Finite Group Databasebeta

← #28306325

Subgroups — #28306325

Order 13568.

684 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 130 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#20290570C211yesmaximal, characteristic
2C4×(C2×#6157)⋊C4C211yesmaximal, characteristic
3#20289588C214yesmaximal
4#20283068C211yesmaximal, characteristic
5C22×C4×#6157—21no—
6C2×C4⋊#88324—21no—
7C2×#13133274C2212yes—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4C2212yes—
10#16811489—21no—
11C2×#13133217C411yescharacteristic, complemented
12C2×#13133197—21no—
13C2×C42×D53C2211yescharacteristic
14#16810528—21no—
15#16810522—21no—
16#16810522C412yescomplemented
17#16810480—21no—
18#16810480C2212yes—
19C53×#384C411yescharacteristic, complemented
20C23×#6157—22no—
21C22×#88324—22no—
22C2×C4×#6157—24no—
23C2×C4×#6157—42no—
24C2×C4×#6157—42no—
25C4⋊#88324—42no—
26#13133274—24no—
27(C2×#6157)⋊C4—24no—
28(C2×#6157)⋊C4—22no—
29(C2×#6157)⋊C4Q814yescomplemented
30C22×Dic106—21no—
31C22×C4×D53D411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53C2311yescharacteristic
34#13133217—42no—
35#13133197—44no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53C2×C412yes—
39C2×#11719921—22no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C42×D53—42no—
44#13133124—22no—
45#13133124—22no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C412yes—
49C2×C42×C53C2×C411yescharacteristic
50C23×D53C2×D411yescharacteristic
51C22×#6157—24no—
52C22×#6157—24no—
53C22×#6157—24no—
54C22×#6157—42no—
55C22×#6157—42no—
56C22×#6157—42no—
57C22×#6157—42no—
58C22×C4×C53—22no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53SD1612yes—
74C2×C4×D53Q1612yes—
75C2×C4×D53C2×Q812yes—
76C2×C4×D53#4112yes—
77C2×Dic106—42no—
78C2×Dic106—42no—
79C2×Dic106—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4×#6157—48no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C42×C53—22no—
88#11719921—44no—
89C4⋊Dic53—44no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4×Dic53—22no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53C22⋊Q812yes—
101C22×D53C2×SD1611yescharacteristic
102C22×D53C2×Q1611yescharacteristic
103C22×D53AGL(1,ℤ/8)12yes—
104C2×#6157—28no—
105C2×#6157—44no—
106C2×#6157—44no—
107C2×#6157—44no—
108C2×#6157—44no—
109C2×#6157—44no—
110C2×#6157—44no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53#9912yes—
119C2×C4×C53#10312yes—
120C2×C4×C53#10412yes—
121C2×C4×C53#10412yes—
122C2×C4×C53C4×Q812yes—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53#10312yes—
134C2×Dic53#12411yescharacteristic
135C2×Dic53C4⋊Q811yescharacteristic
136C4×D53—24no—
137C4×D53—24no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—44no—
141Dic106—84no—
142#88324—82no—
143#9908—531nomaximal, Sylow
144C22×C53#39111yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53#41011yescharacteristic
147C22×C53C2×#10311yescharacteristic
148C22×C53#41811yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53C2×#10411yescharacteristic
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—22no—
158C2×D53—22no—
159C2×D53#47612yes—
160C2×D53#47712yes—
161#6157—48no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—82no—
165C4×C53#32814yes—
166Dic53—21no—
167Dic53—21no—
168Dic53—22no—
169Dic53—41no—
170Dic53—82no—
171#2857—531no—
172#1797—531no—
173C4×#95—531no—
174#1294—534no—
175C2×C53C2×#32811yescharacteristic
176C2×C53#150311yescharacteristic
177C2×C53#150511yescharacteristic
178C2×C53#150711yescharacteristic
179C2×C53#198711yescharacteristic
