Finite Group Databasebeta

← #28299859

Subgroups — #28299859

Order 13568.

562 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C4×C22⋊#88324C211yesmaximal, characteristic
2#20290378C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289535C212yesmaximal, complemented
5#20283063C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133215C411yescharacteristic
13C2×#13133196C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810530C2211yescharacteristic
16#16810528C2211yescharacteristic
17#16810518—21no—
18#16810518—21no—
19#16810492C411yescharacteristic
20#16810481—21no—
21#16810481—21no—
22C2×C22⋊Q8×C53C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—42no—
25C2×#11719940—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28C22⋊#88324—24no—
29C4×#88324—42no—
30C22×Dic106C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133215—42no—
34#13133196—42no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C22⋊Q8×C53—42no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—41no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53SD1612yes—
78C22×Dic53Q1612yes—
79C22×Dic53C2×D411yescharacteristic
80C22×Dic53C2×D411yescharacteristic
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84#11719940—48no—
85C2×#88324—24no—
86C2×#88324—24no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊Dic53—42no—
95C4⋊Dic53—41no—
96#11719919—42no—
97#11719919—42no—
98#11719919—42no—
99#11719919—42no—
100#11719919—42no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#10312yes—
112C23×C53C2×C22⋊C411yescharacteristic
113Q8×C53—82no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—44no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10012yes—
126C2×C4×C53#13112yes—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—24no—
134C2×Dic53—24no—
135C2×Dic53—44no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53#9912yes—
143C2×Dic53#10312yes—
144C2×Dic53#10412yes—
145C2×Dic53#10412yes—
146C2×Dic53C2×SD1611yescharacteristic
147C2×Dic53C2×Q1611yescharacteristic
148C2×Dic53#13712yes—
149Dic106—82no—
150#88324—44no—
151#88324—44no—
152#7687—531nomaximal, Sylow
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53#40811yescharacteristic
161C22×C53C2×#9911yescharacteristic
162C22×C53C2×#10011yescharacteristic
163C22×C53C2×#10311yescharacteristic
164C22×C53#41911yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C22×C53C2×#10411yescharacteristic
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—81no—
170C4×C53—81no—
171Dic53—21no—
172Dic53—21no—
173Dic53—22no—
174Dic53—22no—
175Dic53—81no—
176Dic53—81no—
177Dic53#32814yes—
178#2857—531no—
179C4×C22⋊C8—531no—
180#1728—531no—
181#1272—532no—
182#1272—532no—
183C2×C53—42no—
184C2×C53#147111yescharacteristic
185C2×C53#148511yescharacteristic
186C2×C53#149311yescharacteristic
187C2×C53C2×#32811yescharacteristic
188C2×C53#150811yescharacteristic
189C2×C53#152511yescharacteristic
190C2×C53#154811yescharacteristic
191C22×M16—1061no—
192C2×C4⋊Q8—531no—
193C2×C22⋊Q8—531no—
194C2×C22⋊Q8#615711yescharacteristic
195C22×C42—531no—
196C2×C22⋊C8—1061no—
197C2×C4×C8—1061no—
198#391—531no—
199#391—531no—
200#389—1061no—
201#389—1061no—
202#384—531no—
203#380—1061no—
204#380—1061no—
205#336—532no—
206#336—532no—
207#336—1061no—
208C53#768711yescharacteristic, complemented, Sylow
209C22×Q8—531no—
210C22×Q8C2×#615711yescharacteristic
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C2×#615711yescharacteristic
214C2×M16—1064no—
215C2×M16—2122no—
216C2×M16—2122no—
217C22×C8—1062no—
218C22×C8—1062no—
219C4⋊Q8—2122no—
220C22⋊Q8—42no—
221C22⋊Q8—2122no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4C2×#615711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1061no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—2122no—
237C22⋊C8—1064no—
238C4×C8—2122no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243C24C22⋊#615711yescharacteristic
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—1062no—
248C2×Q8—2121no—
249C2×Q8—2121no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4C22⋊#615711yescharacteristic
269C22×C4C22×#615711yescharacteristic
270M16—2128no—
271C2×C8—1064no—
272C2×C8—1064no—
273C2×C8—2122no—
274C2×C8—2122no—
275C2×C8—2122no—
276C2×C8—2122no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—2121no—
280C4⋊C4—2122no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C4⋊C4—2122no—
285C22⋊C4—42no—
286C22⋊C4—2122no—
287C42—1062no—
288C42—1062no—
289C42—2121no—
290C42—2121no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23#1313328012yes—
298C23C2×C22⋊#615711yescharacteristic
299Q8—82no—
300Q8—4242no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—532no—
318C2×C4—532no—
319C2×C4—1064no—
320C2×C4—1064no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2124no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4#1313327712yes—
335C2×C4#1313330412yes—
336C8—2124no—
337C8—2124no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22#1681159611yescharacteristic
346C22D53⋊#9911yescharacteristic
347C22C2×#1313327711yescharacteristic
348C22C2×#1313328011yescharacteristic
349C22#1681161611yescharacteristic
350C22D53⋊#10411yescharacteristic
351C22D53⋊#10411yescharacteristic
352C4—22no—
353C4—22no—
354C4—81no—
355C4—81no—
356C4—534no—
357C4—1061no—
358C4—1061no—
359C4—1062no—
360C4—1062no—
361C4—4241no—
362C4—4241no—
363C2—42no—
364C2#2028999111yescharacteristic
365C2#2029003711yescharacteristic
366C2#2029006311yescharacteristic
367C2D53⋊#32811yescharacteristic
368C2#2029011311yescharacteristic
369C2#2029016911yescharacteristic
370C2#2029022811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7687
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289535C2semidirect
208. C53#7687semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#883242
#202903782
#202895362
#202895352
#202830632
#768753