Finite Group Databasebeta

← #28304289

Subgroups — #28304289

Order 13568.

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

normal subgroups only

This group has 604 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
1C2×#16811644C211yesmaximal, characteristic
2#20290606C211yesmaximal, characteristic
3C2×#16811468C212yesmaximal
4#20289553C212yesmaximal
5C2×#16810510C211yesmaximal, characteristic
6#16811644—24no—
7C2×C4⋊#88324—21no—
8C2×C22⋊#88324C2212yes—
9C2×C22⋊#88324C2212yes—
10C2×#13133272—21no—
11#16811489—21no—
12#16811489—21no—
13#16811468—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C411yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810510—21no—
18#16810510—21no—
19#16810510—21no—
20#16810510—21no—
21C2×#13133124C411yescharacteristic
22C2×#13133124C411yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#95×C53C411yescharacteristic
25C22×#88324—22no—
26C22×#88324—22no—
27C4⋊#88324—24no—
28C22⋊#88324—22no—
29C22⋊#88324—22no—
30C22⋊#88324—22no—
31C22⋊#88324D414yes—
32#13133272—24no—
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C23×Dic53C2×C411yescharacteristic
36C23×Dic53C2311yescharacteristic
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—22no—
41C2×#11719919—22no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—22no—
45C2×C4×Dic53—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53C2×C411yescharacteristic
59C23×C4×C53C2×C411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62C24×C53C22×C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53M1612yes—
77C22×Dic53—21no—
78C22×Dic53—22no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—22no—
83C22×Dic53—21no—
84C22×Dic53—22no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53C22⋊C411yescharacteristic
92C22×Dic53C22⋊C411yescharacteristic
93C22×Dic53M1612yes—
94C22×Dic53C2×D411yescharacteristic
95C22×Dic53C2×D411yescharacteristic
96C22×Dic53C2×D412yes—
97C22×Dic53#4112yes—
98C2×#88324—24no—
99C2×#88324—42no—
100C2×#88324—42no—
101C2×#88324—42no—
102C2×#88324—42no—
103C2×#88324—42no—
104C4⋊Dic53—41no—
105C4⋊Dic53—42no—
106C4⋊Dic53—41no—
107#11719919—24no—
108#11719919—24no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—22no—
112C4×Dic53—22no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53C4×D412yes—
121C23×C53C2×M1611yescharacteristic
122C23×C53C2×M1611yescharacteristic
123C23×C53#13112yes—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—22no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—24no—
131C2×C4×C53—44no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—82no—
139C2×C4×C53#10112yes—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—82no—
163C2×Dic53#12011yescharacteristic
164C2×Dic53#12112yes—
165C2×Dic53#12311yescharacteristic
166#88324—44no—
167#88324—82no—
168#9086—531nomaximal, Sylow
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53#38611yescharacteristic
178C22×C53#40711yescharacteristic
179C22×C53#40811yescharacteristic
180C22×C53#41011yescharacteristic
181C22×C53#41011yescharacteristic
182C22×C53C2×#10111yescharacteristic
183C22×C53#41311yescharacteristic
184C22×C53#43512yes—
185C22×C53#43612yes—
186C4×C53—22no—
187C4×C53—42no—
188C4×C53—41no—
189C4×C53—81no—
190Dic53—21no—
191Dic53—22no—
192Dic53—21no—
193Dic53—41no—
194Dic53—41no—
195Dic53—42no—
196Dic53—81no—
197C2×#432—531no—
198C2×#387—531no—
199#1817—531no—
200C2×#324—532no—
201#1280—532no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53#146411yescharacteristic
205C2×C53#146711yescharacteristic
206C2×C53#147011yescharacteristic
207C2×C53#147311yescharacteristic
208C2×C53#187011yescharacteristic
209C2×C53#189911yescharacteristic
210C2×C53#190111yescharacteristic
211C22×C4⋊C4—531no—
212C22×C4⋊C4—531no—
213C22×C42—531no—
214#432—1064no—
215C2×C4⋊C8—1061no—
216C2×C22⋊C8—532no—
217C2×C22⋊C8—532no—
218C2×#97—1061no—
219#387—1061no—
220#387—1061no—
221#387—1061no—
222#387—1061no—
223C2×#95—531no—
224C2×#95—531no—
225C2×#95—531no—
226C2×#95#615711yescharacteristic
227#336—1061no—
228#336—1061no—
229#324—1064no—
230C53#908611yescharacteristic, complemented, Sylow
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—531no—
235C23×C4C2×#615711yescharacteristic
236C23×C4C2×#615711yescharacteristic
237C23×C4C2×#615711yescharacteristic
238C22×C8—1062no—
239C22×C8—1062no—
240C2×C4⋊C4—1062no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—1062no—
253C2×C42—1062no—
254C4⋊C8—1064no—
255C22⋊C8—534no—
256C22⋊C8—1062no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259#97—1064no—
260#95—21no—
261#95—21no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268C24C22×#615711yescharacteristic
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—532no—
283C22×C4—532no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—532no—
287C22×C4—1061no—
288C22×C4—1062no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1061no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4#1171993612yes—
302C22×C4C22⋊#615711yescharacteristic
303C22×C4C22⋊#615711yescharacteristic
304C2×C8—1064no—
305C2×C8—2122no—
306C2×C8—2122no—
307C2×C8—2122no—
308C2×C8—2122no—
309C2×C8—2122no—
310C4⋊C4—1064no—
311C4⋊C4—1064no—
312C4⋊C4—2121no—
313C4⋊C4—2122no—
314C4⋊C4—2121no—
315C42—1061no—
316C42—1061no—
317C42—1062no—
318C42—1061no—
319C42—1062no—
320C42—1061no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C2×C22⋊#615711yescharacteristic
326C23D4×#615712yes—
327C23D53⋊M1611yescharacteristic
328C23C2×#1171993611yescharacteristic
329C23#1313330412yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—531no—
346C2×C4—532no—
347C2×C4—531no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1062no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—4242no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2122no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4#1313327812yes—
372C8—4242no—
373C8—2124no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—22no—
382C22#1681157111yescharacteristic
383C22#1681159411yescharacteristic
384C22#1681159711yescharacteristic
385C22#1681160211yescharacteristic
386C22#1681160411yescharacteristic
387C22C2×#1313327811yescharacteristic
388C22#1681161011yescharacteristic
389C22#1681165212yes—
390C22#1681165612yes—
391C4—22no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—1062no—
396C4—1061no—
397C4—1061no—
398C4—4241no—
399C4—2121no—
400C4—2121no—
401C4—2122no—
402C2—22no—
403C2—22no—
404C2#2028996911yescharacteristic
405C2#2028997711yescharacteristic
406C2#2028998811yescharacteristic
407C2#2028999611yescharacteristic
408C2#2029069311yescharacteristic
409C2#2029073911yescharacteristic
410C2#2029074211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC2×#95×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC2×#95#6157
hypocenterC53#9086
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
230. C53#9086semidirect

Maximal subgroups

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

subgroupindex
C2×#168116442
#202906062
C2×#168114682
#202895532
C2×#168105102
#908653