Finite Group Databasebeta

← #27273115

Subgroups — #27273115

Order 13568.

682 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 95 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#20276243C211yesmaximal, characteristic, complemented
2#20276118C212yesmaximal, complemented
3#20272536C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic, complemented
5C53×#2679C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133138—21no—
8C2×#13133138—22no—
9#16810493C2211yescharacteristic, complemented
10#16810385—22no—
11#16810378C2211yescharacteristic, complemented
12#16810378C2212yescomplemented
13C2×#120×C53—22no—
14C2×#120×C53—21no—
15C53×#394—22no—
16C53×#394—21no—
17C53×#390C2211yescharacteristic
18C53×#390C2212yes—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—21no—
23#13133150—42no—
24#13133138—42no—
25#13133138—44no—
26C25×C53—21no—
27C22×D4×C53—21no—
28C22×D4×C53—22no—
29C22×D4×C53D412yes—
30C22×D4×C53D411yescharacteristic
31#120×C53—44no—
32#120×C53—42no—
33#120×C53—44no—
34#120×C53—42no—
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#95×C53—21no—
44C24×C53—22no—
45C24×C53—21no—
46C24×C53—44no—
47C24×C53—42no—
48C24×C53—81no—
49C24×C53—82no—
50C24×C53—81no—
51C2×D4×C53—22no—
52C2×D4×C53—21no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—44no—
58C2×D4×C53—44no—
59C2×D4×C53—44no—
60C2×D4×C53—81no—
61C2×D4×C53—82no—
62C2×D4×C53D814yes—
63C2×D4×C53SD1612yes—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—44no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C42×C53—41no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—42no—
83C4⋊Dic53—42no—
84C2×#88322—22no—
85C2×#88322—41no—
86C2×#88322—41no—
87C2×#88322—42no—
88C2×#88322—42no—
89C2×#88322—42no—
90C23×C53—24no—
91C23×C53—22no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—41no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—41no—
102C23×C53—84no—
103C23×C53—88no—
104C23×C53—88no—
105C23×C53—84no—
106C23×C53—84no—
107C23×C53#12011yescharacteristic
108D4×C53—44no—
109D4×C53—42no—
110D4×C53—82no—
111D4×C53—81no—
112D4×C53—84no—
113D4×C53—88no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53C2×D812yes—
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)12yes—
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×Dic53—82no—
128C2×Dic53—81no—
129#88322—42no—
130#88322—82no—
131#7786—531nomaximal, Sylow
132C22×C53—44no—
133C22×C53—42no—
134C22×C53—44no—
135C22×C53—44no—
136C22×C53—44no—
137C22×C53—816no—
138C22×C53—82no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53—81no—
142C22×C53—82no—
143C22×C53—81no—
144C22×C53—82no—
145C22×C53#44712yes—
146C22×C53#45011yescharacteristic
147C22×C53#45311yescharacteristic
148C22×C53#45512yes—
149C22×C53#45811yescharacteristic
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—81no—
155C4×C53—82no—
156Dic53—161no—
157#2679D5311yescharacteristic
158#2093—531no—
159#2071—532no—
160#1270—531no—
161#1270—532no—
162C2×C53—84no—
163C2×C53—82no—
164C2×C53#160712yes—
165C2×C53#161311yescharacteristic
166C2×C53#161911yescharacteristic
167C2×C53#163912yes—
168C2×C53#165911yescharacteristic
169C2×#120—22no—
170C2×#120—21no—
171C2×#107—1061no—
172C2×#102—1062no—
173C2×#102—1061no—
174#394—22no—
175#394—21no—
176#390C2×D5312yes—
177#390C2×D5311yescharacteristic
178#384—531no—
179#340—1062no—
180#336—532no—
181#336—531no—
182C53#778611yescharacteristic, complemented, Sylow
183C25—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4C22⋊D5312yes—
187C22×D4C22⋊D5311yescharacteristic
188C22×C8—1061no—
189C22×C8—1062no—
190#120—44no—
191#120—42no—
192#120—44no—
193#120—42no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4C22⋊D5312yes—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42C22×D5311yescharacteristic
204#107—2122no—
205#102—2122no—
206#102—2124no—
207#95—21no—
208C24—22no—
209C24—21no—
210C24—44no—
211C24—42no—
212C24—81no—
213C24—82no—
214C24—81no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—81no—
225C2×D4—82no—
226C2×D4#1171992214yes—
227C2×D4#1171992312yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—2121no—
231C22×C4C2×C22⋊D5312yes—
232C22×C4C2×C22⋊D5311yescharacteristic
233C2×C8—1062no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2121no—
238C2×C8—2121no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—84no—
267C23—88no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23#1313318611yescharacteristic
272D4—44no—
273D4—42no—
274D4—82no—
275D4—81no—
276D4—84no—
277D4—88no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—4242no—
288C2×C4—4241no—
289C2×C4C2×#1171992212yes—
290C2×C4C2×#1171992311yescharacteristic
291C2×C4#1313324312yes—
292C2×C4#1313324311yescharacteristic
293C8—4242no—
294C8—2122no—
295C22—44no—
296C22—42no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—816no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22—81no—
307C22—82no—
308C22#1681075712yes—
309C22#1681077411yescharacteristic
310C22#1681079111yescharacteristic
311C22#1681080112yes—
312C22#1681081311yescharacteristic
313C4—22no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—82no—
319C4—8481no—
320C2—84no—
321C2—82no—
322C2#2027370112yes—
323C2#2027373511yescharacteristic
324C2#2027376911yescharacteristic
325C2#2027389412yes—
326C2#2027402111yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2679C2
socleC23×C53#120
hypercenter#2679D53
hypocenterC53#7786
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. #20276243C2semidirect
2. #20276118C2semidirect
3. #20272536C2semidirect
4. #20272536C2semidirect
9. #16810493C22semidirect
11. #16810378C22semidirect
12. #16810378C22semidirect
182. C53#7786semidirect

Maximal subgroups

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

subgroupindex
#202762432
#202761182
#202725362
C53×#26792
#778653