Finite Group Databasebeta

← #27294669

Subgroups — #27294669

Order 13568.

446 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 83 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#20279620C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic, complemented
3#20272720C212yesmaximal, complemented
4#20272717C212yesmaximal
5C4×#101×C53C211yesmaximal, characteristic, complemented
6C2×#13133193—21no—
7C2×C4×D212C411yescharacteristic
8#16810680—21no—
9#16810553C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810540C411yescharacteristic
12#16810540C2211yescharacteristic
13#16810514—21no—
14#16810514—21no—
15#16810489C411yescharacteristic
16#16810480—21no—
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C2×#101×C53—21no—
21C4×M16×C53—21no—
22C53×#343C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C4×D53—21no—
25#13133193—44no—
26C4×D212—41no—
27C4×D212—41no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C4⋊#88322—41no—
38C22⋊#88322—22no—
39#13133128—41no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×Q8×C53D411yescharacteristic
46C2×M16×C53—22no—
47C4×Q8×C53—41no—
48C4×Q8×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53C2311yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55#101×C53—24no—
56#97×C53—41no—
57C4×C8×C53—41no—
58C23×D53—21no—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×D212—41no—
73C2×D212—42no—
74C2×C4×D53—42no—
75C2×C4×D53—42no—
76C2×C4×D53—42no—
77M16×C53—24no—
78M16×C53—42no—
79C2×C8×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89#11719921—42no—
90#11719921—42no—
91#11719921—42no—
92#11719921—41no—
93#11719921—41no—
94#11719921—42no—
95C4⋊Dic53—42no—
96C4⋊Dic53—41no—
97#11719919—42no—
98#11719919—42no—
99C2×#88322—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—22no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108Q8×C53—44no—
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—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#10412yes—
126C2×C4×C53AGL(1,ℤ/8)12yes—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136D212—42no—
137D212—42no—
138C4×D53—84no—
139C8×C53—44no—
140#88322—82no—
141#11778—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41811yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C2×D53—42no—
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—81no—
151C2×D53—81no—
152C2×D53—81no—
153C2×D53—81no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162Dic53—81no—
163Dic53—81no—
164Dic53—81no—
165#2536—531no—
166C4×#101D5311yescharacteristic, complemented
167#1467—531no—
168#1346—532no—
169#1345—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53#150711yescharacteristic
173C2×C53#212011yescharacteristic
174C2×C53#213311yescharacteristic
175D53—82no—
176C2×C22⋊Q8—1061no—
177C2×C4×Q8C2×D5311yescharacteristic
178C2×C4×D4—531no—
179#432—1061no—
180C2×#101—21no—
181C4×M16—21no—
182#398—531no—
183#398—531no—
184#394—531no—
185#394—531no—
186#388—1061no—
187#388—1061no—
188#382—531no—
189#380—1061no—
190#343C2×D5312yes—
191#324—532no—
192#324—1061no—
193C53#1177811yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D5311yescharacteristic
195C22×D4—531no—
196C23×C4—1061no—
197C2×M16—22no—
198C22⋊Q8—2124no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—2121no—
202C4×D4—2121no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C4×D5311yescharacteristic
208C2×C4⋊C4D21211yescharacteristic
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C42C4×D5311yescharacteristic
216C2×C42D21211yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—2121no—
219#101—24no—
220C22⋊C8—1062no—
221#97—41no—
222#97—2121no—
223C4×C8—41no—
224#95—531no—
225#95—531no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24—1061no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—2121no—
234C2×D4—2122no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4—2122no—
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4C2×D21211yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—24no—
249M16—42no—
250C2×C8—42no—
251C2×C8—2122no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C22⋊C4—2121no—
263C22⋊C4—2121no—
264C22⋊C4—2122no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—1062no—
272C23—2121no—
273C23—2121no—
274C23—2121no—
275C23—2121no—
276C23—2121no—
277C23—2121no—
278C23C2×#1171992111yescharacteristic
279Q8—44no—
280D4—2122no—
281D4—2122no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—4244no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4#1313313111yescharacteristic
307C2×C4#1313313111yescharacteristic
308C2×C4#1313314512yes—
309C2×C4#1313324212yes—
310C8—44no—
311C8—4242no—
312C22—21no—
313C22—21no—
314C22—4241no—
315C22—4241no—
316C22—4241no—
317C22—4241no—
318C22—2122no—
319C22—2122no—
320C22—2122no—
321C22C2×#1313313111yescharacteristic
322C22#1681062711yescharacteristic
323C22C2×#1313314511yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—4241no—
333C4—4241no—
334C4—4241no—
335C2—21no—
336C2—21no—
337C2—4242no—
338C2#2027323011yescharacteristic
339C2#2027635711yescharacteristic
340C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#101D53
hypocenterC53#11778
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #20273059C2semidirect
3. #20272720C2semidirect
5. C4×#101×C53C2semidirect
166. C4×#101D53semidirect
193. C53#11778semidirect

Maximal subgroups

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

subgroupindex
#202796202
#202730592
#202727202
#202727172
C4×#101×C532
#1177853