Finite Group Databasebeta

← #28299854

Subgroups — #28299854

Order 13568.

464 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 100 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
2C4×#13133284C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20283079C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133210—21no—
13#16810526C411yescharacteristic
14#16810526C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810519C411yescharacteristic
18#16810517—21no—
19#16810516—21no—
20#16810511—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26C4⋊#88324—24no—
27#13133272—42no—
28C4×#88324—42no—
29#13133210—42no—
30#13133210—42no—
31C2×C4⋊Dic53—21no—
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×C22⋊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—22no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×#88324—24no—
69C2×#88324—24no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—22no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—42no—
77C42×C53—41no—
78C22⋊Dic53—42no—
79C22⋊Dic53—42no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C22⋊Dic53—41no—
83C22⋊Dic53—41no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719919—42no—
89#11719919—42no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—41no—
96C4×Dic53D812yes—
97C4×Dic53SD1612yes—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C2×C22⋊C411yescharacteristic
106D4×C53—82no—
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#10212yes—
117C2×C4×C53#10412yes—
118C2×C4×C53#13112yes—
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—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53#10212yes—
134C2×Dic53#10412yes—
135C2×Dic53C2×D811yescharacteristic
136C2×Dic53C2×SD1611yescharacteristic
137C2×Dic53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53#13711yescharacteristic
139#88324—44no—
140#88324—44no—
141#7686—531nomaximal, Sylow
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53#40811yescharacteristic
150C22×C53#41011yescharacteristic
151C22×C53#41311yescharacteristic
152C22×C53C2×#10211yescharacteristic
153C22×C53#41711yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—81no—
160Dic53—22no—
161Dic53—22no—
162Dic53—21no—
163Dic53—21no—
164Dic53—21no—
165Dic53—21no—
166Dic53—81no—
167Dic53—81no—
168#2860—531no—
169#1767—531no—
170C4×C4⋊C8—531no—
171#1271—532no—
172#1270—532no—
173C2×C53—161no—
174C2×C53#147311yescharacteristic
175C2×C53#148311yescharacteristic
176C2×C53#149211yescharacteristic
177C2×C53#150211yescharacteristic
178C2×C53#150911yescharacteristic
179C2×C53#152411yescharacteristic
180C2×C53#154711yescharacteristic
181C2×#126—1061no—
182C2×C4⋊C8—1061no—
183C2×C4⋊C8—1061no—
184C2×#97—1061no—
185C2×C4×C8—1061no—
186#390—531no—
187#390—531no—
188#390#615711yescharacteristic
189#389—531no—
190#389—531no—
191#389—531no—
192#388—1061no—
193#388—1061no—
194#388—1061no—
195C43—531no—
196#336—532no—
197#336—532no—
198C53#768611yescharacteristic, complemented, Sylow
199C22×D4C2×#615711yescharacteristic
200C22×C8—1062no—
201C22×C8—1062no—
202#126—2122no—
203#126—2122no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C2×#615711yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—531no—
222C2×C42C2×#615711yescharacteristic
223C4⋊C8—1064no—
224C4⋊C8—1064no—
225#97—2122no—
226C4×C8—2122no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C24—21no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4C22⋊#615711yescharacteristic
242C22×C4C22⋊#615711yescharacteristic
243C22×C4C22×#615711yescharacteristic
244C2×C8—1064no—
245C2×C8—1064no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C4⋊C4—22no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2122no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—2122no—
260C22⋊C4—2121no—
261C22⋊C4—2121no—
262C22⋊C4—2121no—
263C22⋊C4—2121no—
264C22⋊C4—2122no—
265C42—41no—
266C42—532no—
267C42—532no—
268C42—1061no—
269C42—1062no—
270C42—1061no—
271C42—1062no—
272C42—1062no—
273C42—2121no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×C22⋊#615711yescharacteristic
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—532no—
293C2×C4—531no—
294C2×C4—531no—
295C2×C4—531no—
296C2×C4—531no—
297C2×C4—532no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4#1313327912yes—
313C2×C4#1313328112yes—
314C2×C4#1313330412yes—
315C8—2124no—
316C8—2124no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1681159711yescharacteristic
325C22#1681160211yescharacteristic
326C22#1681161011yescharacteristic
327C22C2×#1313327911yescharacteristic
328C22#1681161411yescharacteristic
329C22C2×#1313328111yescharacteristic
330C22D53⋊#10411yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—81no—
335C4—1062no—
336C4—1061no—
337C4—1061no—
338C4—1061no—
339C4—1061no—
340C4—1062no—
341C4—4241no—
342C4—4241no—
343C2—161no—
344C2#2028999611yescharacteristic
345C2#2029003111yescharacteristic
346C2#2029006111yescharacteristic
347C2#2029009411yescharacteristic
348C2#2029011711yescharacteristic
349C2#2029016611yescharacteristic
350C2#2029022711yescharacteristic
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#7686
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. C4×#13133284C2semidirect
3. #20289533C2semidirect
198. C53#7686semidirect

Maximal subgroups

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

subgroupindex
#202905012
C4×#131332842
#202895332
#202895312
#202830792
#768653