Finite Group Databasebeta

← #28299784

Subgroups — #28299784

Order 13568.

438 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290430C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20283083C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133212—21no—
13#16810525C411yescharacteristic
14#16810524C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810522C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810515—21no—
19#16810513—21no—
20#16810512—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C4×#88324—24no—
27C22×D212C2×C411yescharacteristic
28#13133212—42no—
29#13133212—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52C23×D53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C2×D212—22no—
64C2×D212—41no—
65C2×D212—41no—
66C2×#88324—24no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—42no—
75C42×C53—41no—
76#11719921—42no—
77#11719921—42no—
78#11719921—42no—
79#11719921—42no—
80#11719921—41no—
81#11719921—41no—
82#11719921—42no—
83C4⋊Dic53—22no—
84C4⋊Dic53—42no—
85#11719919—41no—
86#11719919—41no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C4×Dic53D812yes—
93C4×Dic53SD1612yes—
94C23×C53C2×C22⋊C411yescharacteristic
95C22×D53—41no—
96C22×D53—41no—
97C22×D53—41no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53#10212yes—
115C2×C4×C53#10412yes—
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#10212yes—
130C2×Dic53C2×D811yescharacteristic
131C2×Dic53C2×SD1611yescharacteristic
132C2×Dic53#13512yes—
133D212—82no—
134#88324—44no—
135#88324—82no—
136#7666—531nomaximal, Sylow
137C22×C53#40811yescharacteristic
138C22×C53#41011yescharacteristic
139C22×C53#41311yescharacteristic
140C22×C53C2×#10211yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C22×C53#42111yescharacteristic
144C2×D53—81no—
145C2×D53—81no—
146C2×D53—81no—
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C2×D53—81no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—22no—
157Dic53—22no—
158Dic53—21no—
159Dic53—21no—
160Dic53—41no—
161Dic53—81no—
162#2860—531no—
163#1831—531no—
164#1744—531no—
165#1271—532no—
166#1270—532no—
167C2×C53#147311yescharacteristic
168C2×C53#148211yescharacteristic
169C2×C53#149211yescharacteristic
170C2×C53#150011yescharacteristic
171C2×C53#151711yescharacteristic
172C2×C53#153211yescharacteristic
173C2×C53#154011yescharacteristic
174D53—161no—
175C2×#126—1061no—
176C2×C4⋊C8—1061no—
177C2×C4×C8—1061no—
178#390—531no—
179#390—531no—
180#390—531no—
181#389—531no—
182#389—531no—
183#389#615711yescharacteristic
184#388—1061no—
185#388—1061no—
186#388—1061no—
187C43—531no—
188#336—532no—
189#336—532no—
190#336—1061no—
191#336—1061no—
192C53#766611yescharacteristic, complemented, Sylow
193C22×D4—531no—
194C22×C8—1062no—
195C22×C8—1062no—
196#126—2122no—
197#126—2122no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4C2×#615711yescharacteristic
203C2×C4⋊C4C2×#615711yescharacteristic
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42C2×#615711yescharacteristic
217C4⋊C8—2122no—
218C4×C8—1064no—
219#95—21no—
220#95—1061no—
221#95—1061no—
222C24—1061no—
223C2×D4—1062no—
224C2×D4—2121no—
225C2×D4—2121no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—1061no—
233C22×C4C22⋊#615711yescharacteristic
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22×#615711yescharacteristic
236C2×C8—1064no—
237C2×C8—2122no—
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—1062no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C4⋊C4—2122no—
249C22⋊C4—2122no—
250C22⋊C4—2121no—
251C22⋊C4—2121no—
252C22⋊C4—2122no—
253C22⋊C4—2122no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C42—41no—
257C42—532no—
258C42—532no—
259C42—1062no—
260C42—1062no—
261C42—1062no—
262C42—2121no—
263C42—2121no—
264C23—2121no—
265C23—2121no—
266C23—2121no—
267C23—2121no—
268C23—2121no—
269C23—2121no—
270C23—2121no—
271C23C2×C22⋊#615711yescharacteristic
272D4—4242no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—532no—
286C2×C4—531no—
287C2×C4—531no—
288C2×C4—532no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313327912yes—
302C2×C4#1313328212yes—
303C2×C4#1313330512yes—
304C8—4242no—
305C8—2124no—
306C22—4241no—
307C22—4241no—
308C22—4241no—
309C22—4241no—
310C22—4241no—
311C22—4241no—
312C22—4241no—
313C22#1681159811yescharacteristic
314C22#1681160211yescharacteristic
315C22#1681161011yescharacteristic
316C22C2×#1313327911yescharacteristic
317C22#1681161311yescharacteristic
318C22C2×#1313328211yescharacteristic
319C22#1681162011yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—81no—
324C4—81no—
325C4—1062no—
326C4—1062no—
327C4—1061no—
328C4—1061no—
329C4—4241no—
330C4—2121no—
331C2—8481no—
332C2#2028999811yescharacteristic
333C2#2029002911yescharacteristic
334C2#2029006211yescharacteristic
335C2#2029008811yescharacteristic
336C2#2029014511yescharacteristic
337C2#2029019311yescharacteristic
338C2#2029021311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7666
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #20290632C2semidirect
2. #20290430C2semidirect
3. #20289534C2semidirect
192. C53#7666semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202904302
#202895342
#202895322
#202830832
#766653