180C2×C53#199411yescharacteristic
181C2×C53#199611yescharacteristic
182D53—22no—
183D53—22no—
184C2×C4⋊Q8—531no—
185C2×C22⋊Q8—1061no—
186C22×C42—531no—
187C22×C42—1061no—
188C2×C4⋊C8—1061no—
189C2×C22⋊C8—532no—
190#391—1061no—
191#389—532no—
192#389—1061no—
193#384#615711yescharacteristic, complemented
194#380—532no—
195#380—1061no—
196C2×#95—532no—
197C2×#95—1061no—
198#336—1061no—
199C53#990811yescharacteristic, complemented, Sylow
200C22×Q8—1061no—
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—531no—
204C23×C4—1062no—
205C22×C8—1062no—
206C4⋊Q8—2122no—
207C22⋊Q8—2124no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4C2×#615712yes—
214C2×C22⋊C4—1062no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—1064no—
219C2×C42—2122no—
220C2×C42—2122no—
221C2×C42—2122no—
222C2×C42C2×#615711yescharacteristic
223C4⋊C8—2122no—
224C22⋊C8—1064no—
225#95—534no—
226#95—1062no—
227#95—1062no—
228#95—1062no—
229#95—1064no—
230C24—531no—
231C2×Q8—2122no—
232C2×Q8—2122no—
233C2×Q8—2122no—
234C22×C4—22no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—531no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1064no—
246C22×C4—1064no—
247C22×C4—1064no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4—2122no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22⋊#615711yescharacteristic
259C22×C4C22×#615711yescharacteristic
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—2122no—
267C4⋊C4—2124no—
268C4⋊C4—2122no—
269C22⋊C4—2124no—
270C42—22no—
271C42—1062no—
272C42—2121no—
273C42—2121no—
274C42—2128no—
275C23—532no—
276C23—531no—
277C23—531no—
278C23—532no—
279C23—1061no—
280C23—1061no—
281C23—1061no—
282C23—1061no—
283C23C2×C22⋊#615711yescharacteristic
284Q8—4244no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—531no—
293C2×C4—531no—
294C2×C4—532no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1064no—
300C2×C4—1064no—
301C2×C4—1068no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2124no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2124no—
312C2×C4—2124no—
313C2×C4—2124no—
314C2×C4—2124no—
315C2×C4—2124no—
316C2×C4—2124no—
317C2×C4#1313327612yes—
318C2×C4#1313328012yes—
319C2×C4#1313328112yes—
320C2×C4#1313328212yes—
321C2×C4Q8×#615712yes—
322C8—4242no—
323C22—532no—
324C22—532no—
325C22—1061no—
326C22—1061no—
327C22—1061no—
328C22—1061no—
329C22—1062no—
330C22—1062no—
331C22—1062no—
332C22—1062no—
333C22(C2×#6157)⋊Q811yescharacteristic
334C22C2×#1313327611yescharacteristic
335C22#1681160311yescharacteristic
336C22C2×#1313328011yescharacteristic
337C22#1681161511yescharacteristic
338C22C2×#1313328111yescharacteristic
339C22C2×#1313328211yescharacteristic
340C4—22no—
341C4—22no—
342C4—82no—
343C4—1062no—
344C4—1061no—
345C4—1061no—
346C4—4242no—
347C4—2128no—
348C4—2121no—
349C4#1681147614yes—
350C2—1062no—
351C2—1062no—
352C2C2×#1681147611yescharacteristic
353C2#2029009911yescharacteristic
354C2#2029010411yescharacteristic
355C2#2029011211yescharacteristic
356C2#2029089311yescharacteristic
357C2#2029090011yescharacteristic
358C2#2029090511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#384#6157
hypocenterC53#9908
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
11. C2×#13133217C4semidirect
16. #16810522C4semidirect
19. C53×#384C4semidirect
29. (C2×#6157)⋊C4Q8semidirect
193. #384#6157semidirect
199. C53#9908semidirect

Maximal subgroups

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

subgroupindex
#202905702
C4×(C2×#6157)⋊C42
#202895882
#202830682
#990853