Finite Group Databasebeta

← #27300342

Subgroups — #27300342

Order 13568.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274643C211yesmaximal, characteristic
2#20274633C211yesmaximal, characteristic
3#20272833C214yesmaximal, complemented
4C53×#2311C211yesmaximal, characteristic
5C2×#13133136—21no—
6C2×#13133134C2212yes—
7C2×#13133132—21no—
8C2×#13133132C2212yes—
9#16810587—21no—
10#16810486—21no—
11C22×#41×C53C2211yescharacteristic
12C2×#117×C53C411yescharacteristic, complemented
13C22×C4⋊C4×C53C411yescharacteristic, complemented
14C53×#391—21no—
15C53×#390—21no—
16C53×#386C412yescomplemented
17C53×#386C2212yes—
18C53×#380—21no—
19C53×#380—21no—
20C2×#11719917—22no—
21C2×C22⋊Dic53—22no—
22#13133136—42no—
23#13133134—24no—
24#13133132—24no—
25#13133132—42no—
26C22⋊#88322—42no—
27#13133124—42no—
28C2×#41×C53—24no—
29C2×#41×C53—42no—
30C2×#41×C53—42no—
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#117×C53—42no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—42no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C42×C53—21no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22×C411yescharacteristic
52C24×C53C2×D411yescharacteristic
53#41×C53—416no—
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53#4114yes—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—42no—
78C22×C4×C53—82no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—82no—
89C22⋊C4×C53—82no—
90C42×C53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#11712yes—
111C23×C53#12312yes—
112Q8×C53—44no—
113D4×C53—44no—
114D4×C53—44no—
115D4×C53—44no—
116C2×C4×C53—28no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—44no—
127C2×C4×C53—44no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—84no—
135C2×C4×C53#9912yes—
136C2×C4×C53#9912yes—
137C2×C4×C53#12312yes—
138C2×Dic53—82no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141#88322—82no—
142#12855—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#38111yescharacteristic
161C22×C53C2×#9911yescharacteristic
162C22×C53C2×#9911yescharacteristic
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—41no—
167C4×C53—82no—
168C4×C53—82no—
169Dic53—162no—
170#2311D5311yescharacteristic
171#1785—531no—
172#1782—531no—
173#1390—534no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—42no—
177C2×C53—42no—
178C2×C53—81no—
179C2×C53#181911yescharacteristic
180C2×C53#212211yescharacteristic
181C2×C53#212311yescharacteristic
182C22×#41C2×D5311yescharacteristic
183C2×#117Dic5311yescharacteristic, complemented
184C22×C4⋊C4Dic5311yescharacteristic, complemented
185C2×#101—1061no—
186C2×#100—532no—
187C2×#99—532no—
188C2×#99—1061no—
189#407—1061no—
190#391—21no—
191#390—21no—
192#386Dic5312yescomplemented
193#386C2×D5312yes—
194#381—1061no—
195#380—21no—
196#380—21no—
197C53#1285511yescharacteristic, complemented, Sylow
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8C22⋊D5311yescharacteristic
202C22×D4C2×Dic5311yescharacteristic
203C22×D4C22⋊D5311yescharacteristic
204C22×D4C22×D5311yescharacteristic
205C23×C4C2×Dic5311yescharacteristic
206C23×C4C2×Dic5312yes—
207C23×C4C22⋊D5311yescharacteristic
208C23×C4C22⋊D5311yescharacteristic
209C2×M16—1062no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C2×Dic5312yes—
219C2×C42—21no—
220#101—2122no—
221#100—1064no—
222#99—1064no—
223#99—2122no—
224C22⋊C8—2122no—
225#95—22no—
226#95—22no—
227#95—2122no—
228C24C22⋊Dic5311yescharacteristic
229C24C22×Dic5311yescharacteristic
230C24C2×C22⋊D5311yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1171994514yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—2122no—
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262M16—2122no—
263M16—2122no—
264C2×C8—2122no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊Dic5311yescharacteristic
288C23#1313317612yes—
289C23#1313320012yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—4242no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4#1313313212yes—
317C2×C4#1313313212yes—
318C2×C4#1313320012yes—
319C8—4242no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1681048611yescharacteristic
338C22C2×#1313313211yescharacteristic
339C22C2×#1313313211yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—8482no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#2027478911yescharacteristic
353C2#2027636511yescharacteristic
354C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2311C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#2311D53
hypocenterC53#12855
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #20272833C2semidirect
12. C2×#117×C53C4semidirect
13. C22×C4⋊C4×C53C4semidirect
16. C53×#386C4semidirect
183. C2×#117Dic53semidirect
184. C22×C4⋊C4Dic53semidirect
192. #386Dic53semidirect
197. C53#12855semidirect

Maximal subgroups

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

subgroupindex
#202746432
#202746332
#202728332
C53×#23112
#1285553