Finite Group Databasebeta

← #27294404

Subgroups — #27294404

Order 13568.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#20280528C211yesmaximal, characteristic
2C2×#16810392C211yesmaximal, characteristic
3#20272746C212yesmaximal, complemented
4#20272743C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133196C411yescharacteristic
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810556—21no—
11#16810556—21no—
12#16810543C411yescharacteristic
13#16810543C2211yescharacteristic
14#16810496—21no—
15#16810486—21no—
16#16810483C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—24no—
25C2×#11719917—22no—
26C2×#11719917—41no—
27C2×#11719917—42no—
28C22×#88322—41no—
29#13133196—41no—
30#13133196—41no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—41no—
50C2×C4⋊C4×C53—41no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53C2311yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—81no—
67C22×C4×C53—81no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53SD1612yes—
72C22×C4×C53Q1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×Dic106—41no—
81C2×Dic106—81no—
82C2×C8×C53—42no—
83C2×C8×C53—82no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—42no—
91C4⋊Dic53—41no—
92C4⋊Dic53—81no—
93#11719919—42no—
94#11719919—81no—
95#11719919—81no—
96#11719917—24no—
97#11719917—44no—
98#11719917—42no—
99C2×#88322—44no—
100C2×#88322—42no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#9911yescharacteristic
104C23×C53#9911yescharacteristic
105C23×C53#10312yes—
106C23×C53#10412yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53#10312yes—
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53C2×Q1611yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—42no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142Dic106—82no—
143C8×C53—82no—
144#88322—44no—
145#11722—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53#32812yes—
155C22×C53#32812yes—
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10311yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C4×C53—21no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166Dic53—81no—
167Dic53—81no—
168Dic53—81no—
169Dic53—81no—
170#2627—531no—
171C2×#343—531no—
172C2×#324D5311yescharacteristic
173#1354—532no—
174#1353—532no—
175C2×C53—21no—
176C2×C53—22no—
177C2×C53—22no—
178C2×C53—21no—
179C2×C53C2×#32811yescharacteristic
180C2×C53#212111yescharacteristic
181C2×C53#213311yescharacteristic
182C22×M16—1061no—
183C2×C22⋊Q8—531no—
184C22×C4⋊C4C2×D5311yescharacteristic
185C2×C22⋊C8—21no—
186#400—531no—
187#400—531no—
188#398—1061no—
189#398—1061no—
190#395—531no—
191#395—531no—
192#385—1061no—
193#381—1061no—
194#380—531no—
195#343—532no—
196#343—1061no—
197#324—21no—
198#324C2×D5312yes—
199C53#1172211yescharacteristic, complemented, Sylow
200C22×Q8—531no—
201C23×C4C4×D5311yescharacteristic
202C23×C4D21211yescharacteristic
203C23×C4C22⋊D5311yescharacteristic
204C2×M16—1064no—
205C2×M16—1062no—
206C2×M16—2121no—
207C2×M16—2122no—
208C22×C8—41no—
209C22×C8—2121no—
210C22⋊Q8—2121no—
211C22⋊Q8—2121no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C4×D5311yescharacteristic
220C2×C4⋊C4D21211yescharacteristic
221C2×C4⋊C4C22⋊D5311yescharacteristic
222C2×C4⋊C4C22×D5311yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#95—531no—
228#95—531no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24#1171992111yescharacteristic
236C2×Q8—4241no—
237C2×Q8—2121no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4#1171991312yes—
252C22×C4Dic21212yes—
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4C2×C4×D5311yescharacteristic
257C22×C4C2×D21211yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259M16—1064no—
260M16—2124no—
261M16—2122no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—2122no—
265C2×C8—2124no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—4241no—
271C4⋊C4—4241no—
272C4⋊C4—4241no—
273C4⋊C4—2121no—
274C4⋊C4—2122no—
275C22⋊C4—2121no—
276C22⋊C4—2121no—
277C22⋊C4—2122no—
278C23—41no—
279C23—41no—
280C23#1313313111yescharacteristic
281C23#1313313111yescharacteristic
282C23#1313314212yes—
283C23#1313314312yes—
284C23C2×#1171992111yescharacteristic
285Q8—4242no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2122no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313314212yes—
318C2×C4C2×#1171991311yescharacteristic
319C2×C4C2×Dic21211yescharacteristic
320C8—82no—
321C8—2124no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1681036112yes—
331C22#1681036112yes—
332C22C2×#1313313111yescharacteristic
333C22C2×#1313314211yescharacteristic
334C22C2×#1313314311yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—4241no—
343C4—4241no—
344C4—4241no—
345C4—4241no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1681036111yescharacteristic
351C2#2027635911yescharacteristic
352C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11722
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20272746C2semidirect
4. #20272743C2semidirect
199. C53#11722semidirect

Maximal subgroups

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

subgroupindex
#202805282
C2×#168103922
#202727462
#202727432
C2×C53×#3242
#1172253