Finite Group Databasebeta

← #27288595

Subgroups — #27288595

Order 13568.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#20282987C211yesmaximal, characteristic
2#20274883C211yesmaximal, characteristic
3#20274849C211yesmaximal, characteristic
4#20274848C211yesmaximal, characteristic
5#20272597C211yesmaximal, characteristic
6#20272597C211yesmaximal, characteristic
7C53×#1295C211yesmaximal, characteristic
8C2×#13133151—21no—
9C2×#13133149—21no—
10C2×#13133128C2211yescharacteristic
11#16810565—21no—
12#16810565—21no—
13#16810562—21no—
14#16810544—21no—
15#16810544—21no—
16#16810543—21no—
17#16810493C2211yescharacteristic
18#16810489C2211yescharacteristic
19#16810489C2211yescharacteristic
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810385—21no—
24C2×#97×C53C2211yescharacteristic
25C2×#97×C53C2211yescharacteristic
26C53×#382C2211yescharacteristic
27C22×Dic106—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39#13133151—42no—
40#13133149—42no—
41#13133128—21no—
42#13133128Q812yescomplemented
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#97×C53—21no—
57#97×C53—21no—
58#97×C53Q812yescomplemented
59#97×C53Q812yescomplemented
60#95×C53—21no—
61#95×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C2×Dic106—81no—
73C2×Dic106—81no—
74C2×Dic106—81no—
75C2×C8×C53—22no—
76C2×C8×C53—22no—
77C2×C8×C53—22no—
78C2×C8×C53—22no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53C2×Q811yescharacteristic
84C42×C53C2×Q811yescharacteristic
85C42×C53C2×Q811yescharacteristic
86C42×C53#4111yescharacteristic
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—42no—
94C4⋊Dic53—42no—
95C4⋊Dic53—81no—
96#11719919—81no—
97#11719919—81no—
98C2×#88322—22no—
99C2×#88322—41no—
100C2×#88322—41no—
101C23×C53#12011yescharacteristic
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×C53C22⋊Q811yescharacteristic
115C2×C4×C53C22⋊Q811yescharacteristic
116C2×C4×C53C22⋊Q811yescharacteristic
117C2×C4×C53C4⋊Q811yescharacteristic
118C2×C4×C53C4⋊Q811yescharacteristic
119C2×C4×C53C4⋊Q811yescharacteristic
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135Dic106—84no—
136C8×C53—24no—
137C8×C53—42no—
138#88322—42no—
139#10584—531nomaximal, Sylow
140C22×C53#39511yescharacteristic
141C22×C53#45311yescharacteristic
142C22×C53#45411yescharacteristic
143C22×C53#45411yescharacteristic
144C22×C53#45511yescharacteristic
145C22×C53#45511yescharacteristic
146C22×C53#45611yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—81no—
154C4×C53—81no—
155Dic53—81no—
156Dic53—81no—
157Dic53—81no—
158Dic53—81no—
159Dic53—161no—
160#2844—531no—
161#1847—531no—
162#1836—531no—
163#1836—531no—
164#1295—531no—
165#1295—531no—
166#1295D5311yescharacteristic
167C2×C53#165711yescharacteristic
168C2×C53#165911yescharacteristic
169C2×C53#166411yescharacteristic
170C2×C53#166411yescharacteristic
171C2×C53#206011yescharacteristic
172C2×C53#206111yescharacteristic
173C2×C53#206111yescharacteristic
174C2×#107—1061no—
175C2×#106—1061no—
176C2×#97—531no—
177C2×#97C2×D5311yescharacteristic
178C2×#97C2×D5311yescharacteristic
179#400—1061no—
180#400—1061no—
181#400—1061no—
182#395—1061no—
183#395—1061no—
184#395—1061no—
185#384—531no—
186#382—531no—
187#382—531no—
188#382C2×D5311yescharacteristic
189#340—1061no—
190#340—1061no—
191#340—1061no—
192#340—1061no—
193C53#1058411yescharacteristic, complemented, Sylow
194C22×Q8—1061no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—1061no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4D21211yescharacteristic
209C2×C4⋊C4D21211yescharacteristic
210C2×C42C22×D5311yescharacteristic
211#107—2122no—
212#106—2122no—
213#97—21no—
214#97—21no—
215#97—532no—
216#97—1061no—
217#97Dic10612yescomplemented
218#97Dic10612yescomplemented
219#95—21no—
220#95—21no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C2×Q8—4241no—
230C2×Q8—4241no—
231C2×Q8—4241no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—2121no—
239C22×C4C2×D21211yescharacteristic
240C22×C4D4×D5311yescharacteristic
241C22×C4D4×D5311yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—1062no—
249C2×C8—2121no—
250C2×C8—2121no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—4241no—
254C4⋊C4—4241no—
255C4⋊C4—4241no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C42C2×Dic10611yescharacteristic
265C42C2×Dic10611yescharacteristic
266C42#1171994711yescharacteristic
267C42Q8×D5311yescharacteristic
268C23#1313318711yescharacteristic
269Q8—4244no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—4241no—
283C2×C4—4241no—
284C2×C4—4241no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4#1313319411yescharacteristic
298C2×C4#1313319411yescharacteristic
299C2×C4#1313319711yescharacteristic
300C2×C4#1313321611yescharacteristic
301C2×C4#1313321611yescharacteristic
302C2×C4#1313321711yescharacteristic
303C8—24no—
304C8—42no—
305C8—2122no—
306C22#1681054411yescharacteristic
307C22#1681079211yescharacteristic
308C22#1681079711yescharacteristic
309C22#1681079811yescharacteristic
310C22#1681080211yescharacteristic
311C22#1681080311yescharacteristic
312C22#1681080711yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—81no—
320C4—81no—
321C4—8481no—
322C4—4241no—
323C4—4241no—
324C4—4241no—
325C4—4241no—
326C2#2027401511yescharacteristic
327C2#2027402311yescharacteristic
328C2#2027405011yescharacteristic
329C2#2027405211yescharacteristic
330C2#2027605911yescharacteristic
331C2#2027606411yescharacteristic
332C2#2027606611yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10584
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
42. #13133128Q8semidirect
58. #97×C53Q8semidirect
59. #97×C53Q8semidirect
193. C53#10584semidirect
217. #97Dic106semidirect
218. #97Dic106semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202748832
#202748492
#202748482
#202725972
C53×#12952
#1058453