Finite Group Databasebeta

← #28306257

Subgroups — #28306257

Order 13568.

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

normal subgroups only

This group has 556 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#20290571C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289553C212yesmaximal
4#20289553C212yesmaximal
5C2×#16810509C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324C2212yes—
8C2×C22⋊#88324C2212yes—
9C2×#13133272—21no—
10#16811489—21no—
11#16811489—21no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810509—21no—
15#16810509—21no—
16#16810509—21no—
17#16810509—21no—
18C2×#13133124C411yescharacteristic
19C2×#13133124C411yescharacteristic
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2211yescharacteristic
22C22×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22⋊#88324—22no—
27C22⋊#88324—24no—
28C22⋊#88324Q814yes—
29#13133272—42no—
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2311yescharacteristic
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—42no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C24×C53C22×C411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53M1612yes—
73C22×C4×C53M1612yes—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—22no—
79C22×Dic53—22no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C22⋊C411yescharacteristic
89C22×Dic53C2×D411yescharacteristic
90C22×Dic53C2×D411yescharacteristic
91C22×Dic53C2×Q812yes—
92C22×Dic53#4112yes—
93C2×#88324—24no—
94C2×#88324—42no—
95C2×#88324—42no—
96C2×#88324—42no—
97C2×#88324—42no—
98C2×#88324—42no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—24no—
101C4⋊Dic53—42no—
102C4⋊Dic53—42no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—41no—
106C4×Dic53—41no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53C4×Q812yes—
113C23×C53C2×M1611yescharacteristic
114C23×C53C2×M1611yescharacteristic
115C23×C53#13112yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—44no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C2×Dic53#9912yes—
155C2×Dic53C22⋊Q812yes—
156C2×Dic53#12411yescharacteristic
157C2×Dic53C4⋊Q811yescharacteristic
158#88324—44no—
159#88324—82no—
160#9881—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53#39111yescharacteristic
170C22×C53#40711yescharacteristic
171C22×C53#40811yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53#41011yescharacteristic
174C22×C53#41311yescharacteristic
175C22×C53#41311yescharacteristic
176C22×C53#44612yes—
177C22×C53#44612yes—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—42no—
182Dic53—22no—
183Dic53—22no—
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—81no—
188Dic53—81no—
189C2×#387—531no—
190#1797—531no—
191#1749—531no—
192#1280—532no—
193#1280—532no—
194C2×C53—22no—
195C2×C53—22no—
196C2×C53#146411yescharacteristic
197C2×C53#146511yescharacteristic
198C2×C53#146811yescharacteristic
199C2×C53#147311yescharacteristic
200C2×C53#198511yescharacteristic
201C2×C53#199611yescharacteristic
202C2×C53#199811yescharacteristic
203C22×C4⋊C4—531no—
204C22×C4⋊C4#615711yescharacteristic
205C22×C42—531no—
206C2×C4⋊C8—1061no—
207C2×C22⋊C8—532no—
208C2×C22⋊C8—532no—
209C2×#97—1061no—
210#387—1061no—
211#387—1061no—
212#387—1061no—
213#387—1061no—
214C2×#95—531no—
215C2×#95—531no—
216C2×#95—531no—
217C2×#95—531no—
218#336—1061no—
219#336—1061no—
220C53#988111yescharacteristic, complemented, Sylow
221C23×C4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C23×C4—531no—
225C23×C4C2×#615711yescharacteristic
226C23×C4C2×#615711yescharacteristic
227C23×C4C2×#615711yescharacteristic
228C22×C8—1062no—
229C22×C8—1062no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—2122no—
237C2×C42—531no—
238C2×C42—531no—
239C2×C42—531no—
240C2×C42—531no—
241C2×C42—2122no—
242C4⋊C8—2122no—
243C22⋊C8—534no—
244C22⋊C8—1062no—
245C22⋊C8—1064no—
246#97—2122no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255C24C22×#615711yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—532no—
272C22×C4—532no—
273C22×C4—531no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4#1171994012yes—
288C22×C4#1171994012yes—
289C22×C4C22⋊#615711yescharacteristic
290C22×C4C22⋊#615711yescharacteristic
291C2×C8—1064no—
292C2×C8—2122no—
293C2×C8—2122no—
294C2×C8—2122no—
295C2×C8—2122no—
296C2×C8—2122no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—2122no—
300C4⋊C4—2122no—
301C42—1062no—
302C42—1062no—
303C42—2121no—
304C42—2121no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C22⋊#615711yescharacteristic
310C23Q8×#615712yes—
311C23C2×#1171994011yescharacteristic
312C23C2×#1171994011yescharacteristic
313C23#1313330512yes—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—532no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—532no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4242no—
346C2×C4—4242no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2124no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C8—4242no—
357C8—2124no—
358C22—22no—
359C22—22no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22(C2×#6157)⋊Q811yescharacteristic
367C22#1681159311yescharacteristic
368C22#1681159811yescharacteristic
369C22D53⋊#9911yescharacteristic
370C22#1681160211yescharacteristic
371C22#1681160911yescharacteristic
372C22#1681161011yescharacteristic
373C22#1681167412yes—
374C22#1681167412yes—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—42no—
379C4—1061no—
380C4—1062no—
381C4—1061no—
382C4—1062no—
383C4—4241no—
384C4—4241no—
385C4—2121no—
386C2—22no—
387C2—22no—
388C2#2028996611yescharacteristic
389C2#2028997111yescharacteristic
390C2#2028998311yescharacteristic
391C2#2028999811yescharacteristic
392C2#2029088911yescharacteristic
393C2#2029090611yescharacteristic
394C2#2029090911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#9881
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
220. C53#9881semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
#202895532
C2×#168105092
#988153