Finite Group Databasebeta

← #28299674

Subgroups — #28299674

Order 13568.

556 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290379C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5C4×#13133193C211yesmaximal, characteristic
6C22×#11719936—21no—
7C2×#13133274—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133193C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810504—21no—
17#16810494—21no—
18#16810480C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C4×#11719921—21no—
22C2×C4×Q8×C53C411yescharacteristic
23C2×#11719936—24no—
24C2×#11719936—42no—
25C2×#11719936—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28#13133274—24no—
29#13133272—42no—
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C22×C4×D53D411yescharacteristic
33#13133193—22no—
34#13133193—22no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45C42×D53—42no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53C2×C411yescharacteristic
52C4×Q8×C53—42no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57C23×D53C22⋊C411yescharacteristic
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53M1612yescomplemented
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—22no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—22no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×#88324—24no—
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89#11719936—48no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C42×C53—22no—
93C42×C53—42no—
94#11719921—24no—
95#11719921—42no—
96C4⋊Dic53—21no—
97C4⋊Dic53—41no—
98C4⋊Dic53M1612yescomplemented
99#11719919—24no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C4×Dic53—42no—
106C23×C53C2×C22⋊C411yescharacteristic
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53#10412yes—
114Q8×C53—42no—
115Q8×C53—42no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53#9912yes—
129C2×C4×C53#10412yes—
130C2×C4×C53C2×M1611yescharacteristic
131C2×C4×C53C2×M1611yescharacteristic
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—22no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53AGL(1,ℤ/8)11yescharacteristic
146C2×Dic53#13711yescharacteristic
147C2×Dic53#13712yes—
148C4×D53—24no—
149C4×D53—24no—
150C4×D53—44no—
151C4×D53—42no—
152C4×D53—42no—
153#88324—44no—
154#88324—44no—
155#7634—531nomaximal, Sylow
156C22×C53#40711yescharacteristic
157C22×C53C2×#9911yescharacteristic
158C22×C53#41311yescharacteristic
159C22×C53#41711yescharacteristic
160C22×C53#41911yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—81no—
176Dic53—21no—
177Dic53—21no—
178Dic53—22no—
179Dic53—41no—
180Dic53—42no—
181Dic53—81no—
182C4×C22⋊Q8—531no—
183#1749—531no—
184#1728—531no—
185#1272—532no—
186#1271—532no—
187C2×C53#146511yescharacteristic
188C2×C53#148911yescharacteristic
189C2×C53#149111yescharacteristic
190C2×C53#150411yescharacteristic
191C2×C53#150811yescharacteristic
192C2×C53#152411yescharacteristic
193C2×C53#155911yescharacteristic
194D53—42no—
195C22×M16—1061no—
196C2×C22⋊Q8—531no—
197C2×C4×Q8#615711yescharacteristic
198C22×C42—531no—
199C2×C22⋊C8—1061no—
200C2×#97—1061no—
201#391—531no—
202#389—531no—
203#387—1061no—
204#385—1061no—
205#380—531no—
206C4×C4⋊C4—531no—
207C4×C4⋊C4—1061no—
208C4×C22⋊C4—1061no—
209#336—532no—
210#336—532no—
211#336—1061no—
212C53#763411yescharacteristic, complemented, Sylow
213C22×Q8C2×#615711yescharacteristic
214C23×C4—531no—
215C23×C4—531no—
216C23×C4—531no—
217C2×M16—1064no—
218C2×M16—2122no—
219C2×M16—2122no—
220C22×C8—1062no—
221C22×C8—1062no—
222C22⋊Q8—1062no—
223C22⋊Q8—1062no—
224C4×Q8—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4C2×#615711yescharacteristic
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C42—21no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—531no—
237C2×C42—1061no—
238C2×C42—2122no—
239C2×C42C2×#615711yescharacteristic
240C22⋊C8—1064no—
241#97—2122no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246#95—1061no—
247C24—531no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8#1171993612yescomplemented
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—1062no—
262C22×C4—1061no—
263C22×C4—1062no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4C22⋊#615711yescharacteristic
271C22×C4C22⋊#615711yescharacteristic
272C22×C4C22×#615711yescharacteristic
273M16—2128no—
274C2×C8—1064no—
275C2×C8—1064no—
276C2×C8—2122no—
277C2×C8—2122no—
278C2×C8—2122no—
279C2×C8—2122no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—532no—
283C4⋊C4—1064no—
284C4⋊C4—1061no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C22⋊C4—1064no—
289C22⋊C4—2122no—
290C42—22no—
291C42—42no—
292C42—1062no—
293C42—2121no—
294C42—2121no—
295C42—2122no—
296C23—532no—
297C23—1061no—
298C23—1061no—
299C23—1061no—
300C23—1061no—
301C23—1061no—
302C23—1061no—
303C23C2×C22⋊#615711yescharacteristic
304Q8—42no—
305Q8—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—532no—
321C2×C4—1064no—
322C2×C4—1064no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1062no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—2124no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4#1313327512yes—
340C2×C4#1313328212yes—
341C2×C4D53⋊M1611yescharacteristic
342C2×C4C2×#1171993611yescharacteristic
343C8—2124no—
344C8—2124no—
345C22—1062no—
346C22—1062no—
347C22—1062no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22#1681159411yescharacteristic
353C22C2×#1313327511yescharacteristic
354C22#1681160911yescharacteristic
355C22#1681161411yescharacteristic
356C22#1681161611yescharacteristic
357C22D53⋊#10411yescharacteristic
358C22C2×#1313328211yescharacteristic
359C4—22no—
360C4—22no—
361C4—21no—
362C4—21no—
363C4—42no—
364C4—81no—
365C4—1061no—
366C4—1061no—
367C4—1062no—
368C4—4241no—
369C4—2122no—
370C4—2121no—
371C2—2122no—
372C2#2028997011yescharacteristic
373C2#2029005311yescharacteristic
374C2#2029005811yescharacteristic
375C2#2029010111yescharacteristic
376C2#2029011511yescharacteristic
377C2#2029016711yescharacteristic
378C2#2029025011yescharacteristic
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#7634
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. #20289535C2semidirect
4. #20289534C2semidirect
61. C2×Q8×C53M16semidirect
98. C4⋊Dic53M16semidirect
212. C53#7634semidirect
251. C2×Q8#11719936semidirect

Maximal subgroups

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

subgroupindex
#202904422
#202903792
#202895352
#202895342
C4×#131331932
#763453