Finite Group Databasebeta

← #28305689

Subgroups — #28305689

Order 13568.

774 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 166 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#20290648C211yesmaximal, characteristic, complemented
2C42×#88324C211yesmaximal, characteristic, complemented
3C2×#16811471C212yesmaximal, complemented
4C2×D4⋊#88324C212yesmaximal, complemented
5#20277279C211yesmaximal, characteristic
6C2×#13133284—22no—
7C2×C4⋊#88324C2212yes—
8C2×C4×#88324—22no—
9C2×C4×#88324C2212yes—
10#16811471—24no—
11D4⋊#88324—24no—
12C2×#13133213C411yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15C2×#13133177—22no—
16#16810525C411yescharacteristic
17#16810523—22no—
18#16810492C411yescharacteristic
19C42×Dic53C2211yescharacteristic
20C53×#390C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23#13133284—44no—
24C4⋊#88324—22no—
25C4⋊#88324—22no—
26C4×#88324—28no—
27C4×#88324—22no—
28C4×#88324D414yescomplemented
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—22no—
31C23×Dic53—21no—
32C22×C4×D53—21no—
33#13133213—22no—
34#13133213C814yescomplemented
35D4×Dic53—22no—
36D4×Dic53—22no—
37#13133179—22no—
38#13133179—22no—
39#13133177—44no—
40#13133177—44no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919—22no—
43C2×C22⋊Dic53—21no—
44C2×#11719921—22no—
45C2×#11719921—21no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D412yes—
48C2×C4×Dic53D412yes—
49C2×C4×Dic53C2311yescharacteristic
50C22×D4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—22no—
53C2×C42×C53C2×C411yescharacteristic
54C24×C53—21no—
55C23×D53—21no—
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53—22no—
59C2×D4×C53C2×C812yes—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22×C411yescharacteristic
63C2×C22⋊D53—28no—
64C2×C22⋊D53—44no—
65C2×C22⋊D53—44no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53C22⋊C412yes—
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—21no—
77C2×D212C2×C812yes—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53—42no—
81C2×C4×D53—42no—
82C2×#88324—28no—
83C2×#88324—24no—
84C2×#88324—42no—
85C2×#88324—44no—
86C4⋊C4×C53—22no—
87C22⋊C4×C53—44no—
88C42×C53—22no—
89C22⋊Dic53—42no—
90#11719921—44no—
91#11719921—42no—
92C4⋊Dic53—22no—
93#11719919—44no—
94C4×Dic53—24no—
95C4×Dic53C2×C812yes—
96C4×Dic53M1612yes—
97C4×Dic53D814yes—
98C4×Dic53SD1614yes—
99C4×Dic53C2×D412yes—
100C4×Dic53#4112yes—
101C23×C53—22no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53C2×C22⋊C411yescharacteristic
107C22×D53—22no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—42no—
111C22×D53—42no—
112D4×C53—24no—
113D4×C53—42no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53#10214yes—
119C2×C4×C53C4×D412yes—
120C2×C4×C53C22×C811yescharacteristic
121C2×C4×C53C2×M1611yescharacteristic
122C2×C4×C53#13112yes—
123C22⋊D53—416no—
124C2×Dic53—22no—
125C2×Dic53—41no—
126C2×Dic53—42no—
127C2×Dic53—82no—
128C2×Dic53—81no—
129C2×Dic53—81no—
130C2×Dic53C22⋊C814yes—
131C2×Dic53#10214yes—
132C2×Dic53#10414yes—
133C2×Dic53#12112yes—
134C2×Dic53#12411yescharacteristic
135C2×Dic53#12711yescharacteristic
136C2×Dic53C2×D812yes—
137C2×Dic53C2×SD1612yes—
138D212—24no—
139D212—42no—
140C4×D53—82no—
141C4×D53—81no—
142C4×D53—81no—
143#88324—28no—
144#88324—82no—
145#9638—531nomaximal, Sylow
146C22×C53—42no—
147C22×C53—44no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—82no—
151C22×C53#39011yescharacteristic
152C22×C53C2×C22⋊C811yescharacteristic
153C22×C53#40811yescharacteristic
154C22×C53C2×#10212yes—
155C22×C53C2×#10412yes—
156C2×D53—42no—
157C2×D53—44no—
158C2×D53—81no—
159C2×D53—81no—
160C2×D53—82no—
161C4×C53—24no—
162C4×C53—81no—
163C4×C53C8×D412yes—
164C4×C53#43612yes—
165Dic53—22no—
166Dic53—81no—
167Dic53D4⋊C818yescomplemented
168Dic53#48612yes—
169Dic53#49211yescharacteristic
170Dic53#49311yescharacteristic
