Finite Group Databasebeta

← #27294908

Subgroups — #27294908

Order 13568.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274711C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272745C212yesmaximal
4#20272721C212yesmaximal
5C53×#2399C211yesmaximal, characteristic
6#16810680—21no—
7#16810647—21no—
8C2×#13133136—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C4×Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398C411yescharacteristic
16C53×#398C2211yescharacteristic
17C53×#395—21no—
18C53×#395C411yescharacteristic
19C53×#395C2211yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—42no—
28C22⋊#88322—22no—
29#13133128—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C4×Q8×C53—41no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C42×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C2×Q8×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53—42no—
62C2×Q8×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C42×C53—24no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—41no—
92C42×C53—41no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C2×#88322—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98Q8×C53—42no—
99Q8×C53—42no—
100Q8×C53—42no—
101Q8×C53—42no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
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×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C53#13711yescharacteristic
133#88322—82no—
134#88322—82no—
135#11831—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53C2×#9911yescharacteristic
139C22×C53#41711yescharacteristic
140C22×C53#42111yescharacteristic
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—42no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—81no—
153C4×C53#35612yes—
154#2399D5311yescharacteristic
155#1801—531no—
156#1467—531no—
157#1354—532no—
158#1346—532no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53#151311yescharacteristic
162C2×C53#212011yescharacteristic
163C2×C53C2×#35611yescharacteristic
164C2×C4×Q8Dic5311yescharacteristic
165C2×C4×Q8C2×D5311yescharacteristic
166#432—1061no—
167#423—1061no—
168C2×#101—1061no—
169#398—21no—
170#398Dic5311yescharacteristic
171#398C2×D5311yescharacteristic
172#395—21no—
173#395Dic5311yescharacteristic
174#395C2×D5311yescharacteristic
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177#376Dic5311yescharacteristic
178#343—532no—
179#324—532no—
180#324—1061no—
181C53#1183111yescharacteristic, complemented, Sylow
182C22×Q8C2×Dic5311yescharacteristic
183C22×Q8C22⋊D5311yescharacteristic
184C2×M16—1062no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×Dic5311yescharacteristic
195C2×C4⋊C4C2×Dic5311yescharacteristic
196C2×C4⋊C4C22⋊D5311yescharacteristic
197C2×C4⋊C4C22×D5311yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×Dic5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205C4⋊C8—2121no—
206#101—2122no—
207C22⋊C8—1062no—
208#97—2121no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×Dic5311yescharacteristic
215#95C2×Dic5311yescharacteristic
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22×Dic5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C23C2×C22⋊Dic5311yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
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#1313313211yescharacteristic
289C2×C4#1313313211yescharacteristic
290C2×C4#1313324211yescharacteristic
291C2×C4#1313324711yescharacteristic
292C8—4242no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22C2×#1313313211yescharacteristic
297C22#1681062411yescharacteristic
298C22#1681064011yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4#1681042212yes—
312C2—21no—
313C2—21no—
314C2#2027325811yescharacteristic
315C2#2027635611yescharacteristic
316C2C2×#1681042211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2399C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2399D53
hypocenterC53#11831
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2399

Complements

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

NH ≅ G/Ntype
181. C53#11831semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730592
#202727452
#202727212
C53×#23992
#1183153