Finite Group Databasebeta

← #27297158

Subgroups — #27297158

Order 13568.

352 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276065C211yesmaximal, characteristic
2#20273756C211yesmaximal, characteristic
3#20273688C211yesmaximal, characteristic
4#20272745C211yesmaximal, characteristic
5#20272742C211yesmaximal, characteristic
6#20272659C211yesmaximal, characteristic
7C53×#2612C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810958—21no—
10#16810779—21no—
11#16810663—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#400C2211yescharacteristic
19C53×#398—21no—
20C53×#395C2211yescharacteristic
21C53×#391—21no—
22C53×#389—21no—
23C53×#387—21no—
24C53×#387—21no—
25C53×#382C2211yescharacteristic
26C53×#376—21no—
27C2×#11719925—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133150—41no—
35#13133150—41no—
36#13133148—41no—
37#13133141—41no—
38#13133140—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×Q8×C53D411yescharacteristic
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53C2311yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53—21no—
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C2×Q8×C53—42no—
63C2×Q8×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—81no—
82C42×C53—21no—
83C42×C53—42no—
84C42×C53—42no—
85C42×C53Q1612yescomplemented
86C22⋊Dic53—41no—
87#11719925—82no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92#11719919—41no—
93#11719919—41no—
94C4×Dic53—41no—
95#11719917—22no—
96#11719917—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C23×C53#12011yescharacteristic
101Q8×C53—42no—
102Q8×C53—81no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53C2×Q1611yescharacteristic
128C2×C4×C53#13711yescharacteristic
129C2×Dic53—41no—
130C2×Dic53—41no—
131Dic106—81no—
132Dic106—81no—
133#88322—42no—
134#88322—81no—
135#88322—81no—
136#12258—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53#45111yescharacteristic
140C22×C53#45611yescharacteristic
141C22×C53B(4,𝔽2)11yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—42no—
149C4×C53—81no—
150C4×C53—81no—
151C4×C53—81no—
152Dic53—81no—
153Dic53—81no—
154#2612D5311yescharacteristic
155#2061—531no—
156#1617—531no—
157#1605—531no—
158#1354—531no—
159#1353—531no—
160#1324—531no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53#160511yescharacteristic
164C2×C53#219711yescharacteristic
165C2×C53#220411yescharacteristic
166#544—531no—
167#484—1061no—
168#451—1061no—
169#428—1061no—
170#400C2×D5311yescharacteristic
171#398—21no—
172#395C2×D5311yescharacteristic
173#391—21no—
174#389—21no—
175#387—21no—
176#387—21no—
177#382C2×D5311yescharacteristic
178#376—21no—
179#343—531no—
180#339—1061no—
181#328—1061no—
182#328—1061no—
183#324—531no—
184#324—531no—
185C53#1225811yescharacteristic, complemented, Sylow
186C22×Q8C22⋊D5311yescharacteristic
187C2×Q16—2121no—
188C2×M16—1061no—
189C4⋊Q8—2121no—
190#125—2121no—
191C22⋊Q8—1061no—
192C22⋊Q8—1061no—
193#117—1061no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D5311yescharacteristic
199C2×C4⋊C4C22⋊D5311yescharacteristic
200C2×C4⋊C4C22×D5311yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊D5311yescharacteristic
204#107—2121no—
205#107—2121no—
206#106—2121no—
207#103—2121no—
208#103—2121no—
209C22⋊C8—1061no—
210C22⋊C8—1061no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C22⋊D5311yescharacteristic
220#95C22⋊D5311yescharacteristic
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—1061no—
224C2×Q8—1061no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233C22×C4C2×C22⋊D5311yescharacteristic
234Q16—4242no—
235M16—1062no—
236M16—2121no—
237C2×C8—2121no—
238C2×C8—2121no—
239C2×C8—2121no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—1061no—
248C4⋊C4—1061no—
249C4⋊C4—1061no—
250C4⋊C4—1061no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C22⋊C4—2121no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42—2121no—
258C42#1171992512yescomplemented
259C23#1313318611yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—4241no—
263Q8—4241no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4C2×#1171992511yescharacteristic
291C2×C4#1313324811yescharacteristic
292C8—4241no—
293C8—4241no—
294C8—2122no—
295C22—21no—
296C22—21no—
297C22#1681077911yescharacteristic
298C22#1681080611yescharacteristic
299C22#1681081011yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—4241no—
311C4—4241no—
312C2—21no—
313C2—21no—
314C2#2027368911yescharacteristic
315C2#2027670511yescharacteristic
316C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2612C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2612D53
hypocenterC53#12258
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2612

Complements

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

NH ≅ G/Ntype
85. C42×C53Q16semidirect
185. C53#12258semidirect
258. C42#11719925semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202737562
#202736882
#202727452
#202727422
#202726592
C53×#26122
#1225853