Finite Group Databasebeta

← #28299996

Subgroups — #28299996

Order 13568.

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

normal subgroups only

This group has 494 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#20290606C211yesmaximal, characteristic
2#20290569C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283065C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×#13133194—21no—
14C22×C4×Dic53C2211yescharacteristic
15#16810527—21no—
16#16810522C2211yescharacteristic
17#16810520—21no—
18#16810519C2211yescharacteristic
19#16810492C411yescharacteristic
20#16810483C411yescharacteristic
21#16810481—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C22×Dic106—21no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133216—42no—
32#13133194—44no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—21no—
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53SD1612yes—
72C22×Dic53Q1612yes—
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×Dic106—42no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—22no—
89C4⋊Dic53—42no—
90C4⋊Dic53—41no—
91#11719919—42no—
92#11719919—42no—
93#11719919—42no—
94#11719919—42no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53#10312yes—
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—44no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53#10112yes—
119C2×C4×C53#13112yes—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—44no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53#9912yes—
138C2×Dic53#10312yes—
139C2×Dic53C2×SD1611yescharacteristic
140C2×Dic53C2×Q1611yescharacteristic
141C2×Dic53AGL(1,ℤ/8)12yes—
142Dic106—84no—
143#88324—82no—
144#88324—82no—
145#7725—531nomaximal, Sylow
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53#40811yescharacteristic
154C22×C53C2×#9911yescharacteristic
155C22×C53C2×#10111yescharacteristic
156C22×C53C2×#10311yescharacteristic
157C22×C53#41811yescharacteristic
158C22×C53#42111yescharacteristic
159C22×C53#42111yescharacteristic
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—81no—
163C4×C53—81no—
164Dic53—22no—
165Dic53—22no—
166Dic53—21no—
167Dic53—22no—
168Dic53—21no—
169Dic53—81no—
170Dic53—81no—
171#2857—531no—
172#1817—531no—
173#1797—531no—
174#1271—532no—
175#1271—532no—
176C2×C53—42no—
177C2×C53#147211yescharacteristic
178C2×C53#148111yescharacteristic
179C2×C53#149611yescharacteristic
180C2×C53#151011yescharacteristic
181C2×C53#152011yescharacteristic
182C2×C53#153711yescharacteristic
183C2×C53#154911yescharacteristic
184C2×C4⋊Q8—531no—
185C2×C22⋊Q8—1061no—
186C22×C42—531no—
187C2×C4⋊C8—1061no—
188C2×C22⋊C8—1061no—
189C2×C22⋊C8—1061no—
190#391—1061no—
191#389—531no—
192#389—531no—
193#389—1061no—
194#384—531no—
195#380—531no—
196#380—1061no—
197#380#615711yescharacteristic
198#336—532no—
199#336—532no—
200#336—1061no—
201C53#772511yescharacteristic, complemented, Sylow
202C22×Q8—1061no—
203C23×C4—531no—
204C23×C4—531no—
205C23×C4C2×#615711yescharacteristic
206C22×C8—1062no—
207C22×C8—1062no—
208C4⋊Q8—2122no—
209C22⋊Q8—2124no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4C2×#615711yescharacteristic
218C2×C4⋊C4C2×#615711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—1061no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—2122no—
226C4⋊C8—2122no—
227C22⋊C8—2122no—
228C22⋊C8—2122no—
229#95—21no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C24C22⋊#615711yescharacteristic
234C2×Q8—2122no—
235C2×Q8—2122no—
236C2×Q8—2122no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—532no—
247C22×C4—532no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4—2121no—
255C22×C4C22⋊#615711yescharacteristic
256C22×C4C22×#615711yescharacteristic
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—1062no—
266C4⋊C4—2121no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C4⋊C4—2122no—
270C4⋊C4—2122no—
271C4⋊C4—2122no—
272C22⋊C4—42no—
273C22⋊C4—2122no—
274C42—1062no—
275C42—1062no—
276C42—2121no—
277C42—2121no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#1313328012yes—
285C23C2×C22⋊#615711yescharacteristic
286Q8—4244no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—532no—
299C2×C4—532no—
300C2×C4—531no—
301C2×C4—531no—
302C2×C4—532no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2124no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4#1313327812yes—
321C2×C4#1313330412yes—
322C8—4242no—
323C8—4242no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#1681159611yescharacteristic
332C22D53⋊#9911yescharacteristic
333C22C2×#1313327811yescharacteristic
334C22C2×#1313328011yescharacteristic
335C22#1681161511yescharacteristic
336C22#1681162011yescharacteristic
337C22#1681162011yescharacteristic
338C4—22no—
339C4—22no—
340C4—81no—
341C4—81no—
342C4—1062no—
343C4—1062no—
344C4—1061no—
345C4—1062no—
346C4—1061no—
347C4—4241no—
348C4—4241no—
349C2—42no—
350C2#2028999411yescharacteristic
351C2#2029002311yescharacteristic
352C2#2029007411yescharacteristic
353C2#2029012111yescharacteristic
354C2#2029015311yescharacteristic
355C2#2029020611yescharacteristic
356C2#2029023011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7725
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
201. C53#7725semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202905692
#202895342
#202895332
#202830652
#772553