Finite Group Databasebeta

← #27277724

Subgroups — #27277724

Order 13568.

480 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277655C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272601C212yesmaximal, complemented
4#20272568C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810516—21no—
8#16810509—21no—
9#16810503—21no—
10#16810491C411yescharacteristic, complemented
11#16810486C2211yescharacteristic
12#16810483C411yescharacteristic, complemented
13C22⋊C4×Dic53—21no—
14C4×C22⋊Dic53C2211yescharacteristic
15#16810469C411yescharacteristic, complemented
16C2×#13133124C411yescharacteristic, complemented
17#16810349—21no—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53C2212yes—
22C53×#342—21no—
23C23×Dic53—21no—
24C2×#11719917—22no—
25C2×C4⋊Dic53—21no—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—21no—
29C2×C4×Dic53—21no—
30C2×C4×Dic53—21no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—21no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124C2×C411yescharacteristic
39#13133124C2×C411yescharacteristic
40#13133124C2×C411yescharacteristic
41#13133124C2×C411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53—22no—
52C2×C42×C53C2×C411yescharacteristic
53#99×C53—24no—
54#95×C53—22no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—81no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C22×C4×C53#4112yes—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—81no—
79C4⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—42no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—82no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—81no—
90C4⋊Dic53—81no—
91C4×Dic53—42no—
92C4×Dic53—81no—
93#11719917—82no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12312yes—
106D4×C53—44no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—44no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—82no—
122C2×C4×C53—82no—
123C2×C4×C53#11711yescharacteristic
124C2×C4×C53#11711yescharacteristic
125C2×C4×C53#12312yes—
126C2×Dic53—22no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—84no—
139#88322—82no—
140#8777—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53#38111yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C22×C53#42111yescharacteristic
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—84no—
163Dic53—42no—
164Dic53—81no—
165Dic53—81no—
166Dic53—81no—
167#2353—531no—
168C4×#99D5311yescharacteristic
169#1465—531no—
170#1297—532no—
171#1284—532no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—42no—
175C2×C53—42no—
176C2×C53#182611yescharacteristic
177C2×C53#182711yescharacteristic
178C2×C53#182811yescharacteristic
179C2×C4×D4C2×D5311yescharacteristic
180C2×#99—21no—
181#407—532no—
182#388—1061no—
183#387—1061no—
184#386—1061no—
185#383—531no—
186#381—531no—
187#380—531no—
188C4×C22⋊C4—531no—
189C4×C22⋊C4—1061no—
190C4×C22⋊C4C2×D5312yes—
191#376—531no—
192C2×#95—531no—
193#342—21no—
194#324—1061no—
195#323—1061no—
196C53#877711yescharacteristic, complemented, Sylow
197C22×D4C22⋊D5311yescharacteristic
198C23×C4—1061no—
199C23×C4C4×D5311yescharacteristic
200C23×C4C22×D5311yescharacteristic
201C2×M16—1062no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4D21211yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C22⋊C4D21211yescharacteristic
211C2×C22⋊C4C22⋊D5311yescharacteristic
212C2×C42—22no—
213C2×C42—1061no—
214C2×C42—1061no—
215C2×C42C4×D5311yescharacteristic
216#99—24no—
217C22⋊C8—1062no—
218C22⋊C8—1062no—
219#95—22no—
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24#1171992111yescharacteristic
230C24C2×C22⋊D5311yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—4241no—
244C22×C4—2122no—
245C22×C4—2122no—
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4C2×C4×D5311yescharacteristic
250C22×C4C2×D21211yescharacteristic
251C22×C4#1171994712yes—
252C22×C4#1171994712yes—
253M16—4242no—
254C2×C8—2122no—
255C4⋊C4—22no—
256C4⋊C4—4241no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—4241no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C42—82no—
267C42—4241no—
268C42—2122no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#1313314512yes—
278C23C2×#1171992111yescharacteristic
279C23#1313319812yes—
280D4—44no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—1062no—
298C2×C4—4244no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4#1313317311yescharacteristic
311C2×C4#1313317311yescharacteristic
312C2×C4#1313320112yes—
313C8—4242no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22#1681048411yescharacteristic
327C22C2×#1313314511yescharacteristic
328C22#1681064111yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—84no—
336C4—4241no—
337C4—4241no—
338C4—4241no—
339C4—2122no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2#2027481511yescharacteristic
345C2#2027481611yescharacteristic
346C2#2027482111yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC4×#99×C53C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenterC4×#99D53
hypocenterC53#8777
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20272601C2semidirect
4. #20272568C2semidirect
10. #16810491C4semidirect
12. #16810483C4semidirect
15. #16810469C4semidirect
16. C2×#13133124C4semidirect
196. C53#8777semidirect

Maximal subgroups

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

subgroupindex
#202776552
#202730532
#202726012
#202725682
C4×#99×C532
#877753