×C53C2×#45611yescharacteristic
201C2×C53#302611yescharacteristic
202#544—534no—
203#544—1064no—
204#544—1062no—
205C2×C4⋊Q8—1062no—
206C2×#125—1062no—
207C2×C22⋊Q8—532no—
208C2×C22⋊Q8—1061no—
209C2×C22⋊Q8C2×D5311yescharacteristic
210C2×#117—1061no—
211C22×C4⋊C4C2×D5311yescharacteristic
212#483—534no—
213#483—1062no—
214#470—1064no—
215#451—534no—
216#451—1062no—
217#407—532no—
218#398C2×D5312yes—
219#395C2×D5312yes—
220#389—22no—
221#385—21no—
222#385—21no—
223C4×C22⋊C4C2×D5311yescharacteristic
224#328—538no—
225#324—534no—
226C53#1832711yescharacteristic, complemented, Sylow
227C22×Q8—532no—
228C22×Q8—1061no—
229C22×Q8C22×D5311yescharacteristic
230C23×C4C22⋊D5311yescharacteristic
231C23×C4C22⋊D5311yescharacteristic
232C23×C4C22×D5311yescharacteristic
233C2×Q16—1064no—
234C2×M16—1062no—
235C4⋊Q8—1068no—
236C4⋊Q8—1068no—
237#125—1068no—
238#125—1068no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—534no—
242C22⋊Q8—534no—
243C22⋊Q8—1064no—
244C22⋊Q8—1062no—
245C22⋊Q8—1062no—
246C22⋊Q8—1064no—
247#117—1064no—
248#117—1064no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—532no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1062no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4C22⋊D5312yes—
261C2×C4⋊C4C22×D5311yescharacteristic
262C2×C4⋊C4C22×D5312yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1062no—
266C2×C42—21no—
267C2×C42—1062no—
268#106—1064no—
269#103—1064no—
270#103—1064no—
271C22⋊C8—534no—
272C22⋊C8—1062no—
273#95—22no—
274#95—22no—
275#95C22×D5312yes—
276C24C2×C22⋊D5311yescharacteristic
277C2×Q8—22no—
278C2×Q8—81no—
279C2×Q8—534no—
280C2×Q8—1064no—
281C2×Q8—1062no—
282C2×Q8—4242no—
283C2×Q8—4242no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4D4×D5312yes—
297C22×C4D4×D5312yes—
298C22×C4C2×C22⋊D5312yes—
299C22×C4C2×C22⋊D5312yes—
300C22×C4C2×C22⋊D5311yescharacteristic
301C22×C4C23×D5311yescharacteristic
302Q16—4244no—
303M16—4242no—
304C2×C8—1064no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C4⋊C4—534no—
312C4⋊C4—1064no—
313C4⋊C4—1064no—
314C4⋊C4—1064no—
315C4⋊C4—1068no—
316C4⋊C4—1062no—
317C4⋊C4—1064no—
318C4⋊C4—1064no—
319C22⋊C4—24no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—1064no—
323C22⋊C4—1064no—
324C42—81no—
325C42—1068no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23#1313318511yescharacteristic
330C23#1313318511yescharacteristic
331C23C22×C22⋊D5311yescharacteristic
332Q8—44no—
333Q8—4242no—
334Q8—4244no—
335Q8—2124no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—1064no—
356C2×C4—1064no—
357C2×C4—2124no—
358C2×C4—2124no—
359C2×C4#1313318512yes—
360C2×C4#1313324714yes—
361C2×C4C2×D4×D5312yes—
362C8—2124no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#1681080512yes—
372C22C2×#1313318511yescharacteristic
373C22C2×#1313324712yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—81no—
380C4—81no—
381C4—2124no—
382C4—2124no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2#2028394811yescharacteristic
387C2C2×#1681080511yescharacteristic
388C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2583C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#2583D53
hypocenterC53#18327
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #20273722C2semidirect
4. #20273688C2semidirect
226. C53#18327semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202855362
#202737222
#202736882
#202732342
#202730452
C53×#25832
#1832753