Finite Group Databasebeta

← #28299481

Subgroups — #28299481

Order 13568.

500 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290643C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20277177C211yesmaximal, characteristic
6C2×C4×#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×D4×Dic53C411yescharacteristic
13C2×#13133173—21no—
14#16810526C2211yescharacteristic
15#16810521C411yescharacteristic
16#16810519C2211yescharacteristic
17#16810517—21no—
18#16810511—21no—
19C4⋊C4×Dic53C411yescharacteristic
20C22⋊C4×Dic53—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4×#88324—24no—
26C23×Dic53—21no—
27D4×Dic53—22no—
28D4×Dic53C814yescomplemented
29#13133173—44no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×D4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C24×C53—21no—
53C2×D4×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53C2×C812yes—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—24no—
63C22×Dic53—21no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×#88324—24no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C42×C53—41no—
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C4⋊Dic53—21no—
84C4⋊Dic53—41no—
85C4⋊Dic53C2×C812yes—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—22no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C4×Dic53—42no—
94C4×Dic53C2×C812yes—
95C4×Dic53M1612yes—
96C23×C53—22no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104D4×C53—42no—
105D4×C53—42no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53#10412yes—
117C2×C4×C53C22×C811yescharacteristic
118C2×C4×C53C2×M1611yescharacteristic
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—22no—
128C2×Dic53—48no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53C22⋊C812yes—
135C2×Dic53C22⋊C812yes—
136C2×Dic53#13512yes—
137C2×Dic53AGL(1,ℤ/8)12yes—
138#88324—44no—
139#88324—82no—
140#7580—531nomaximal, Sylow
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53C2×C22⋊C811yescharacteristic
149C22×C53#41011yescharacteristic
150C22×C53#41311yescharacteristic
151C22×C53#41611yescharacteristic
152C22×C53#41811yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C22×C53#42111yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—81no—
160Dic53—22no—
161Dic53—22no—
162Dic53—22no—
163Dic53—22no—
164Dic53—42no—
165Dic53—81no—
166#2293—531no—
167#1839—531no—
168#1744—531no—
169#1271—532no—
170#1270—532no—
171C2×C53—82no—
172C2×C53#145811yescharacteristic
173C2×C53#147611yescharacteristic
174C2×C53#147811yescharacteristic
175C2×C53#150511yescharacteristic
176C2×C53#152111yescharacteristic
177C2×C53#153211yescharacteristic
178C2×C53#155311yescharacteristic
179C2×C4×D4—531no—
180C2×#117—1061no—
181C2×C4×C8—1061no—
182#390—531no—
183#390#615711yescharacteristic
184#389—531no—
185#389—531no—
186#388—1061no—
187#388—1061no—
188C4×C4⋊C4—531no—
189C4×C22⋊C4—1061no—
190C43—531no—
191#336—532no—
192#336—532no—
193#336—1061no—
194#336—1061no—
195#336—1061no—
196C53#758011yescharacteristic, complemented, Sylow
197C22×D4C2×#615711yescharacteristic
198C23×C4—1061no—
199C22×C8—1062no—
200C22×C8—1062no—
201C4×D4—534no—
202C4×D4—1062no—
203#117—2124no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4C2×#615711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—1061no—
221C2×C42C2×#615711yescharacteristic
222C4×C8—1064no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C24—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4D53⋊C812yes—
231C22×C4—21no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—1064no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—2122no—
240C22×C4—2122no—
241C22×C4C22⋊#615711yescharacteristic
242C22×C4C22⋊#615711yescharacteristic
243C22×C4C22×#615711yescharacteristic
244C2×C8—1064no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—532no—
252C4⋊C4—1061no—
253C4⋊C4—2121no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—1064no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C42—41no—
262C42—532no—
263C42—532no—
264C42—1062no—
265C42—1062no—
266C42—1062no—
267C42—2121no—
268C42—2121no—
269C42—2122no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#615711yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—532no—
291C2×C4—532no—
292C2×C4—532no—
293C2×C4—532no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1062no—
303C2×C4—2128no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4#1313328112yes—
310C2×C4C2×D53⋊C811yescharacteristic
311C2×C4D53⋊M1611yescharacteristic
312C8—4242no—
313C8—2124no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22D53⋊(C22⋊C8)11yescharacteristic
322C22#1681160311yescharacteristic
323C22#1681161011yescharacteristic
324C22#1681161311yescharacteristic
325C22#1681161511yescharacteristic
326C22C2×#1313328111yescharacteristic
327C22#1681162011yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—81no—
333C4—1062no—
334C4—1062no—
335C4—1062no—
336C4—1062no—
337C4—4241no—
338C4—2122no—
339C2—82no—
340C2#2028994711yescharacteristic
341C2#2029000711yescharacteristic
342C2#2029001511yescharacteristic
343C2#2029010411yescharacteristic
344C2#2029015611yescharacteristic
345C2#2029019111yescharacteristic
346C2#2029023911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#7580
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20290430C2semidirect
3. #20289534C2semidirect
28. D4×Dic53C8semidirect
196. C53#7580semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202904302
#202895342
#202895312
#202771772
#758053