Finite Group Databasebeta

← #28299533

Subgroups — #28299533

Order 13568.

482 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290631C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20277173C211yesmaximal, characteristic
6C2×#13133284—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133179C411yescharacteristic
13C2×#13133175—21no—
14#16810525C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810515—21no—
18#16810513—21no—
19C4⋊C4×Dic53C411yescharacteristic
20#16810470—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22×D212C2×C411yescharacteristic
27C22×C4×D53—21no—
28#13133179—22no—
29#13133179C814yescomplemented
30#13133175—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×C4×Dic53—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×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
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—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53C2×C812yes—
80C42×C53—41no—
81#11719921—24no—
82#11719921—42no—
83#11719921—42no—
84#11719921—41no—
85#11719921—42no—
86#11719921—41no—
87C4⋊Dic53—42no—
88C4⋊Dic53—41no—
89C4×Dic53—22no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C4×Dic53—42no—
96C4×Dic53C2×C812yes—
97C4×Dic53M1612yes—
98C23×C53C2×C22⋊C411yescharacteristic
99C22×D53—22no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—41no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53C22×C811yescharacteristic
121C2×C4×C53C2×M1611yescharacteristic
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53C22⋊C812yes—
134C2×Dic53C22⋊C812yes—
135C2×Dic53AGL(1,ℤ/8)11yescharacteristic
136C2×Dic53AGL(1,ℤ/8)12yes—
137C2×Dic53#13711yescharacteristic
138D212—42no—
139D212—42no—
140C4×D53—48no—
141#88324—82no—
142#88324—82no—
143#7594—531nomaximal, Sylow
144C22×C53C2×C22⋊C811yescharacteristic
145C22×C53#41011yescharacteristic
146C22×C53#41311yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#41811yescharacteristic
149C22×C53#42111yescharacteristic
150C22×C53#42111yescharacteristic
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—81no—
155C2×D53—81no—
156C2×D53—81no—
157C2×D53—81no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—42no—
162C4×C53—41no—
163C4×C53—81no—
164Dic53—22no—
165Dic53—22no—
166Dic53—21no—
167Dic53—21no—
168Dic53—22no—
169Dic53—81no—
170#2293—531no—
171#1839—531no—
172#1831—531no—
173#1271—532no—
174#1270—532no—
175C2×C53#145811yescharacteristic
176C2×C53#147711yescharacteristic
177C2×C53#147811yescharacteristic
178C2×C53#151211yescharacteristic
179C2×C53#151411yescharacteristic
180C2×C53#153511yescharacteristic
181C2×C53#155711yescharacteristic
182D53—82no—
183C2×C4×D4—531no—
184C2×#117—1061no—
185C2×C4⋊C8—1061no—
186#390—531no—
187#390—531no—
188#389—531no—
189#389#615711yescharacteristic
190#388—1061no—
191#388—1061no—
192C4×C4⋊C4—531no—
193C4×C22⋊C4—1061no—
194C43—531no—
195#336—532no—
196#336—532no—
197#336—1061no—
198#336—1061no—
199#336—1061no—
200C53#759411yescharacteristic, complemented, Sylow
201C22×D4—531no—
202C23×C4—1061no—
203C22×C8—1062no—
204C22×C8—1062no—
205C4×D4—534no—
206C4×D4—1062no—
207#117—2124no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C2×#615711yescharacteristic
212C2×C4⋊C4C2×#615711yescharacteristic
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—1061no—
225C2×C42C2×#615711yescharacteristic
226C4⋊C8—2122no—
227#95—21no—
228#95—1061no—
229#95—1061no—
230C24—1061no—
231C2×D4—532no—
232C2×D4—1061no—
233C2×D4—2121no—
234C2×D4—2121no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—1064no—
242C22×C4—1061no—
243C22×C4—2122no—
244C22×C4—2122no—
245C22×C4C22⋊#615711yescharacteristic
246C22×C4C22⋊#615711yescharacteristic
247C22×C4C22×#615711yescharacteristic
248C2×C8—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C2×C8—2122no—
252C2×C8—2122no—
253C2×C8—2122no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—2121no—
258C4⋊C4—2122no—
259C4⋊C4D53⋊C812yes—
260C22⋊C4—1064no—
261C22⋊C4—2122no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C22⋊C4—2121no—
265C22⋊C4—2121no—
266C42—41no—
267C42—532no—
268C42—532no—
269C42—1061no—
270C42—1062no—
271C42—1061no—
272C42—2121no—
273C42—2121no—
274C42—2121no—
275C42—2122no—
276C23—1062no—
277C23—2121no—
278C23—2121no—
279C23—2121no—
280C23—2121no—
281C23—2121no—
282C23—2121no—
283C23C2×C22⋊#615711yescharacteristic
284D4—2122no—
285D4—2122no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—532no—
301C2×C4—532no—
302C2×C4—531no—
303C2×C4—531no—
304C2×C4—532no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—2128no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4C2×D53⋊C811yescharacteristic
318C2×C4D53⋊M1611yescharacteristic
319C8—4242no—
320C8—4242no—
321C22—4241no—
322C22—4241no—
323C22—4241no—
324C22—4241no—
325C22—2122no—
326C22—2122no—
327C22—2122no—
328C22D53⋊(C22⋊C8)11yescharacteristic
329C22#1681160411yescharacteristic
330C22#1681161011yescharacteristic
331C22#1681161411yescharacteristic
332C22#1681161511yescharacteristic
333C22#1681162011yescharacteristic
334C22#1681162211yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—42no—
340C4—81no—
341C4—1061no—
342C4—1061no—
343C4—1062no—
344C4—1062no—
345C4—1062no—
346C4—4241no—
347C2—4242no—
348C2#2028994911yescharacteristic
349C2#2029001311yescharacteristic
350C2#2029001511yescharacteristic
351C2#2029012911yescharacteristic
352C2#2029013611yescharacteristic
353C2#2029020211yescharacteristic
354C2#2029024711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7594
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20290631C2semidirect
3. #20289533C2semidirect
29. #13133179C8semidirect
200. C53#7594semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202906312
#202895332
#202895322
#202771732
#759453