Finite Group Databasebeta

← #27294303

Subgroups — #27294303

Order 13568.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274454C211yesmaximal, characteristic, complemented
2#20273029C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2619C211yesmaximal, characteristic
5C22×#11719917—21no—
6#16810676—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×#127×C53C411yescharacteristic
12C2×#121×C53—21no—
13C2×#117×C53C2211yescharacteristic
14C53×#394—21no—
15C53×#394C2212yes—
16C53×#392—21no—
17C53×#392C412yes—
18C53×#390—21no—
19C53×#382C411yescharacteristic
20C2×#11719917—24no—
21C2×#11719917—22no—
22C2×#11719917—41no—
23C2×#11719917—42no—
24C22×#88322—41no—
25C4⋊#88322—41no—
26C22⋊#88322—22no—
27C4×#88322—41no—
28C22×D4×C53—22no—
29C22×D4×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53C2×C412yes—
32C23×C4×C53D411yescharacteristic
33#127×C53—41no—
34#127×C53—41no—
35#121×C53—42no—
36#121×C53—42no—
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49#95×C53—21no—
50#95×C53C2×C412yes—
51C24×C53—22no—
52C24×C53C22⋊C411yescharacteristic
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—44no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—42no—
59C2×D4×C53—42no—
60C2×D4×C53—82no—
61C2×D4×C53—82no—
62C2×D4×C53—82no—
63C2×D4×C53—82no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53D812yes—
72C22×C4×C53SD1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—82no—
84C22⋊C4×C53—82no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89#11719917—24no—
90#11719917—44no—
91#11719917—42no—
92C2×#88322—42no—
93C2×#88322—44no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53#10212yes—
107C23×C53C2×C22⋊C411yescharacteristic
108D4×C53—84no—
109D4×C53—84no—
110D4×C53—88no—
111D4×C53—82no—
112D4×C53—82no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#10212yes—
133C2×C4×C53#10412yes—
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136#88322—44no—
137#88322—82no—
138#11699—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—22no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—81no—
146C22×C53—82no—
147C22×C53—82no—
148C22×C53—82no—
149C22×C53—82no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53C2×#9911yescharacteristic
154C22×C53C2×#10211yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—82no—
163C4×C53#32712yes—
164C4×C53#32712yes—
165#2619D5311yescharacteristic
166#1742—531no—
167#1458—531no—
168#1345—534no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53—162no—
173C2×C53C2×#32711yescharacteristic
174C2×C53#212211yescharacteristic
175C2×C53#213411yescharacteristic
176C22×M16—1061no—
177C2×#127Dic5311yescharacteristic
178C2×#121—21no—
179C2×#117C2×D5311yescharacteristic
180#431—1061no—
181#394—21no—
182#394C2×D5312yes—
183#392—21no—
184#392Dic5312yes—
185#390—21no—
186#382Dic5311yescharacteristic
187#343—532no—
188#343—1061no—
189#324—532no—
190#323—1061no—
191C53#1169911yescharacteristic, complemented, Sylow
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C2×Dic5312yes—
196C23×C4C22⋊D5311yescharacteristic
197C2×M16—1064no—
198C2×M16—1062no—
199C2×M16—2121no—
200C2×M16—2122no—
201C22×C8—2121no—
202#127—41no—
203#127—41no—
204#121—42no—
205#121—42no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C4⋊C4C22×D5311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C42C2×Dic5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—2121no—
219C22⋊C8—1062no—
220C4×C8—2121no—
221#95—21no—
222#95C2×Dic5312yes—
223C24—22no—
224C24C22⋊Dic5311yescharacteristic
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4#1171992212yes—
241C22×C4#1171992312yes—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22×Dic5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—1064no—
249M16—2124no—
250M16—2122no—
251C2×C8—2122no—
252C2×C8—2122no—
253C2×C8—2124no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23#1313313812yes—
279C23C2×C22⋊Dic5311yescharacteristic
280D4—84no—
281D4—84no—
282D4—88no—
283D4—82no—
284D4—82no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4#1313313211yescharacteristic
303C2×C4#1313313211yescharacteristic
304C2×C4#1313313812yes—
305C2×C4#1313314412yes—
306C2×C4C2×#1171992211yescharacteristic
307C2×C4C2×#1171992311yescharacteristic
308C8—4242no—
309C8—2124no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22C2×#1313313211yescharacteristic
325C22C2×#1313313811yescharacteristic
326C22C2×#1313314411yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4#1681035712yes—
335C4#1681035712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—162no—
340C2C2×#1681035711yescharacteristic
341C2#2027636511yescharacteristic
342C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2619C2
socleC22×C53C2×#99
cosocleC2×C4⋊C4×C53C23
hypercenter#2619D53
hypocenterC53#11699
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #20274454C2semidirect
2. #20273029C2semidirect
3. #20272718C2semidirect
191. C53#11699semidirect

Maximal subgroups

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

subgroupindex
#202744542
#202730292
#202727182
C53×#26192
#1169953