Finite Group Databasebeta

← #28299614

Subgroups — #28299614

Order 13568.

550 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290570C211yesmaximal, characteristic
2#20290379C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5C4×#13133193C211yesmaximal, characteristic
6C22×#11719936—21no—
7C2×C4⋊#88324—21no—
8C2×#13133274—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—
28C4⋊#88324—42no—
29#13133274—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—21no—
77C2×C4×D53—21no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53SD1612yes—
83C2×C4×D53Q1612yes—
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90#11719936—48no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—41no—
93C42×C53—22no—
94C42×C53—42no—
95#11719921—24no—
96#11719921—42no—
97C4⋊Dic53—21no—
98C4⋊Dic53—41no—
99C4⋊Dic53M1612yescomplemented
100#11719919—24no—
101#11719919—41no—
102#11719919—41no—
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×D53—21no—
113C22×D53—21no—
114C22×D53#10312yes—
115Q8×C53—42no—
116Q8×C53—42no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
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#10312yes—
130C2×C4×C53#10412yes—
131C2×C4×C53C2×M1611yescharacteristic
132C2×C4×C53C2×M1611yescharacteristic
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—22no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53C2×SD1611yescharacteristic
147C2×Dic53C2×Q1611yescharacteristic
148C2×Dic53AGL(1,ℤ/8)11yescharacteristic
149C2×Dic53#13711yescharacteristic
150C4×D53—24no—
151C4×D53—24no—
152C4×D53—42no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—42no—
156#88324—44no—
157#88324—82no—
158#7617—531nomaximal, Sylow
159C22×C53#40711yescharacteristic
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—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—41no—
177C4×C53—81no—
178C4×C53#32814yes—
179Dic53—21no—
180Dic53—21no—
181Dic53—21no—
182Dic53—21no—
183Dic53—41no—
184Dic53—42no—
185Dic53—81no—
186C4×C22⋊Q8—531no—
187#1797—531no—
188#1728—531no—
189#1272—532no—
190#1271—532no—
191C2×C53#146511yescharacteristic
192C2×C53#148711yescharacteristic
193C2×C53#148911yescharacteristic
194C2×C53C2×#32811yescharacteristic
195C2×C53#151311yescharacteristic
196C2×C53#152911yescharacteristic
197C2×C53#155011yescharacteristic
198D53—42no—
199C22×M16—1061no—
200C2×C22⋊Q8—531no—
201C2×C4×Q8#615711yescharacteristic
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—531no—
211C4×C4⋊C4—1061no—
212C4×C22⋊C4—1061no—
213#336—532no—
214#336—532no—
215#336—1061no—
216C53#761711yescharacteristic, complemented, Sylow
217C22×Q8C2×#615711yescharacteristic
218C23×C4—531no—
219C23×C4—531no—
220C23×C4—531no—
221C2×M16—1064no—
222C2×M16—2122no—
223C2×M16—2122no—
224C22×C8—1062no—
225C22×C8—1062no—
226C22⋊Q8—1062no—
227C22⋊Q8—1062no—
228C4×Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
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—1061no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251C24—531no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8#1171993612yescomplemented
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—532no—
261C22×C4—532no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1061no—
268C22×C4—1061no—
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
278M16—2128no—
279C2×C8—1064no—
280C2×C8—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C2×C8—2122no—
284C2×C8—2122no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—532no—
288C4⋊C4—1064no—
289C4⋊C4—1061no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C22⋊C4—1064no—
294C22⋊C4—2122no—
295C42—22no—
296C42—42no—
297C42—1062no—
298C42—2121no—
299C42—2121no—
300C42—2122no—
301C23—532no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23C2×C22⋊#615711yescharacteristic
309Q8—42no—
310Q8—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1064no—
331C2×C4—1064no—
332C2×C4—1062no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4#1313327512yes—
346C2×C4#1313328012yes—
347C2×C4#1313328212yes—
348C2×C4D53⋊M1611yescharacteristic
349C2×C4C2×#1171993611yescharacteristic
350C8—4242no—
351C8—2124no—
352C22—1062no—
353C22—1062no—
354C22—1062no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22#1681159411yescharacteristic
360C22C2×#1313327511yescharacteristic
361C22#1681160911yescharacteristic
362C22C2×#1313328011yescharacteristic
363C22#1681161411yescharacteristic
364C22C2×#1313328211yescharacteristic
365C22#1681162011yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—42no—
370C4—81no—
371C4—1061no—
372C4—1061no—
373C4—1061no—
374C4—1061no—
375C4—4241no—
376C4—2122no—
377C4—2121no—
378C4#1681147714yes—
379C2—2122no—
380C2#2028997011yescharacteristic
381C2#2029004411yescharacteristic
382C2#2029005311yescharacteristic
383C2C2×#1681147711yescharacteristic
384C2#2029013011yescharacteristic
385C2#2029018211yescharacteristic
386C2#2029023211yescharacteristic
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#7617
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
99. C4⋊Dic53M16semidirect
216. C53#7617semidirect
255. C2×Q8#11719936semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202903792
#202895352
#202895342
C4×#131331932
#761753