Finite Group Databasebeta

← #28300020

Subgroups — #28300020

Order 13568.

420 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290632C211yesmaximal, characteristic, complemented
2#20290631C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20283079C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133210—21no—
13#16810526C411yescharacteristic
14#16810526C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810519C411yescharacteristic
18#16810517—21no—
19#16810516—21no—
20#16810511—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—42no—
27#13133210—42no—
28#13133210—42no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×D4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C24×C53—21no—
53C2×D4×C53—22no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—22no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C42×C53—41no—
76C22⋊Dic53—42no—
77C22⋊Dic53—42no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—22no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C4×Dic53D812yes—
95C4×Dic53SD1612yes—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104D4×C53—82no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
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—41no—
115C2×C4×C53#10212yes—
116C2×C4×C53#13112yes—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53#10212yes—
133C2×Dic53C2×D811yescharacteristic
134C2×Dic53C2×SD1611yescharacteristic
135C2×Dic53AGL(1,ℤ/8)11yescharacteristic
136C2×Dic53#13711yescharacteristic
137#88324—82no—
138#88324—82no—
139#7732—531nomaximal, Sylow
140C22×C53—81no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53#40811yescharacteristic
148C22×C53#41011yescharacteristic
149C22×C53#41311yescharacteristic
150C22×C53C2×#10211yescharacteristic
151C22×C53#41711yescharacteristic
152C22×C53#42111yescharacteristic
153C22×C53#42111yescharacteristic
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—81no—
158Dic53—22no—
159Dic53—21no—
160Dic53—21no—
161Dic53—21no—
162Dic53—21no—
163Dic53—41no—
164Dic53—81no—
165Dic53—81no—
166#2860—531no—
167#1831—531no—
168#1831—531no—
169#1271—532no—
170#1270—532no—
171C2×C53—161no—
172C2×C53#147311yescharacteristic
173C2×C53#148011yescharacteristic
174C2×C53#149511yescharacteristic
175C2×C53#151211yescharacteristic
176C2×C53#151711yescharacteristic
177C2×C53#153511yescharacteristic
178C2×C53#154711yescharacteristic
179C2×#126—1061no—
180C2×C4⋊C8—1061no—
181C2×C4⋊C8—1061no—
182#390—531no—
183#390—531no—
184#390#615711yescharacteristic
185#389—531no—
186#389—531no—
187#389—531no—
188#388—1061no—
189#388—1061no—
190#388—1061no—
191C43—531no—
192#336—532no—
193#336—532no—
194#336—1061no—
195#336—1061no—
196C53#773211yescharacteristic, complemented, Sylow
197C22×D4C2×#615711yescharacteristic
198C22×C8—1062no—
199C22×C8—1062no—
200#126—2122no—
201#126—2122no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C2×#615711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42C2×#615711yescharacteristic
221C4⋊C8—2122no—
222C4⋊C8—2122no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C24—21no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4C22⋊#615711yescharacteristic
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22×#615711yescharacteristic
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C4⋊C4—22no—
247C4⋊C4—2121no—
248C4⋊C4—2121no—
249C4⋊C4—2122no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2122no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—2122no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C22⋊C4—2121no—
259C22⋊C4—2121no—
260C22⋊C4—2122no—
261C42—41no—
262C42—532no—
263C42—532no—
264C42—1061no—
265C42—1061no—
266C42—1062no—
267C42—2121no—
268C42—2121no—
269C42—2121no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#615711yescharacteristic
278D4—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—532no—
290C2×C4—531no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—531no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4#1313327912yes—
310C2×C4#1313330412yes—
311C8—4242no—
312C8—4242no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#1681159711yescharacteristic
321C22#1681160211yescharacteristic
322C22#1681161011yescharacteristic
323C22C2×#1313327911yescharacteristic
324C22#1681161411yescharacteristic
325C22#1681162011yescharacteristic
326C22#1681162111yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—81no—
331C4—1061no—
332C4—1061no—
333C4—1061no—
334C4—1061no—
335C4—1062no—
336C4—4241no—
337C4—4241no—
338C4—2121no—
339C2—161no—
340C2#2028999611yescharacteristic
341C2#2029002111yescharacteristic
342C2#2029007111yescharacteristic
343C2#2029012611yescharacteristic
344C2#2029014511yescharacteristic
345C2#2029020011yescharacteristic
346C2#2029022711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#7732
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290632C2semidirect
2. #20290631C2semidirect
3. #20289533C2semidirect
196. C53#7732semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202906312
#202895332
#202895312
#202830792
#773253