Finite Group Databasebeta

← #28302323

Subgroups — #28302323

Order 13568.

814 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 120 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#20290442C211yesmaximal, characteristic
2#20290442C211yesmaximal, characteristic
3C2×#16811480C214yesmaximal, complemented
4C2×#16811178C211yesmaximal, characteristic
5C2×#13133274—21no—
6C2×#13133274—21no—
7C2×#13133272C2212yes—
8C2×#13133272C2212yes—
9#16811489—21no—
10#16811489—21no—
11#16811480—28no—
12#16811178—28no—
13C2×#13133217C411yescharacteristic
14C2×#13133208C2212yes—
15C2×#13133197—21no—
16C2×#13133193—21no—
17C2×Q8×Dic53—21no—
18C2×#13133182—21no—
19C2×#13133175C412yes—
20C2×C42×D53C2211yescharacteristic
21C2×C4⋊Q8×C53C411yescharacteristic
22C22×#88324—22no—
23C22×#88324—22no—
24#13133274—24no—
25#13133274—24no—
26#13133272—24no—
27#13133272—24no—
28C22×Dic106—21no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133217—24no—
33#13133208—24no—
34#13133208D418yes—
35#13133197—28no—
36#13133193—28no—
37Q8×Dic53—28no—
38#13133182—28no—
39#13133175—24no—
40#13133175—24no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53C2×C412yes—
44C2×#11719919—22no—
45C2×#11719921—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50C42×D53—24no—
51C22×Q8×C53—21no—
52C4⋊Q8×C53—24no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C412yes—
55C2×C42×C53C2×C411yescharacteristic
56C23×D53C22⋊C411yescharacteristic
57C2×Q8×C53—24no—
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×Dic106—24no—
78C2×Dic106—42no—
79C2×Dic106—42no—
80C2×#88324—24no—
81C2×#88324—24no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53C22⋊C414yes—
89C42×C53—22no—
90#11719921—28no—
91C4⋊Dic53—24no—
92C4⋊Dic53—22no—
93C4⋊Dic53—28no—
94C4⋊Dic53C22⋊C414yes—
95#11719919—28no—
96C4×Dic53—28no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53C2×D414yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53#10112yes—
108Q8×C53—48no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—24no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53#10412yes—
119C2×C4×C53#10412yes—
120C2×C4×C53C2×C22⋊C412yes—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—24no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53#10112yes—
131C2×Dic53#12012yes—
132C2×Dic53#12012yes—
133C4×D53—28no—
134C4×D53—44no—
135C4×D53—44no—
136Dic106—48no—
137#88324—44no—
138#88324—44no—
139#8338—531nomaximal, Sylow
140C22×C53#37911yescharacteristic
141C22×C53C2×#10111yescharacteristic
142C22×C53C2×#10111yescharacteristic
143C22×C53C2×#10411yescharacteristic
144C22×C53C2×#10411yescharacteristic
145C22×C53#42111yescharacteristic
146C22×C53#42111yescharacteristic
147C2×D53—22no—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—44no—
158Dic53—22no—
159Dic53—22no—
160Dic53—22no—
161Dic53—44no—
162C2×#533—531no—
163#1749—531no—
164#1749—531no—
165C2×#330—534no—
166C2×C53#152611yescharacteristic
167C2×C53#153111yescharacteristic
168C2×C53#153111yescharacteristic
169C2×C53#153711yescharacteristic
170C2×C53#178411yescharacteristic
171C2×C53#178911yescharacteristic
172C2×C53#178911yescharacteristic
173D53—42no—
174#533—1068no—
175C2×C4⋊Q8—531no—
176C2×C4⋊Q8#615711yescharacteristic
177C2×#125—532no—
178C2×C22⋊Q8—1061no—
179C2×C22⋊Q8—1061no—
180C2×C4×Q8—1061no—
181C2×C4×Q8—1061no—
182C2×#117—532no—
183C22×C42—531no—
184C2×C22⋊C8—1061no—
185C2×C22⋊C8—1061no—
186C2×#97—532no—
187C2×#97—532no—
188#336—1061no—
189#336—1061no—
190#330—1068no—
191C53#833811yescharacteristic, complemented, Sylow
192C22×Q8—21no—
193C22×Q8—1061no—
194C23×C4—531no—
195C23×C4—531no—
196C23×C4—531no—
197C22×C8—1062no—
198C22×C8—1062no—
199C4⋊Q8—24no—
200C4⋊Q8—1064no—
201#125—538no—
202#125—1064no—
203C22⋊Q8—1068no—
204C22⋊Q8—1068no—
205C4×Q8—1068no—
206C4×Q8—1068no—
207#117—1064no—
208#117—1064no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C2×#615712yes—
215C2×C22⋊C4—1062no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—1064no—
220C2×C42—1062no—
221C2×C42C2×#615711yescharacteristic
222C22⋊C8—1064no—
223C22⋊C8—1064no—
224#97—1064no—
225#97—1064no—
226C24—531no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—1064no—
231C2×Q8—2122no—
232C2×Q8—2122no—
233C22×C4—22no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—2121no—
246C22×C4—2121no—
247C22×C4C22⋊#615711yescharacteristic
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22×#615711yescharacteristic
250C2×C8—1064no—
251C2×C8—1064no—
252C2×C8—2122no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—534no—
259C4⋊C4—1068no—
260C4⋊C4—1068no—
261C4⋊C4—1064no—
262C4⋊C4—1062no—
263C4⋊C4C22⋊#615714yes—
264C22⋊C4—1068no—
265C42—22no—
266C42—534no—
267C42—1062no—
268C42—1062no—
269C42—1068no—
270C23—532no—
271C23—1061no—
272C23—1061no—
273C23—1061no—
274C23—1061no—
275C23—1061no—
276C23—1061no—
277C23C2×C22⋊#615711yescharacteristic
278Q8—48no—
279Q8—2128no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—532no—
292C2×C4—1068no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1064no—
300C2×C4—2124no—
301C2×C4—2124no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4#1313328112yes—
305C2×C4#1313328212yes—
306C2×C4C2×C22⋊#615712yes—
307C8—2124no—
308C8—2124no—
309C22—1062no—
310C22—1062no—
311C22—1062no—
312C22—2121no—
313C22—2121no—
314C22—2121no—
315C22—2121no—
316C22#1681156411yescharacteristic
317C22D53⋊#10111yescharacteristic
318C22D53⋊#10111yescharacteristic
319C22C2×#1313328111yescharacteristic
320C22C2×#1313328211yescharacteristic
321C22#1681162111yescharacteristic
322C22#1681162211yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—1062no—
328C4—1062no—
329C4—1062no—
330C4—2124no—
331C2—2122no—
332C2#2029017311yescharacteristic
333C2#2029018811yescharacteristic
334C2#2029018911yescharacteristic
335C2#2029020711yescharacteristic
336C2#2029055011yescharacteristic
337C2#2029055611yescharacteristic
338C2#2029055811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#8338
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16811480C2semidirect
191. C53#8338semidirect

Maximal subgroups

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

subgroupindex
#202904422
C2×#168114802
C2×#168111782
#833853