Finite Group Databasebeta

← #27294732

Subgroups — #27294732

Order 13568.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280077C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic
3#20272746C212yesmaximal
4#20272743C212yesmaximal
5C53×#1467C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810562C411yescharacteristic
10#16810562C2211yescharacteristic
11#16810543C411yescharacteristic
12#16810543C2211yescharacteristic
13#16810530—21no—
14#16810519—21no—
15#16810507—21no—
16#16810507—21no—
17#16810489C411yescharacteristic
18#16810392C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—22no—
25C4×Dic106—41no—
26C4×Dic106—41no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C4×Dic53—21no—
33#13133136—24no—
34#13133128—41no—
35C4×#88322—41no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×Q8×C53D411yescharacteristic
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56#97×C53—41no—
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106—41no—
72C2×Dic106—42no—
73C2×C8×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C4⋊Dic53—42no—
84C4⋊Dic53—41no—
85#11719919—42no—
86#11719919—42no—
87#11719919—41no—
88#11719919—41no—
89C4×Dic53—42no—
90#11719917—24no—
91#11719917—42no—
92C2×#88322—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—44no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#10412yes—
112C2×C4×C53#13712yes—
113C2×Dic53—22no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124Dic106—42no—
125Dic106—42no—
126C8×C53—82no—
127#88322—44no—
128#11792—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41911yescharacteristic
133C22×C53C2×#10411yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142Dic53—42no—
143Dic53—81no—
144Dic53—81no—
145Dic53—81no—
146#2581—531no—
147C4×#101—531no—
148#1467D5311yescharacteristic
149#1354—532no—
150#1353—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#150811yescharacteristic
154C2×C53#212011yescharacteristic
155C2×C53#213311yescharacteristic
156C2×C4×Q8—531no—
157C2×C4×Q8C2×D5311yescharacteristic
158#432—21no—
159C2×#101—1061no—
160C4×M16—1061no—
161#400—531no—
162#400—531no—
163#395—531no—
164#395—531no—
165#391—1061no—
166#389—1061no—
167#387—1061no—
168#387—1061no—
169#382—531no—
170#343—532no—
171#324—21no—
172#324C2×D5312yes—
173C53#1179211yescharacteristic, complemented, Sylow
174C22×Q8—531no—
175C22×Q8D21211yescharacteristic
176C2×M16—1062no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—2121no—
180C4×Q8—2121no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4C4×D5311yescharacteristic
187C2×C4⋊C4C22⋊D5311yescharacteristic
188C2×C4⋊C4C22×D5311yescharacteristic
189C2×C42—1061no—
190C2×C42C4×D5311yescharacteristic
191C2×C42D21211yescharacteristic
192C2×C42C22⋊D5311yescharacteristic
193C4⋊C8—41no—
194#101—1064no—
195C22⋊C8—22no—
196#97—41no—
197#97—2121no—
198C4×C8—2121no—
199#95—531no—
200#95—531no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—2121no—
212C2×Q8—2122no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4#1171992111yescharacteristic
218C22×C4#1171992111yescharacteristic
219C22×C4#1171992111yescharacteristic
220C22×C4#1171992111yescharacteristic
221C22×C4C2×C4×D5311yescharacteristic
222C22×C4C2×D21211yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224M16—1064no—
225M16—2122no—
226C2×C8—42no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—2121no—
232C4⋊C4—2122no—
233C4⋊C4—2122no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C4⋊C4—2122no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—2122no—
244C23C2×#1171992111yescharacteristic
245Q8—44no—
246Q8—2122no—
247Q8—2122no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—1062no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2122no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4#1313313111yescharacteristic
274C2×C4#1313313111yescharacteristic
275C2×C4#1313314312yes—
276C2×C4#1313325012yes—
277C8—82no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22C2×#1313313111yescharacteristic
282C22#1681063311yescharacteristic
283C22C2×#1313314311yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—4241no—
293C4—4241no—
294C4—4241no—
295C4—2122no—
296C2—21no—
297C2—21no—
298C2#2027323611yescharacteristic
299C2#2027635411yescharacteristic
300C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11792
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
173. C53#11792semidirect

Maximal subgroups

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

subgroupindex
#202800772
C4×#131331362
#202727462
#202727432
C53×#14672
#1179253