Finite Group Databasebeta

← #28299651

Subgroups — #28299651

Order 13568.

510 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290501C211yesmaximal, characteristic, complemented
2#20290432C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20277177C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133272—21no—
8C2×#13133272—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—
26#13133272—24no—
27#13133272—42no—
28C23×Dic53—21no—
29D4×Dic53—22no—
30D4×Dic53—22no—
31#13133173—44no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×D4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C42×C53C2×C411yescharacteristic
54C24×C53—21no—
55C2×D4×C53—21no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53M1612yes—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—24no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C2×#88324—24no—
73C2×#88324—24no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—42no—
81C42×C53—41no—
82C22⋊Dic53—24no—
83C22⋊Dic53—41no—
84C22⋊Dic53—42no—
85C22⋊Dic53—41no—
86C4⋊Dic53—21no—
87C4⋊Dic53—41no—
88C4⋊Dic53M1612yes—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—22no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—22no—
95C4×Dic53—22no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—41no—
99C4×Dic53—42no—
100C23×C53—22no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53C2×C22⋊C411yescharacteristic
108D4×C53—42no—
109D4×C53—42no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#10412yes—
121C2×C4×C53C2×M1611yescharacteristic
122C2×C4×C53C2×M1611yescharacteristic
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—48no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53#10412yes—
140C2×Dic53AGL(1,ℤ/8)11yescharacteristic
141C2×Dic53AGL(1,ℤ/8)12yes—
142C2×Dic53#13711yescharacteristic
143#88324—44no—
144#88324—44no—
145#7628—531nomaximal, Sylow
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53#40711yescharacteristic
154C22×C53#41011yescharacteristic
155C22×C53#41311yescharacteristic
156C22×C53#41711yescharacteristic
157C22×C53#41811yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C22×C53C2×#10411yescharacteristic
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—81no—
165Dic53—22no—
166Dic53—22no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—42no—
171Dic53—81no—
172#2293—531no—
173#1767—531no—
174#1745—531no—
175#1271—532no—
176#1270—532no—
177C2×C53—82no—
178C2×C53#146411yescharacteristic
179C2×C53#148911yescharacteristic
180C2×C53#149011yescharacteristic
181C2×C53#150511yescharacteristic
182C2×C53#150911yescharacteristic
183C2×C53#152411yescharacteristic
184C2×C53#155711yescharacteristic
185C2×C4×D4—531no—
186C2×#117—1061no—
187C2×C4⋊C8—1061no—
188C2×#97—1061no—
189C2×#97—1061no—
190#390—531no—
191#390#615711yescharacteristic
192#389—531no—
193#389—531no—
194#388—1061no—
195#388—1061no—
196C4×C4⋊C4—531no—
197C4×C22⋊C4—1061no—
198C43—531no—
199#336—532no—
200#336—532no—
201#336—1061no—
202C53#762811yescharacteristic, complemented, Sylow
203C22×D4C2×#615711yescharacteristic
204C23×C4—1061no—
205C22×C8—1062no—
206C22×C8—1062no—
207C4×D4—1062no—
208C4×D4—1062no—
209#117—2124no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C2×#615711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C42—531no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—1061no—
227C2×C42C2×#615711yescharacteristic
228C4⋊C8—1064no—
229#97—1064no—
230#97—2122no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234C24—21no—
235C2×D4—21no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4#1171993612yes—
239C22×C4—21no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1064no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—2122no—
248C22×C4—2122no—
249C22×C4C22⋊#615711yescharacteristic
250C22×C4C22⋊#615711yescharacteristic
251C22×C4C22×#615711yescharacteristic
252C2×C8—1064no—
253C2×C8—1064no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—41no—
259C4⋊C4—532no—
260C4⋊C4—1061no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—1064no—
267C22⋊C4—2122no—
268C22⋊C4—2121no—
269C22⋊C4—2121no—
270C42—41no—
271C42—1061no—
272C42—1062no—
273C42—1062no—
274C42—1061no—
275C42—1062no—
276C42—1061no—
277C42—1061no—
278C42—2121no—
279C42—2122no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊#615711yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—531no—
301C2×C4—531no—
302C2×C4—532no—
303C2×C4—532no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1062no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2128no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4#1313328112yes—
321C2×C4D53⋊M1611yescharacteristic
322C2×C4C2×#1171993611yescharacteristic
323C8—2124no—
324C8—2124no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#1681159411yescharacteristic
333C22#1681160311yescharacteristic
334C22#1681161011yescharacteristic
335C22#1681161411yescharacteristic
336C22#1681161511yescharacteristic
337C22C2×#1313328111yescharacteristic
338C22D53⋊#10411yescharacteristic
339C4—22no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—81no—
344C4—1062no—
345C4—1062no—
346C4—1061no—
347C4—1061no—
348C4—4241no—
349C4—2121no—
350C4—2122no—
351C2—82no—
352C2#2028996711yescharacteristic
353C2#2029005111yescharacteristic
354C2#2029005511yescharacteristic
355C2#2029010411yescharacteristic
356C2#2029011611yescharacteristic
357C2#2029016611yescharacteristic
358C2#2029024711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#7628
LCSG ▷ C22×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. #20290501C2semidirect
2. #20290432C2semidirect
3. #20289534C2semidirect
202. C53#7628semidirect

Maximal subgroups

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

subgroupindex
#202905012
#202904322
#202895342
#202895312
#202771772
#762853