Finite Group Databasebeta

← #27358014

Subgroups — #27358014

Order 13568.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 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#20285487C212yesmaximal
3#20273756C214yesmaximal, complemented
4#20273688C214yesmaximal, complemented
5#20273286C212yesmaximal
6C2×#16810349C211yesmaximal, characteristic
7C53×#2627C211yesmaximal, characteristic
8#16811254—24no—
9#16811254—22no—
10#16811254C2214yes—
11C2×#13133216—22no—
12C2×#13133208—22no—
13C2×#13133196—21no—
14C2×#13133196C2212yes—
15C2×#13133176—21no—
16#16810958C2214yes—
17#16810956—22no—
18#16810862—24no—
19#16810779C2214yes—
20#16810778—22no—
21C2×C22⋊#88322C2212yes—
22#16810359C2218yes—
23#16810349C2214yes—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#400C2212yes—
27C53×#398—22no—
28C53×#395C2212yes—
29C53×#385—21no—
30C53×#381—21no—
31C53×#380C2211yescharacteristic
32C22×Dic106—21no—
33C22×Dic106D412yes—
34C2×#11719925—24no—
35C22×#88322—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#13133150—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—21no—
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—
77#95×C53—22no—
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—22no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53—81no—
93C22×C4×C53—81no—
94C22×C4×C53Q1614yescomplemented
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2411yescharacteristic
101C22×Dic53—22no—
102C22×Dic53—22no—
103C2×Dic106—24no—
104C2×Dic106—22no—
105C2×Dic106—82no—
106C2×Dic106—82no—
107C2×Dic106C2×D414yes—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—81no—
113C4⋊C4×C53—82no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—42no—
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—
129C2×#88322—24no—
130C2×#88322—82no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53#12011yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53C2×Q1612yes—
136C23×C53#13712yes—
137C23×C53C22×D411yescharacteristic
138Q8×C53—44no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
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—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—81no—
158C2×C4×C53#12012yes—
159C2×C4×C53C2×Q1614yes—
160C2×C4×C53C22×D412yes—
161C2×Dic53—24no—
162C2×Dic53—24no—
163C2×Dic53—44no—
164C2×Dic53—44no—
165Dic106—44no—
166Dic106—82no—
167Dic106—84no—
168#88322—44no—
169#18295—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53#45112yes—
179C22×C53#45112yes—
180C22×C53C2×#12011yescharacteristic
181C22×C53C22×Q1611yescharacteristic
182C22×C53C2×#13711yescharacteristic
183C4×C53—24no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—42no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190Dic53—44no—
191Dic53—44no—
192C2×#544—531no—
193#3189—532no—
194#2627D5311yescharacteristic
195#1617—534no—
196#1605—534no—
197#1520—532no—
198C2×#324—531no—
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—22no—
202C2×C53—22no—
203C2×C53C2×#45111yescharacteristic
204C2×C53#302011yescharacteristic
205C2×C53#302611yescharacteristic
206#544—534no—
207#544—1064no—
208#544—1062no—
209C2×C4⋊Q8—1062no—
210C2×#125—1062no—
211C2×C22⋊Q8—532no—
212C2×C22⋊Q8—1061no—
213C2×C22⋊Q8C2×D5311yescharacteristic
214C2×#117—1061no—
215C22×C4⋊C4C2×D5311yescharacteristic
216#484—534no—
217#484—1062no—
218#467—1064no—
219#451—534no—
220#451—1062no—
221C2×C22⋊C8—532no—
222#400C2×D5312yes—
223#398—22no—
224#395C2×D5312yes—
225#385—21no—
226#381—21no—
227#380C2×D5311yescharacteristic
228#328—538no—
229#324—534no—
230C53#1829511yescharacteristic, complemented, Sylow
231C22×Q8—532no—
232C22×Q8—1061no—
233C22×Q8C22×D5311yescharacteristic
234C23×C4C22⋊D5311yescharacteristic
235C23×C4C22⋊D5311yescharacteristic
236C23×C4C22×D5311yescharacteristic
237C2×Q16—1064no—
238C22×C8—1062no—
239C4⋊Q8—1068no—
240C4⋊Q8—1068no—
241#125—1068no—
242#125—1068no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—534no—
246C22⋊Q8—534no—
247C22⋊Q8—1064no—
248C22⋊Q8—1062no—
249C22⋊Q8—1062no—
250C22⋊Q8—1064no—
251#117—1064no—
252#117—1064no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—532no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1062no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4C22⋊D5312yes—
265C2×C4⋊C4C22×D5311yescharacteristic
266C2×C4⋊C4C22×D5312yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1062no—
270C2×C42—1062no—
271#107—1064no—
272#103—1064no—
273#103—1064no—
274C22⋊C8—534no—
275C22⋊C8—1062no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95C22×D5312yes—
280C24C2×C22⋊D5311yescharacteristic
281C2×Q8—22no—
282C2×Q8—81no—
283C2×Q8—534no—
284C2×Q8—1064no—
285C2×Q8—1062no—
286C2×Q8—4242no—
287C2×Q8—4242no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4#1171992514yescomplemented
301C22×C4D4×D5312yes—
302C22×C4D4×D5312yes—
303C22×C4C2×C22⋊D5312yes—
304C22×C4C2×C22⋊D5312yes—
305C22×C4C2×C22⋊D5311yescharacteristic
306C22×C4C23×D5311yescharacteristic
307Q16—4244no—
308C2×C8—1064no—
309C2×C8—4242no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C4⋊C4—534no—
317C4⋊C4—1064no—
318C4⋊C4—1064no—
319C4⋊C4—1064no—
320C4⋊C4—1068no—
321C4⋊C4—1062no—
322C4⋊C4—1064no—
323C4⋊C4—1064no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—1064no—
328C22⋊C4—1064no—
329C42—1068no—
330C23—41no—
331C23—41no—
332C23#1313318511yescharacteristic
333C23#1313318511yescharacteristic
334C23C2×#1171992512yes—
335C23#1313324712yes—
336C23C22×C22⋊D5311yescharacteristic
337Q8—44no—
338Q8—4242no—
339Q8—4244no—
340Q8—2124no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—1064no—
361C2×C4—1064no—
362C2×C4—2124no—
363C2×C4—2124no—
364C2×C4#1313318512yes—
365C2×C4C2×#1171992514yes—
366C2×C4C2×D4×D5312yes—
367C8—2124no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1681077812yes—
377C22#1681077812yes—
378C22C2×#1313318511yescharacteristic
379C22C22×#1171992511yescharacteristic
380C22C2×#1313324711yescharacteristic
381C4—24no—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—2124no—
389C4—2124no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1681077811yescharacteristic
395C2#2028407811yescharacteristic
396C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2627C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#2627D53
hypocenterC53#18295
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #20273756C2semidirect
4. #20273688C2semidirect
94. C22×C4×C53Q16semidirect
230. C53#18295semidirect
300. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202854872
#202737562
#202736882
#202732862
C2×#168103492
C53×#26272
#1829553