Finite Group Databasebeta

← #27357224

Subgroups — #27357224

Order 13568.

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

normal subgroups only

This group has 890 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
1#20288525C211yesmaximal, characteristic
2#20284754C212yesmaximal
3#20273756C214yesmaximal
4#20273688C214yesmaximal
5#20273278C212yesmaximal
6#20273039C211yesmaximal, characteristic
7C53×#2581C211yesmaximal, characteristic
8Q8×Dic106—22no—
9#16811352—22no—
10#16811349—24no—
11#16811254—21no—
12#16811254C2214yes—
13#16811180—22no—
14#16811180C2212yes—
15#16811101—21no—
16#16810958C2214yes—
17#16810918—22no—
18#16810833—22no—
19#16810779C2214yes—
20C4×#11719925—24no—
21#16810676C2212yes—
22#16810359C2218yes—
23#16810349C2214yes—
24C2×C4×Q8×C53C2211yescharacteristic
25C2×C4×Q8×C53C2211yescharacteristic
26C53×#400C2212yes—
27C53×#395C2212yes—
28C53×#391—21no—
29C53×#389—21no—
30C53×#387—22no—
31C53×#382C2211yescharacteristic
32C2×#11719925—24no—
33#13133217—22no—
34#13133217—22no—
35#13133217D412yes—
36#13133217D412yes—
37#13133216—24no—
38#13133216—28no—
39#13133209—24no—
40#13133209—22no—
41#13133208—24no—
42#13133208—28no—
43#13133196—22no—
44#13133196—24no—
45#13133196C2314yes—
46Q8×Dic53—24no—
47Q8×Dic53—24no—
48C4×Dic106—24no—
49C4×Dic106—24no—
50C4×Dic106—22no—
51C4×Dic106D414yes—
52#13133176—24no—
53#13133176—22no—
54#13133150—24no—
55C4⋊#88322—22no—
56#13133141—24no—
57#13133140—24no—
58C22⋊#88322C2314yes—
59C4×#88322—22no—
60C22×Q8×C53D411yescharacteristic
61C22×Q8×C53C2311yescharacteristic
62C4×Q8×C53—22no—
63C4×Q8×C53—22no—
64C4×Q8×C53—22no—
65C4×Q8×C53—22no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C4⋊C4×C53C2311yescharacteristic
71C2×C4⋊C4×C53C2312yes—
72C2×C42×C53—21no—
73C2×C42×C53D412yes—
74C2×C42×C53C2311yescharacteristic
75#95×C53—21no—
76#95×C53—22no—
77#95×C53—22no—
78#95×C53—22no—
79#95×C53C2312yes—
80C2×Q8×C53—22no—
81C2×Q8×C53—22no—
82C2×Q8×C53—22no—
83C2×Q8×C53—42no—
84C2×Q8×C53Q1614yescomplemented
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—21no—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C2×Dic106—22no—
94C2×Dic106—24no—
95C2×Dic106C2×D414yes—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—44no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—42no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—42no—
106C42×C53C2×D412yes—
107C42×C53C2×D412yes—
108C22⋊Dic53—24no—
109#11719925—48no—
110C4⋊Dic53—24no—
111C4⋊Dic53—24no—
112C4⋊Dic53—24no—
113C4⋊Dic53—22no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53C2×D414yes—
117#11719919—28no—
118#11719919—24no—
119#11719919—24no—
120C4×Dic53—28no—
121C4×Dic53—24no—
122C2×#88322—24no—
123C23×C53C22×D411yescharacteristic
124Q8×C53—28no—
125Q8×C53—24no—
126Q8×C53—42no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12012yes—
144C2×C4×C53C2×Q1614yes—
145C2×C4×C53C2×Q1612yes—
146C2×C4×C53#13712yes—
147C2×C4×C53C22×D412yes—
148C2×Dic53—24no—
149C2×Dic53—24no—
150Dic106—44no—
151Dic106—44no—
152Dic106—84no—
153#88322—44no—
154#18221—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53C2×#12011yescharacteristic
158C22×C53C22×Q1611yescharacteristic
159C22×C53C2×#13711yescharacteristic
160C4×C53—24no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—42no—
165C4×C53—42no—
166C4×C53—82no—
167C4×C53—81no—
168C4×C53#45114yes—
169Dic53—44no—
170Dic53—44no—
171#3502—531no—
172#3104—532no—
173#2581D5311yescharacteristic
174#1617—534no—
175#1605—534no—
