Finite Group Databasebeta

← #28306258

Subgroups — #28306258

Order 13568.

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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#20290571C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289553C214yesmaximal
4C2×#16810528C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×C22⋊#88324C2212yes—
7C2×C22⋊#88324C2212yes—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489—21no—
11C23×Dic106C411yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13#16810528—21no—
14#16810528—21no—
15#16810528—21no—
16#16810528—21no—
17C2×#13133124C412yes—
18C2×#13133124C2212yes—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—42no—
23C22⋊#88324—22no—
24C22⋊#88324—24no—
25C22⋊#88324Q814yescomplemented
26#13133272—42no—
27C22×Dic106—21no—
28C22×Dic106—21no—
29C22×Dic106—24no—
30C22×Dic106—42no—
31C22×Dic106—42no—
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C412yes—
34C23×Dic53C2311yescharacteristic
35C2×C4×Dic53—42no—
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40#13133124—22no—
41#13133124—22no—
42#13133124—22no—
43#13133124—22no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C412yes—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—42no—
49C24×C53C22×C411yescharacteristic
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—24no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53M1614yes—
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×Q812yes—
74C22×Dic53#4112yes—
75C2×Dic106—24no—
76C2×Dic106—24no—
77C2×Dic106—416no—
78C2×Dic106—42no—
79C2×Dic106—42no—
80C2×Dic106—42no—
81C2×Dic106—42no—
82C2×#88324—24no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—42no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53C4×Q812yes—
100C23×C53C2×M1612yes—
101C23×C53#13112yes—
102C2×C4×C53—28no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—84no—
111C2×C4×C53#9912yes—
112C2×C4×C53#10012yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—24no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—44no—
126C2×Dic53—44no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—44no—
130C2×Dic53#10112yes—
131C2×Dic53C22⋊Q812yes—
132C2×Dic53#12411yescharacteristic
133C2×Dic53C4⋊Q811yescharacteristic
134Dic106—48no—
135Dic106—48no—
136#88324—44no—
137#88324—82no—
138#9882—531nomaximal, Sylow
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53#39111yescharacteristic
146C22×C53#40711yescharacteristic
147C22×C53#40811yescharacteristic
148C22×C53C2×#9911yescharacteristic
149C22×C53#41011yescharacteristic
150C22×C53C2×#10011yescharacteristic
151C22×C53C2×#10111yescharacteristic
152C22×C53#44614yes—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—82no—
156Dic53—22no—
157Dic53—22no—
158Dic53—21no—
159Dic53—21no—
160Dic53—44no—
161Dic53—41no—
162C2×#391—531no—
163#1797—531no—
164#1749—531no—
165#1280—534no—
166C2×C53—24no—
167C2×C53#146211yescharacteristic
168C2×C53#146711yescharacteristic
169C2×C53#146911yescharacteristic
170C2×C53#147211yescharacteristic
171C2×C53#198511yescharacteristic
172C2×C53#199611yescharacteristic
173C2×C53#199711yescharacteristic
174C23×Q8—531no—
175C22×C4⋊C4#615711yescharacteristic
176C22×C42—531no—
177C2×C4⋊C8—1061no—
178C2×C22⋊C8—532no—
179C2×C22⋊C8—532no—
180C2×#97—1061no—
181#391—1061no—
182#391—1061no—
183#391—1061no—
184#391—1061no—
185C2×#95—532no—
186C2×#95—532no—
187#336—1061no—
188#336—1061no—
189C53#988211yescharacteristic, complemented, Sylow
190C22×Q8—1064no—
191C22×Q8—1061no—
192C22×Q8—1061no—
193C22×Q8—2122no—
194C22×Q8—2122no—
195C23×C4—531no—
196C23×C4—531no—
197C23×C4—532no—
198C23×C4C2×#615712yes—
199C23×C4C2×#615711yescharacteristic
200C22×C8—1062no—
201C22×C8—1062no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—2122no—
210C4⋊C8—2122no—
211C22⋊C8—534no—
212C22⋊C8—1062no—
213C22⋊C8—1064no—
214#97—2122no—
215#95—1062no—
216#95—1062no—
217#95—1062no—
218#95—1062no—
219C24C22×#615711yescharacteristic
220C2×Q8—1064no—
221C2×Q8—1064no—
222C2×Q8—21216no—
223C2×Q8—2122no—
224C2×Q8—2122no—
225C2×Q8—2122no—
226C2×Q8—2122no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—534no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—2121no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4C22⋊#615711yescharacteristic
251C22×C4C22⋊#615711yescharacteristic
252C2×C8—1064no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42—1062no—
261C42—1062no—
262C42—2121no—
263C42—2121no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C2×C22⋊#615711yescharacteristic
269C23Q8×#615712yes—
270C23D53⋊M1612yes—
271C23#1313330412yes—
272Q8—2128no—
273Q8—2128no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—532no—
284C2×C4—531no—
285C2×C4—531no—
286C2×C4—532no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1064no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4—2124no—
304C2×C4#1313327612yes—
305C2×C4#1313327712yes—
306C8—4242no—
307C8—2124no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22(C2×#6157)⋊Q811yescharacteristic
315C22#1681159511yescharacteristic
316C22#1681159711yescharacteristic
317C22C2×#1313327611yescharacteristic
318C22#1681160311yescharacteristic
319C22C2×#1313327711yescharacteristic
320C22D53⋊#10111yescharacteristic
321C22#1681167514yes—
322C4—22no—
323C4—22no—
324C4—82no—
325C4—1061no—
326C4—1062no—
327C4—1061no—
328C4—1062no—
329C4—2124no—
330C4—2121no—
331C2—24no—
332C2#2028996211yescharacteristic
333C2#2028997911yescharacteristic
334C2#2028998511yescharacteristic
335C2#2028999311yescharacteristic
336C2#2029088711yescharacteristic
337C2#2029090511yescharacteristic
338C2#2029090711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#9882
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
25. C22⋊#88324Q8semidirect
189. C53#9882semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
#202895532
C2×#168105282
#988253