Finite Group Databasebeta

← #28299820

Subgroups — #28299820

Order 13568.

458 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C4×C4⋊#88324C211yesmaximal, characteristic, complemented
2#20290432C211yesmaximal, characteristic, complemented
3#20289536C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20283086C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133272—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133208C411yescharacteristic
13C2×#13133204C411yescharacteristic
14#16810529C411yescharacteristic
15#16810528C2211yescharacteristic
16#16810526C2211yescharacteristic
17#16810516—21no—
18#16810516—21no—
19#16810511—21no—
20#16810511—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26#13133272—24no—
27C4×#88324—42no—
28C22×Dic106C2×C411yescharacteristic
29#13133208—42no—
30#13133204—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊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—
47#13133124—21no—
48C22×D4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C42×C53C2×C411yescharacteristic
53C24×C53—21no—
54C2×D4×C53—22no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×Dic106—22no—
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×#88324—24no—
71C2×#88324—24no—
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—22no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87#11719919—42no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103D4×C53—82no—
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#10412yes—
114C2×C4×C53#10412yes—
115C2×C4×C53#13112yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53#10012yes—
132C2×Dic53#13512yes—
133C2×Dic53AGL(1,ℤ/8)11yescharacteristic
134C2×Dic53#13711yescharacteristic
135Dic106—82no—
136#88324—44no—
137#88324—44no—
138#7676—531nomaximal, Sylow
139C22×C53—81no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53#40811yescharacteristic
147C22×C53#41011yescharacteristic
148C22×C53C2×#10011yescharacteristic
149C22×C53#41611yescharacteristic
150C22×C53#41711yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—81no—
157Dic53—22no—
158Dic53—22no—
159Dic53—21no—
160Dic53—21no—
161Dic53—41no—
162Dic53—81no—
163Dic53—81no—
164#2861—531no—
165C4×C4⋊C8—531no—
166#1745—531no—
167#1272—532no—
168#1270—532no—
169C2×C53—161no—
170C2×C53#146911yescharacteristic
171C2×C53#148311yescharacteristic
172C2×C53#149411yescharacteristic
173C2×C53#150011yescharacteristic
174C2×C53#150911yescharacteristic
175C2×C53#152511yescharacteristic
176C2×C53#155111yescharacteristic
177C2×#125—531no—
178C2×#124—531no—
179C2×C4⋊C8—1061no—
180C2×#97—1061no—
181C2×C4×C8—1061no—
182#391—531no—
183#391—531no—
184#390—531no—
185#390#615711yescharacteristic
186#388—1061no—
187#388—1061no—
188#388—1061no—
189#388—1061no—
190C43—531no—
191#336—532no—
192#336—532no—
193#336—1061no—
194C53#767611yescharacteristic, complemented, Sylow
195C22×Q8—531no—
196C22×D4C2×#615711yescharacteristic
197C22×C8—1062no—
198C22×C8—1062no—
199#125—2122no—
200#124—2122no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C2×#615711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42C2×#615711yescharacteristic
218C4⋊C8—1064no—
219#97—1064no—
220C4×C8—2122no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C24—21no—
226C2×Q8—1062no—
227C2×Q8—2121no—
228C2×Q8—2121no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4C22⋊#615711yescharacteristic
240C22×C4C22⋊#615711yescharacteristic
241C22×C4C22×#615711yescharacteristic
242C2×C8—1064no—
243C2×C8—1064no—
244C2×C8—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C4⋊C4—41no—
249C4⋊C4—2121no—
250C4⋊C4—2122no—
251C4⋊C4—2122no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C22⋊C4—2122no—
259C42—22no—
260C42—1061no—
261C42—1062no—
262C42—1062no—
263C42—1061no—
264C42—1062no—
265C42—1062no—
266C42—2121no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊#615711yescharacteristic
275Q8—4242no—
276D4—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—532no—
287C2×C4—532no—
288C2×C4—531no—
289C2×C4—531no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313328112yes—
306C2×C4#1313328112yes—
307C2×C4#1313330412yes—
308C8—2124no—
309C8—2124no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#1681159611yescharacteristic
318C22#1681160211yescharacteristic
319C22D53⋊#10011yescharacteristic
320C22#1681161311yescharacteristic
321C22#1681161411yescharacteristic
322C22C2×#1313328111yescharacteristic
323C22C2×#1313328111yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—81no—
328C4—1062no—
329C4—1061no—
330C4—1061no—
331C4—1062no—
332C4—4241no—
333C4—4241no—
334C4—2121no—
335C2—161no—
336C2#2028998411yescharacteristic
337C2#2029003011yescharacteristic
338C2#2029006611yescharacteristic
339C2#2029008611yescharacteristic
340C2#2029011711yescharacteristic
341C2#2029017011yescharacteristic
342C2#2029023511yescharacteristic
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#7676
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. C4×C4⋊#88324C2semidirect
2. #20290432C2semidirect
3. #20289536C2semidirect
194. C53#7676semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#883242
#202904322
#202895362
#202895312
#202830862
#767653