176#1518—532no—
177#1461—531no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53C2×#45111yescharacteristic
181C2×C53#302011yescharacteristic
182C2×C53#302511yescharacteristic
183Q82—1062no—
184#557—1064no—
185#557—1062no—
186#544—534no—
187#544—1061no—
188#533—532no—
189#533—1062no—
190#519—1061no—
191C2×C4×Q8C2×D5311yescharacteristic
192C2×C4×Q8C2×D5311yescharacteristic
193#484—534no—
194#477—1062no—
195#462—1062no—
196#451—534no—
197C4×Q16—1064no—
198#431—532no—
199#400C2×D5312yes—
200#395C2×D5312yes—
201#391—21no—
202#389—21no—
203#387—22no—
204#382C2×D5311yescharacteristic
205#328—538no—
206#324—534no—
207C53#1822111yescharacteristic, complemented, Sylow
208C22×Q8C22⋊D5311yescharacteristic
209C22×Q8C22×D5311yescharacteristic
210C2×Q16—1064no—
211C4⋊Q8—532no—
212C4⋊Q8—532no—
213C4⋊Q8—1062no—
214C4⋊Q8—1062no—
215C4⋊Q8—1068no—
216C4⋊Q8—1064no—
217#125—1064no—
218#125—1068no—
219#125—1064no—
220#125—1062no—
221C22⋊Q8—534no—
222C22⋊Q8—1062no—
223C22⋊Q8—1064no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—534no—
229C4×Q8—1064no—
230C4×Q8—1064no—
231C4×Q8—1064no—
232C4×Q8—1064no—
233C4×Q8—1062no—
234#117—1062no—
235#117—1064no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22⋊D5311yescharacteristic
240C2×C4⋊C4C22×D5311yescharacteristic
241C2×C4⋊C4C22×D5312yes—
242C2×C42—21no—
243C2×C42C22⋊D5312yes—
244C2×C42C22×D5311yescharacteristic
245#107—1064no—
246C4⋊C8—1062no—
247#103—1064no—
248#103—1064no—
249C22⋊C8—534no—
250C4×C8—1062no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22×D5312yes—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—42no—
260C2×Q8—534no—
261C2×Q8—1064no—
262C2×Q8—1062no—
263C2×Q8#1171992514yescomplemented
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4C2×C22⋊D5312yes—
268C22×C4C2×C22⋊D5312yes—
269C22×C4C2×C22⋊D5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271C22×C4C23×D5311yescharacteristic
272Q16—2128no—
273C2×C8—1064no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—534no—
282C4⋊C4—1064no—
283C4⋊C4—1064no—
284C4⋊C4—1064no—
285C4⋊C4—1068no—
286C4⋊C4—1062no—
287C4⋊C4—1062no—
288C4⋊C4—1064no—
289C4⋊C4—1062no—
290C4⋊C4—1064no—
291C22⋊C4—1064no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—1068no—
296C42—1064no—
297C42D4×D5312yes—
298C42D4×D5312yes—
299C23C22×C22⋊D5311yescharacteristic
300Q8—28no—
301Q8—24no—
302Q8—42no—
303Q8—4244no—
304Q8—2124no—
305Q8—2124no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—1064no—
321C2×C4—1064no—
322C2×C4#1313318511yescharacteristic
323C2×C4#1313318511yescharacteristic
324C2×C4#1313318512yes—
325C2×C4C2×#1171992514yes—
326C2×C4C2×#1171992512yes—
327C2×C4#1313324712yes—
328C2×C4C2×D4×D5312yes—
329C8—2124no—
330C22—21no—
331C22—21no—
332C22C2×#1313318511yescharacteristic
333C22C22×#1171992511yescharacteristic
334C22C2×#1313324711yescharacteristic
335C4—24no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—81no—
343C4—2124no—
344C4—2124no—
345C4#1681077814yes—
346C2—21no—
347C2—21no—
348C2C2×#1681077811yescharacteristic
349C2#2028407811yescharacteristic
350C2#2028410611yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2581C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#2581D53
hypocenterC53#18221
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
84. C2×Q8×C53Q16semidirect
207. C53#18221semidirect
263. C2×Q8#11719925semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202847542
#202737562
#202736882
#202732782
#202730392
C53×#25812
#1822153