171C4×#127—531no—
172#1842—531no—
173C42×C8—531no—
174C2×D4⋊C8—532no—
175C2×D4⋊C8—532no—
176C2×C53—82no—
177C2×C53C2×D4⋊C812yes—
178C2×C53#149212yes—
179C2×C53#195411yescharacteristic
180C2×C53#195711yescharacteristic
181C2×C53#196211yescharacteristic
182D53—82no—
183C2×#127—531no—
184C2×C4×D4—531no—
185C2×C4×D4—531no—
186C2×C4×D4—1062no—
187C2×C4⋊C8—532no—
188C2×C4⋊C8—1062no—
189C2×C4×C8—532no—
190C2×C4×C8—1062no—
191#390—531no—
192#390—1062no—
193#390#615711yescharacteristic
194#384—531no—
195C43—531no—
196D4⋊C8—1064no—
197D4⋊C8—1064no—
198C53#963811yescharacteristic, complemented, Sylow
199C22×D4—531no—
200C22×D4—1062no—
201C22×D4C2×#615711yescharacteristic
202C23×C4—1061no—
203C23×C4—1061no—
204C22×C8—1062no—
205C22×C8—1062no—
206#127—534no—
207#127—1062no—
208C4×D4—1062no—
209C4×D4—1062no—
210C4×D4—1062no—
211C4×D4—1062no—
212C4×D4—2124no—
213C4×D4—2124no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4C2×#615711yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1062no—
221C2×C42—532no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—532no—
225C2×C42C2×#615711yescharacteristic
226C4⋊C8—1062no—
227C4⋊C8—1062no—
228C4⋊C8—2124no—
229C4×C8—534no—
230C4×C8—1062no—
231C4×C8—1068no—
232C24—21no—
233C24—1061no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—22no—
237C2×D4—532no—
238C2×D4—1068no—
239C2×D4—1062no—
240C2×D4—1061no—
241C2×D4—1062no—
242C2×D4—2124no—
243C2×D4—2124no—
244C2×D4D53⋊C812yes—
245C22×C4—21no—
246C22×C4—532no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4—2122no—
253C22×C4—2122no—
254C22×C4—2121no—
255C22×C4—2121no—
256C22×C4—2122no—
257C22×C4—2122no—
258C22×C4C22⋊#615712yes—
259C22×C4C22×#615711yescharacteristic
260C2×C8—1064no—
261C2×C8—1068no—
262C2×C8—2122no—
263C2×C8—2124no—
264C4⋊C4—22no—
265C4⋊C4—1062no—
266C4⋊C4—2124no—
267C22⋊C4—44no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C22⋊C4—2124no—
271C42—22no—
272C42—534no—
273C42—534no—
274C42—532no—
275C42—532no—
276C42—532no—
277C42—532no—
278C42—1064no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23—1062no—
285C23—2121no—
286C23—2121no—
287C23—2122no—
288C23—2122no—
289C23C2×C22⋊#615711yescharacteristic
290D4—24no—
291D4—42no—
292D4—1064no—
293D4—21216no—
294D4—2122no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—531no—
300C2×C4—531no—
301C2×C4—534no—
302C2×C4—534no—
303C2×C4—532no—
304C2×C4—532no—
305C2×C4—534no—
306C2×C4—532no—
307C2×C4—1062no—
308C2×C4—4241no—
309C2×C4—4242no—
310C2×C4—4241no—
311C2×C4—4241no—
312C2×C4—4242no—
313C2×C4—4241no—
314C2×C4—2121no—
315C2×C4—2122no—
316C2×C4#1313327914yes—
317C2×C4D4×#615712yes—
318C2×C4C2×D53⋊C811yescharacteristic
319C2×C4D53⋊M1611yescharacteristic
320C2×C4#1313330412yes—
321C8—1068no—
322C8—4242no—
323C22—42no—
324C22—44no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22—4242no—
329C22—4241no—
330C22—4241no—
331C22—2122no—
332C22—2124no—
333C22#1681157511yescharacteristic
334C22D53⋊(C22⋊C8)11yescharacteristic
335C22#1681159611yescharacteristic
336C22C2×#1313327912yes—
337C22D53⋊#10412yes—
338C4—24no—
339C4—81no—
340C4—531no—
341C4—532no—
342C4—531no—
343C4—538no—
344C4—1062no—
345C4—4241no—
346C4#1681165012yes—
347C4#1681165612yes—
348C2—82no—
349C2—4242no—
350C2D53⋊(D4⋊C8)12yes—
351C2#2029006012yes—
352C2#2029083811yescharacteristic
353C2#2029084311yescharacteristic
354C2#2029085011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#9638
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290648C2semidirect
2. C42×#88324C2semidirect
3. C2×#16811471C2semidirect
4. C2×D4⋊#88324C2semidirect
28. C4×#88324D4semidirect
34. #13133213C8semidirect
167. Dic53D4⋊C8semidirect
198. C53#9638semidirect

Maximal subgroups

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

subgroupindex
#202906482
C42×#883242
C2×#168114712
C2×D4⋊#883242
#202772792
#963853