Finite Group Databasebeta

← #28299696

Subgroups — #28299696

Order 13568.

582 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290570C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#389×D53C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×C4⋊#88324—21no—
8C2×#13133274—21no—
9C2×#13133274—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53C411yescharacteristic
14C2×C4⋊C4×D53C411yescharacteristic
15C2×C42×D53C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810520—21no—
19#16810519C2211yescharacteristic
20#16810518—21no—
21#95×D53—21no—
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C4⋊#88324—42no—
27#13133274—42no—
28#13133274—42no—
29C22×C4×D53—21no—
30C22×C4×D53—21no—
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C4⋊C4×D53—22no—
42C4⋊C4×D53—22no—
43C4⋊C4×D53—42no—
44C4⋊C4×D53—42no—
45C4⋊C4×D53—42no—
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49C42×D53—42no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57#95×C53—21no—
58C23×D53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×C4×D53—22no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—22no—
73C2×C4×D53—24no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×C4×D53—42no—
82C2×C4×D53—42no—
83C2×C4×D53D812yes—
84C2×C4×D53SD1612yes—
85C2×C4×D53SD1612yes—
86C2×C4×D53Q1612yes—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53M1612yes—
97C42×C53—41no—
98C4⋊Dic53—21no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53M1612yes—
103#11719919—24no—
104#11719919—42no—
105#11719919—42no—
106#11719919—41no—
107#11719919—41no—
108C4×Dic53—22no—
109C4×Dic53—22no—
110C4×Dic53—41no—
111C23×C53C2×C22⋊C411yescharacteristic
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53#10112yes—
117C22×D53#10212yes—
118C22×D53#10312yes—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53#10212yes—
132C2×C4×C53#10312yes—
133C2×C4×C53C2×M1611yescharacteristic
134C2×C4×C53C2×M1611yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—22no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53#9912yes—
148C2×Dic53C2×D811yescharacteristic
149C2×Dic53C2×SD1611yescharacteristic
150C2×Dic53C2×SD1611yescharacteristic
151C2×Dic53C2×Q1611yescharacteristic
152C4×D53—24no—
153C4×D53—24no—
154C4×D53—42no—
155C4×D53—42no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—48no—
159C4×D53—84no—
160#88324—82no—
161#88324—82no—
162#7641—531nomaximal, Sylow
163C22×C53#40711yescharacteristic
164C22×C53C2×#9911yescharacteristic
165C22×C53C2×#10111yescharacteristic
166C22×C53C2×#10211yescharacteristic
167C22×C53C2×#10311yescharacteristic
168C22×C53#42111yescharacteristic
169C22×C53#42111yescharacteristic
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—22no—
176C2×D53#33214yes—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—42no—
182C4×C53—81no—
183Dic53—22no—
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53—21no—
188Dic53—42no—
189Dic53—81no—
190C2×#389—531no—
191#1797—531no—
192#1797—531no—
193#1271—532no—
194#1271—532no—
195C2×C53#146311yescharacteristic
196C2×C53#148611yescharacteristic
197C2×C53#148711yescharacteristic
198C2×C53#151911yescharacteristic
199C2×C53#152011yescharacteristic
200C2×C53#153411yescharacteristic
201C2×C53C2×#33211yescharacteristic
202D53—24no—
203C22×C4⋊C4—531no—
204C22×C4⋊C4—531no—
205C22×C4⋊C4—1061no—
206C22×C42—531no—
207C2×C4⋊C8—1061no—
208C2×C4⋊C8—1061no—
209C2×C22⋊C8—1061no—
210C2×C22⋊C8—1061no—
211#389—531no—
212#389—531no—
213#389—531no—
214#389—1061no—
215#389—1061no—
216#389#615711yescharacteristic
217C2×#95—1061no—
218#336—532no—
219#336—532no—
220C53#764111yescharacteristic, complemented, Sylow
221C23×C4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C23×C4—1061no—
225C23×C4—1061no—
226C22×C8—1062no—
227C22×C8—1062no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4—2122no—
239C2×C4⋊C4—2122no—
240C2×C4⋊C4—2122no—
241C2×C4⋊C4C2×#615711yescharacteristic
242C2×C4⋊C4C2×#615711yescharacteristic
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—2122no—
247C2×C42C2×#615711yescharacteristic
248C4⋊C8—2122no—
249C4⋊C8—2122no—
250C22⋊C8—2122no—
251C22⋊C8—2122no—
252#95—21no—
253#95—1061no—
254#95—1061no—
255#95—1061no—
256C24—531no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—532no—
263C22×C4—532no—
264C22×C4—532no—
265C22×C4—532no—
266C22×C4—1062no—
267C22×C4—1062no—
268C22×C4—1064no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2122no—
278C22×C4—2122no—
279C22×C4—2122no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4C22⋊#615711yescharacteristic
283C22×C4C22⋊#615711yescharacteristic
284C22×C4C22×#615711yescharacteristic
285C2×C8—2122no—
286C2×C8—2122no—
287C2×C8—2122no—
288C2×C8—2122no—
289C2×C8—2122no—
290C2×C8—2122no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—532no—
295C4⋊C4—1064no—
296C4⋊C4—1061no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C4⋊C4—2121no—
302C4⋊C4—2122no—
303C4⋊C4—2122no—
304C4⋊C4#1171993612yes—
305C42—41no—
306C42—1062no—
307C42—1062no—
308C42—2121no—
309C23—532no—
310C23—532no—
311C23—532no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23C2×C22⋊#615711yescharacteristic
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—532no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—1064no—
335C2×C4—1064no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1062no—
343C2×C4—4244no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2128no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4#1313327912yes—
355C2×C4#1313328012yes—
356C2×C4D53⋊M1611yescharacteristic
357C2×C4C2×#1171993611yescharacteristic
358C8—4242no—
359C8—4242no—
360C22—534no—
361C22—1062no—
362C22—1062no—
363C22—1062no—
364C22—1062no—
365C22—1062no—
366C22—1062no—
367C22#1681159411yescharacteristic
368C22D53⋊#9911yescharacteristic
369C22D53⋊#10111yescharacteristic
370C22C2×#1313327911yescharacteristic
371C22C2×#1313328011yescharacteristic
372C22#1681162011yescharacteristic
373C22#1681162011yescharacteristic
374C4—22no—
375C4—22no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—81no—
380C4—1061no—
381C4—1061no—
382C4—1061no—
383C4—1061no—
384C4—1062no—
385C4—4241no—
386C4—2122no—
387C2—1064no—
388C2#2028996411yescharacteristic
389C2#2029004111yescharacteristic
390C2#2029004411yescharacteristic
391C2#2029015011yescharacteristic
392C2#2029015311yescharacteristic
393C2#2029019811yescharacteristic
394C2D53⋊#33211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7641
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
220. C53#7641semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202895342
#202895332
#389×D532
#764153