Finite Group Databasebeta

← #27302389

Subgroups — #27302389

Order 13568.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16810413C214yesmaximal
2#20274707C211yesmaximal, characteristic
3C4×#13133132C211yesmaximal, characteristic
4C53×#2486C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×C22⋊Dic53—21no—
9#16810413—28no—
10#16810392—21no—
11#16810390—21no—
12C2×#123×C53C2212yes—
13C2×C4×D4×C53C2211yescharacteristic
14C22×C4⋊C4×C53C411yescharacteristic, complemented
15C53×#398—21no—
16C53×#397—21no—
17C53×#389—21no—
18C53×#381C412yescomplemented
19C53×#380—21no—
20C53×#380C411yescharacteristic, complemented
21C2×#11719917—22no—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—41no—
24C4⋊#88322—41no—
25C4⋊#88322—41no—
26#13133134—24no—
27#13133132—22no—
28#13133132D414yes—
29#13133124—41no—
30C22×D4×C53C2311yescharacteristic
31C23×C4×C53C2×C412yes—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53D411yescharacteristic
34#123×C53—24no—
35#123×C53—24no—
36C4×D4×C53—41no—
37C4×D4×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—22no—
51#95×C53—21no—
52#95×C53—22no—
53C24×C53C22×C411yescharacteristic
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C2×D412yes—
58C2×D4×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—24no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—44no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—24no—
90C22⋊Dic53—42no—
91C4×Dic53—82no—
92#11719917—42no—
93#11719917—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C4×D412yes—
105C23×C53#12112yes—
106D4×C53—42no—
107D4×C53—42no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—81no—
127C2×C4×C53—84no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53#12711yescharacteristic
132C2×Dic53—44no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135#88322—82no—
136#13220—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53#39011yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C22×C53#41011yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—82no—
159C4×C53—82no—
160C4×C53#35212yescomplemented
161Dic53—84no—
162#2486D5311yescharacteristic
163C2×#352—534no—
164#1800—531no—
165C4×#99—531no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—42no—
169C2×C53—42no—
170C2×C53#196411yescharacteristic
171C2×C53C2×#35211yescharacteristic
172C2×C53#212311yescharacteristic
173C2×#123C2×D5312yes—
174C2×C4×D4C2×D5311yescharacteristic
175C22×C4⋊C4Dic5311yescharacteristic, complemented
176#423—1061no—
177C2×#100—532no—
178C2×#99—532no—
179#398—21no—
180#397—21no—
181#389—21no—
182#381Dic5312yescomplemented
183#380—21no—
184#380Dic5311yescharacteristic, complemented
185C4×C22⋊C4—1061no—
186#352—1068no—
187#343—1061no—
188#342—1061no—
189C53#1322011yescharacteristic, complemented, Sylow
190C22×D4C22×D5311yescharacteristic
191C23×C4C2×Dic5312yes—
192C23×C4C2×Dic5311yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C2×M16—1062no—
195#123—24no—
196#123—24no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4C2×Dic5312yes—
209C2×C22⋊C4C2×Dic5311yescharacteristic
210C2×C22⋊C4C22⋊D5311yescharacteristic
211C2×C42—2121no—
212C2×C42C22⋊D5311yescharacteristic
213C4⋊C8—2121no—
214C4⋊C8—2121no—
215#100—1064no—
216#99—534no—
217#99—1062no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95—2121no—
222C24C22×Dic5311yescharacteristic
223C24C2×C22⋊D5311yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D5312yes—
227C2×D4#1171994712yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—2122no—
240C22×C4C22⋊Dic5311yescharacteristic
241C22×C4C22⋊Dic5311yescharacteristic
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—2122no—
246M16—2122no—
247C2×C8—2122no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—24no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—44no—
259C22⋊C4—1064no—
260C22⋊C4—2122no—
261C42—21no—
262C42—21no—
263C42—4242no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊Dic5311yescharacteristic
273C23D4×Dic5312yes—
274C23#1313318812yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—4242no—
298C2×C4—4242no—
299C2×C4—2124no—
300C2×C4#1313313211yescharacteristic
301C2×C4#1313313211yescharacteristic
302C2×C4#1313320411yescharacteristic
303C2×C4#1313321311yescharacteristic
304C8—4242no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1681052611yescharacteristic
318C22C2×#1313313211yescharacteristic
319C22#1681059711yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—82no—
328C4—4244no—
329C4#1681041312yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2027555211yescharacteristic
335C2C2×#1681041311yescharacteristic
336C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2486C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2486D53
hypocenterC53#13220
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C53C4semidirect
18. C53×#381C4semidirect
20. C53×#380C4semidirect
160. C4×C53#352semidirect
175. C22×C4⋊C4Dic53semidirect
182. #381Dic53semidirect
184. #380Dic53semidirect
189. C53#13220semidirect
329. C4#16810413semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202747072
C4×#131331322
C53×#24862
#1322053