Finite Group Databasebeta

← #28299516

Subgroups — #28299516

Order 13568.

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

normal subgroups only

This group has 578 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#20290572C211yesmaximal, characteristic
2#20290376C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5C4×#13133197C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×#13133284—21no—
8C2×#13133274—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133197C411yescharacteristic
13C2×C4×Dic106C411yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810528C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810508—21no—
18#16810497—21no—
19#16810480C411yescharacteristic
20C4×C4⋊Dic53—21no—
21C4×#11719921—21no—
22C4×C4⋊C4×C53C411yescharacteristic
23C2×D53⋊C8—24no—
24C2×D53⋊C8—42no—
25C2×D53⋊C8—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28#13133284—42no—
29#13133274—42no—
30C22×Dic106C2×C411yescharacteristic
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34#13133197—22no—
35#13133197C814yescomplemented
36C4×Dic106—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47C42×D53—42no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C42×C53—21no—
55C2×C42×C53C2×C411yescharacteristic
56#95×C53—21no—
57C23×D53C22⋊C411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53C2×C812yes—
77C2×C4×D53M1612yes—
78C2×C4×D53SD1612yes—
79C2×C4×D53Q1612yes—
80C2×Dic106—21no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×Dic106C2×C812yescomplemented
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90D53⋊C8—48no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53C2×C812yescomplemented
94C42×C53—22no—
95C42×C53—42no—
96#11719921—24no—
97#11719921—42no—
98C4⋊Dic53—24no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102#11719919—42no—
103C4×Dic53—22no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106C4×Dic53—42no—
107C23×C53C2×C22⋊C411yescharacteristic
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C22⋊C812yes—
113C22×D53#10312yes—
114C22×D53#10412yes—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53#9912yes—
128C2×C4×C53#10312yes—
129C2×C4×C53C22×C811yescharacteristic
130C2×C4×C53C2×M1611yescharacteristic
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—22no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53C22⋊C812yes—
144C2×Dic53C2×SD1611yescharacteristic
145C2×Dic53C2×Q1611yescharacteristic
146C2×Dic53AGL(1,ℤ/8)11yescharacteristic
147C2×Dic53#13711yescharacteristic
148C4×D53—24no—
149C4×D53—24no—
150C4×D53—42no—
151C4×D53—42no—
152C4×D53—42no—
153C4×D53—42no—
154Dic106—42no—
155Dic106—42no—
156#88324—44no—
157#88324—82no—
158#7589—531nomaximal, Sylow
159C22×C53C2×C22⋊C811yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53#41311yescharacteristic
162C22×C53C2×#10311yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53#42111yescharacteristic
166C2×D53—22no—
167C2×D53—22no—
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53Q8⋊C814yes—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—42no—
178C4×C53—81no—
179Dic53—22no—
180Dic53—21no—
181Dic53—21no—
182Dic53—21no—
183Dic53—21no—
184Dic53—42no—
185Dic53—81no—
186C4×C22⋊Q8—531no—
187#1797—531no—
188C2×#336—531no—
189#1272—532no—
190#1271—532no—
191C2×C53#145911yescharacteristic
192C2×C53C2×Q8⋊C811yescharacteristic
193C2×C53#147711yescharacteristic
194C2×C53#150411yescharacteristic
195C2×C53#152011yescharacteristic
196C2×C53#153211yescharacteristic
197C2×C53#155011yescharacteristic
198D53—24no—
199C23×C8—1061no—
200C2×C22⋊Q8—531no—
201C2×C4×Q8—531no—
202C22×C42—531no—
203C2×C4⋊C8—1061no—
204C2×C22⋊C8—1061no—
205#391—531no—
206#389—531no—
207#387—1061no—
208#385—1061no—
209#380—531no—
210C4×C4⋊C4—1061no—
211C4×C4⋊C4#615711yescharacteristic
212C4×C22⋊C4—1061no—
213#336—532no—
214#336—532no—
215#336—1061no—
216C53#758911yescharacteristic, complemented, Sylow
217C22×Q8—531no—
218C23×C4—531no—
219C23×C4—531no—
220C23×C4—531no—
221C22×C8—1062no—
222C22×C8—1064no—
223C22×C8—1062no—
224C22×C8—2122no—
225C22×C8—2122no—
226C22⋊Q8—534no—
227C22⋊Q8—1062no—
228C4×Q8—2122no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4C2×#615711yescharacteristic
234C2×C4⋊C4C2×#615711yescharacteristic
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C42—21no—
238C2×C42—531no—
239C2×C42—531no—
240C2×C42—531no—
241C2×C42—1061no—
242C2×C42—2122no—
243C2×C42C2×#615711yescharacteristic
244C4⋊C8—2122no—
245C22⋊C8—2122no—
246#95—21no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251C24—531no—
252C2×Q8—532no—
253C2×Q8—1061no—
254C2×Q8—2121no—
255C2×Q8—2121no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—532no—
264C22×C4—532no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1062no—
268C22×C4—1062no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4C22⋊#615711yescharacteristic
276C22×C4C22⋊#615711yescharacteristic
277C22×C4C22×#615711yescharacteristic
278C2×C8—1064no—
279C2×C8—2122no—
280C2×C8—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C2×C8—2122no—
284C2×C8—2128no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—1064no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C4⋊C4—2122no—
292C4⋊C4D53⋊C812yescomplemented
293C22⋊C4—1064no—
294C22⋊C4—2122no—
295C42—22no—
296C42—42no—
297C42—1062no—
298C42—2121no—
299C42—2121no—
300C42—2122no—
301C23—532no—
302C23—532no—
303C23—532no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23C2×C22⋊#615711yescharacteristic
309Q8—2122no—
310Q8—2122no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—532no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—1064no—
329C2×C4—1064no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1062no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4#1313327612yes—
347C2×C4#1313328012yes—
348C2×C4C2×D53⋊C811yescharacteristic
349C2×C4D53⋊M1611yescharacteristic
350C8—4242no—
351C8—2124no—
352C22—534no—
353C22—1062no—
354C22—1062no—
355C22—1062no—
356C22—1062no—
357C22—1062no—
358C22—1062no—
359C22D53⋊(C22⋊C8)11yescharacteristic
360C22C2×#1313327611yescharacteristic
361C22#1681160911yescharacteristic
362C22C2×#1313328011yescharacteristic
363C22#1681161411yescharacteristic
364C22D53⋊#10411yescharacteristic
365C22#1681162111yescharacteristic
366C4—22no—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—81no—
372C4—1061no—
373C4—1061no—
374C4—1061no—
375C4—1061no—
376C4—1062no—
377C4—4241no—
378C4—2122no—
379C2—1064no—
380C2#2028995211yescharacteristic
381C2D53⋊(Q8⋊C8)11yescharacteristic
382C2#2029001111yescharacteristic
383C2#2029010311yescharacteristic
384C2#2029015411yescharacteristic
385C2#2029019011yescharacteristic
386C2#2029023211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7589
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289533C2semidirect
35. #13133197C8semidirect
83. C2×Dic106C2×C8semidirect
93. C4⋊C4×C53C2×C8semidirect
216. C53#7589semidirect
292. C4⋊C4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202903762
#202895362
#202895332
C4×#131331972
#758953