Finite Group Databasebeta

← #27282042

Subgroups — #27282042

Order 13568.

684 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283744C211yesmaximal, characteristic
2#20274639C212yesmaximal, complemented
3#20273053C211yesmaximal, characteristic
4#20272621C212yesmaximal, complemented
5C53×#2409C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133136—24no—
9#16810587C2212yes—
10#16810587C2212yes—
11C2×C22⋊#88322—21no—
12#16810349—21no—
13#16810347—21no—
14C23×Q8×C53C2211yescharacteristic
15C2×C22⋊Q8×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#395—21no—
18C53×#393—21no—
19C53×#391—21no—
20C53×#386C412yes—
21C53×#385C412yes—
22C53×#380C2211yescharacteristic
23C2×#11719917—22no—
24C2×#11719917D414yes—
25C22×#88322—22no—
26C4⋊#88322—22no—
27#13133136—48no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34C22×Q8×C53—21no—
35C22×Q8×C53—22no—
36C22×Q8×C53—21no—
37C22×Q8×C53—22no—
38C22×Q8×C53—44no—
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53C2×C412yes—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C22⋊Q8×C53—44no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53C2×C412yes—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—22no—
55#95×C53—22no—
56#95×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22×C411yescharacteristic
59C2×Q8×C53—24no—
60C2×Q8×C53—28no—
61C2×Q8×C53—42no—
62C2×Q8×C53—44no—
63C2×Q8×C53—48no—
64C2×Q8×C53—44no—
65C2×Q8×C53—41no—
66C2×Q8×C53—42no—
67C2×Q8×C53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—41no—
71C22×C4×C53—22no—
72C22×C4×C53—24no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—44no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53#4112yes—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—44no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—42no—
93C42×C53—21no—
94C42×C53—21no—
95#11719917—44no—
96#11719917—44no—
97C2×#88322—24no—
98C2×#88322—42no—
99C2×#88322—82no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53#10112yescomplemented
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53C4×D412yes—
111Q8×C53—48no—
112Q8×C53—44no—
113Q8×C53—44no—
114D4×C53—44no—
115C2×C4×C53—22no—
116C2×C4×C53—24no—
117C2×C4×C53—28no—
118C2×C4×C53—24no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—44no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—81no—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12112yes—
138C2×C4×C53#12311yescharacteristic
139#88322—44no—
140#88322—82no—
141#9505—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—81no—
154C22×C53#38611yescharacteristic
155C22×C53C2×#10111yescharacteristic
156C22×C53#41311yescharacteristic
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—44no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—82no—
164C4×C53B(4,𝔽2)12yes—
165#2973—531no—
166#2409D5311yescharacteristic
167#1784—532no—
168#1465—531no—
169#1305—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53—42no—
174C2×C53#188911yescharacteristic
175C2×C53#190111yescharacteristic
176C2×C53#191511yescharacteristic
177C23×Q8C2×D5311yescharacteristic
178C2×C22⋊Q8—21no—
179C2×C4×D4C2×D5311yescharacteristic
180#435—1064no—
181#432—1061no—
182C2×#101—1064no—
183#407—532no—
184#407—532no—
185C2×C22⋊C8—1061no—
186#395—21no—
187#393—21no—
188#391—21no—
189#386Dic5312yes—
190#385Dic5312yes—
191#380C2×D5311yescharacteristic
192#324—1061no—
193#323—1061no—
194C53#950511yescharacteristic, complemented, Sylow
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—44no—
200C22×D4C2×Dic5311yescharacteristic
201C23×C4C2×Dic5312yes—
202C23×C4C22⋊D5311yescharacteristic
203C23×C4C22×D5311yescharacteristic
204C2×M16—534no—
205C2×M16—1062no—
206C22×C8—1062no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4C2×Dic5312yes—
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—1062no—
219#101—2128no—
220C22⋊C8—1062no—
221C22⋊C8—1062no—
222C22⋊C8—1062no—
223C22⋊C8—1062no—
224C22⋊C8—1062no—
225#97—1062no—
226#95—21no—
227#95—22no—
228#95—22no—
229C24C22⋊Dic5311yescharacteristic
230C24C22×Dic5311yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22⋊Dic5311yescharacteristic
254C22×C4D4×D5312yes—
255C22×C4#1171994712yes—
256C22×C4C2×C22⋊D5311yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258M16—2124no—
259M16—2124no—
260C2×C8—1064no—
261C2×C8—4242no—
262C2×C8—2122no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1313313612yescomplemented
281C23C2×C22⋊Dic5311yescharacteristic
282C23D4×Dic5312yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4#1313318511yescharacteristic
309C2×C4#1313318912yes—
310C2×C4#1313319811yescharacteristic
311C8—4242no—
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1681050311yescharacteristic
326C22C2×#1313313611yescharacteristic
327C22#1681060711yescharacteristic
328C4—22no—
329C4—21no—
330C4—41no—
331C4—44no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1681080912yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#2027512411yescharacteristic
341C2#2027518411yescharacteristic
342C2#2027526811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2409C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2409D53
hypocenterC53#9505
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #20274639C2semidirect
4. #20272621C2semidirect
108. C23×C53#101semidirect
194. C53#9505semidirect
280. C23#13133136semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202746392
#202730532
#202726212
C53×#24092
#950553