Finite Group Databasebeta

← #27280171

Subgroups — #27280171

Order 13568.

578 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 119 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#20277318C211yesmaximal, characteristic, complemented
2#20273283C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic, complemented
4#20272582C212yesmaximal, complemented
5C53×#2618C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810509—24no—
8#16810486—21no—
9#16810483C2212yes—
10C4×C22⋊Dic53—21no—
11#16810357—24no—
12#16810349—22no—
13C2×#123×C53—21no—
14C2×#121×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#398C412yescomplemented
17C53×#396—22no—
18C53×#394C412yescomplemented
19C53×#386—21no—
20C53×#380C2211yescharacteristic
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—22no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—24no—
29#13133124D414yes—
30C22×D4×C53—21no—
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35#123×C53—44no—
36#121×C53—22no—
37#121×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—21no—
49#95×C53—22no—
50#95×C53C2×C412yes—
51C24×C53—21no—
52C24×C53C2×D411yescharacteristic
53C2×D4×C53—22no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—81no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—44no—
80C4⋊C4×C53—81no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—82no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—82no—
88C4×Dic53—82no—
89C2×#88322—42no—
90C2×#88322—82no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53#12011yescharacteristic
100C23×C53#12311yescharacteristic
101C23×C53#13512yes—
102C23×C53AGL(1,ℤ/8)12yes—
103D4×C53—82no—
104D4×C53—82no—
105D4×C53—82no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—81no—
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12112yes—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—44no—
130#88322—82no—
131#9206—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—82no—
141C22×C53—82no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53#38611yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53#41811yescharacteristic
148C22×C53#44912yes—
149C22×C53#48212yes—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—42no—
153C4×C53—42no—
154C4×C53—82no—
155C4×C53—81no—
156Dic53—84no—
157#2618D5311yescharacteristic
158#2313—531no—
159#1519—532no—
160C2×#324—531no—
161#1289—532no—
162C2×C53—22no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—22no—
166C2×C53—161no—
167C2×C53#187511yescharacteristic
168C2×C53#191211yescharacteristic
169C2×C53#191311yescharacteristic
170C2×#123—21no—
171C2×#121C2×D5311yescharacteristic
172C22×C4⋊C4C2×D5311yescharacteristic
173C2×C22⋊C8—532no—
174#398Dic5312yescomplemented
175#396—22no—
176#394Dic5312yescomplemented
177#387—1064no—
178#386—21no—
179#381—1061no—
180#380—532no—
181#380C2×D5311yescharacteristic
182C4×C22⋊C4—1061no—
183#327—1064no—
184#324—1062no—
185C53#920611yescharacteristic, complemented, Sylow
186C22×D4—21no—
187C22×D4C2×Dic5311yescharacteristic
188C23×C4C22⋊D5311yescharacteristic
189C23×C4C22⋊D5311yescharacteristic
190C23×C4C22×D5311yescharacteristic
191C22×C8—1062no—
192#123—44no—
193#121—22no—
194#121—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4C2×Dic5311yescharacteristic
201C2×C4⋊C4C2×Dic5312yes—
202C2×C4⋊C4C22⋊D5312yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1062no—
208C2×C42—1062no—
209C22⋊C8—1062no—
210C22⋊C8—1062no—
211#95—22no—
212#95—534no—
213#95—1062no—
214#95—1064no—
215#95C2×Dic5312yes—
216C24—21no—
217C24C2×C22⋊D5311yescharacteristic
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—81no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1064no—
235C22×C4C22⋊Dic5312yes—
236C22×C4C22⋊Dic5312yes—
237C22×C4D4×D5312yes—
238C22×C4#1171994712yes—
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C2×C8—4242no—
242C2×C8—2122no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—44no—
247C4⋊C4—81no—
248C4⋊C4—4242no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—2122no—
254C22⋊C4—2122no—
255C42—4242no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic5311yescharacteristic
264C23#1313318511yescharacteristic
265C23#1313319811yescharacteristic
266C23#1313323712yes—
267C23#1313324312yes—
268D4—82no—
269D4—82no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—2124no—
291C2×C4—2124no—
292C2×C4—2124no—
293C2×C4D4×Dic5312yes—
294C2×C4#1313318912yes—
295C8—4242no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#1681050311yescharacteristic
310C22#1681061911yescharacteristic
311C22#1681062811yescharacteristic
312C22#1681076612yes—
313C22#1681094412yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—81no—
320C4—4244no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—161no—
326C2#2027503811yescharacteristic
327C2#2027525311yescharacteristic
328C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2618C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2618D53
hypocenterC53#9206
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. #20277318C2semidirect
3. C2×#16810349C2semidirect
4. #20272582C2semidirect
16. C53×#398C4semidirect
18. C53×#394C4semidirect
174. #398Dic53semidirect
176. #394Dic53semidirect
185. C53#9206semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202732832
C2×#168103492
#202725822
C53×#26182
#920653