Finite Group Databasebeta

← #28299482

Subgroups — #28299482

Order 13568.

500 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290642C211yesmaximal, characteristic, complemented
2#20290429C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20277176C211yesmaximal, characteristic
6C2×C4×#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×C4×D212C411yescharacteristic
13C2×#13133174—21no—
14#16810524C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810519C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810512—21no—
19#16810512—21no—
20C4×#11719921—21no—
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4×#88324—24no—
26C22×D212C2×C411yescharacteristic
27C22×C4×D53—21no—
28C4×D212—22no—
29C4×D212C814yescomplemented
30#13133174—44no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52C23×D53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C2×D212—21no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212C2×C812yes—
67C2×C4×D53—24no—
68C2×C4×D53—42no—
69C2×C4×D53—42no—
70C2×#88324—24no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C42×C53—42no—
79C42×C53C2×C812yes—
80C42×C53M1612yes—
81#11719921—24no—
82#11719921—42no—
83#11719921—42no—
84#11719921—42no—
85#11719921—42no—
86C4⋊Dic53—21no—
87C4⋊Dic53C2×C812yes—
88#11719919—42no—
89#11719919—42no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C4×Dic53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97C22×D53—22no—
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53C22⋊C812yes—
117C2×C4×C53C22⋊C812yes—
118C2×C4×C53C22×C811yescharacteristic
119C2×C4×C53C2×M1611yescharacteristic
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—22no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53#10412yes—
133C2×Dic53#13512yes—
134C2×Dic53AGL(1,ℤ/8)12yes—
135D212—42no—
136D212—42no—
137C4×D53—48no—
138#88324—44no—
139#88324—82no—
140#7580—531nomaximal, Sylow
141C22×C53C2×C22⋊C811yescharacteristic
142C22×C53#41011yescharacteristic
143C22×C53#41311yescharacteristic
144C22×C53#41611yescharacteristic
145C22×C53#41811yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C22×C53#42111yescharacteristic
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—81no—
152C2×D53—81no—
153C2×D53—81no—
154C2×D53—81no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—81no—
160C4×C53—81no—
161Dic53—22no—
162Dic53—22no—
163Dic53—22no—
164Dic53—41no—
165Dic53—42no—
166#2293—531no—
167#1839—531no—
168#1744—531no—
169#1271—532no—
170#1270—532no—
171C2×C53#145811yescharacteristic
172C2×C53#147611yescharacteristic
173C2×C53#147811yescharacteristic
174C2×C53#150511yescharacteristic
175C2×C53#152111yescharacteristic
176C2×C53#153211yescharacteristic
177C2×C53#155311yescharacteristic
178D53—82no—
179C2×C4×D4—531no—
180C2×#117—1061no—
181C2×C4×C8—1061no—
182#390—531no—
183#390—531no—
184#389—531no—
185#389—531no—
186#388—1061no—
187#388—1061no—
188C4×C4⋊C4#615711yescharacteristic
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×D4—531no—
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—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C2×#615711yescharacteristic
208C2×C4⋊C4C2×#615711yescharacteristic
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C42—21no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42C2×#615711yescharacteristic
222C4×C8—1064no—
223#95—21no—
224#95—1061no—
225#95—1061no—
226C24—1061no—
227C2×D4—532no—
228C2×D4—1061no—
229C2×D4—2121no—
230C2×D4—2121no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—1061no—
238C22×C4—1064no—
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—41no—
252C4⋊C4—532no—
253C4⋊C4—1061no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C22⋊C4—1064no—
257C22⋊C4—2122no—
258C22⋊C4—2122no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C42—42no—
262C42—1062no—
263C42—1062no—
264C42—1062no—
265C42—2121no—
266C42—2121no—
267C42—2121no—
268C42D53⋊C812yes—
269C42#1171993612yes—
270C23—1062no—
271C23—2121no—
272C23—2121no—
273C23—2121no—
274C23—2121no—
275C23—2121no—
276C23—2121no—
277C23C2×C22⋊#615711yescharacteristic
278D4—2122no—
279D4—2122no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—532no—
293C2×C4—532no—
294C2×C4—532no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1062no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—2128no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4#1313327412yes—
309C2×C4#1313327412yes—
310C2×C4C2×D53⋊C811yescharacteristic
311C2×C4C2×#1171993611yescharacteristic
312C8—4242no—
313C8—2124no—
314C22—4241no—
315C22—4241no—
316C22—4241no—
317C22—4241no—
318C22—2122no—
319C22—2122no—
320C22—2122no—
321C22C2×#1313327411yescharacteristic
322C22#1681160211yescharacteristic
323C22#1681160911yescharacteristic
324C22#1681161311yescharacteristic
325C22#1681161511yescharacteristic
326C22D53⋊#10411yescharacteristic
327C22#1681162211yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—81no—
333C4—81no—
334C4—1062no—
335C4—1062no—
336C4—1062no—
337C4—2121no—
338C4—2122no—
339C2—4242no—
340C2#2028994811yescharacteristic
341C2#2029000811yescharacteristic
342C2#2029001611yescharacteristic
343C2#2029010511yescharacteristic
344C2#2029015911yescharacteristic
345C2#2029019211yescharacteristic
346C2#2029023811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7580
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290642C2semidirect
2. #20290429C2semidirect
3. #20289534C2semidirect
29. C4×D212C8semidirect
196. C53#7580semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202904292
#202895342
#202895322
#202771762
#758053