Finite Group Databasebeta

← #28306263

Subgroups — #28306263

Order 13568.

600 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290571C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289553C214yesmaximal
4C2×#16810522C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×C22⋊#88324C2212yes—
7C2×C22⋊#88324C2212yes—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489—21no—
11C22×C4⋊Dic53C412yes—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522—21no—
15#16810522—21no—
16#16810522—21no—
17#16810522—21no—
18C2×#13133124C2212yes—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—42no—
23C22⋊#88324—22no—
24C22⋊#88324—24no—
25C22⋊#88324Q814yes—
26#13133272—42no—
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C412yes—
29C23×Dic53C2311yescharacteristic
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—44no—
35C2×C4⋊Dic53—42no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—22no—
42#13133124—22no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2×C412yes—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—24no—
48C24×C53C22×C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53M1614yes—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—24no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×Q812yes—
73C22×Dic53#4112yes—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—24no—
82C4⋊Dic53—44no—
83C4⋊Dic53—42no—
84C4⋊Dic53—42no—
85C4⋊Dic53—44no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53C4×Q812yes—
96C23×C53C2×M1612yes—
97C23×C53#13112yes—
98C2×C4×C53—28no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—24no—
104C2×C4×C53—44no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—44no—
108C2×C4×C53#9912yes—
109C2×C4×C53#10112yes—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—44no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—84no—
126C2×Dic53#10012yes—
127C2×Dic53C22⋊Q812yes—
128C2×Dic53#12411yescharacteristic
129C2×Dic53C4⋊Q811yescharacteristic
130#88324—44no—
131#88324—82no—
132#9883—531nomaximal, Sylow
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53#39111yescharacteristic
140C22×C53#40711yescharacteristic
141C22×C53#40811yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53#41011yescharacteristic
144C22×C53C2×#10011yescharacteristic
145C22×C53C2×#10111yescharacteristic
146C22×C53#44614yes—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—44no—
150Dic53—22no—
151Dic53—22no—
152Dic53—21no—
153Dic53—21no—
154Dic53—41no—
155Dic53—82no—
156C2×#389—531no—
157#1797—531no—
158#1749—531no—
159#1280—534no—
160C2×C53—24no—
161C2×C53#146311yescharacteristic
162C2×C53#146611yescharacteristic
163C2×C53#147011yescharacteristic
164C2×C53#147111yescharacteristic
165C2×C53#198511yescharacteristic
166C2×C53#199611yescharacteristic
167C2×C53#199711yescharacteristic
168C22×C4⋊C4—532no—
169C22×C4⋊C4—531no—
170C22×C4⋊C4#615711yescharacteristic
171C22×C42—531no—
172C2×C4⋊C8—1061no—
173C2×C22⋊C8—532no—
174C2×C22⋊C8—532no—
175C2×#97—1061no—
176#389—1061no—
177#389—1061no—
178#389—1061no—
179#389—1061no—
180C2×#95—532no—
181#336—1061no—
182#336—1061no—
183C53#988311yescharacteristic, complemented, Sylow
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—532no—
187C23×C4C2×#615712yes—
188C23×C4C2×#615711yescharacteristic
189C22×C8—1062no—
190C22×C8—1062no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—2122no—
199C2×C4⋊C4—2124no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—2122no—
205C4⋊C8—2122no—
206C22⋊C8—534no—
207C22⋊C8—1062no—
208C22⋊C8—1064no—
209#97—2122no—
210#95—1062no—
211#95—1062no—
212C24C22×#615711yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—532no—
224C22×C4—532no—
225C22×C4—531no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—1064no—
229C22×C4—1062no—
230C22×C4—2121no—
231C22×C4—2121no—
232C22×C4—2121no—
233C22×C4—2122no—
234C22×C4—2122no—
235C22×C4#1171994014yes—
236C22×C4C22⋊#615711yescharacteristic
237C22×C4C22⋊#615711yescharacteristic
238C2×C8—1064no—
239C2×C8—2122no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—2122no—
247C4⋊C4—2124no—
248C4⋊C4—2124no—
249C4⋊C4—2122no—
250C42—1062no—
251C42—1062no—
252C42—2121no—
253C42—2121no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23C2×C22⋊#615711yescharacteristic
259C23Q8×#615712yes—
260C23C2×#1171994012yes—
261C23#1313330512yes—
262C2×C4—28no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—532no—
273C2×C4—531no—
274C2×C4—531no—
275C2×C4—532no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—4244no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2124no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4#1313327612yes—
293C2×C4#1313327812yes—
294C8—4242no—
295C8—2124no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22(C2×#6157)⋊Q811yescharacteristic
303C22#1681159311yescharacteristic
304C22#1681159811yescharacteristic
305C22C2×#1313327611yescharacteristic
306C22#1681160311yescharacteristic
307C22D53⋊#10011yescharacteristic
308C22C2×#1313327811yescharacteristic
309C22#1681167414yes—
310C4—22no—
311C4—22no—
312C4—44no—
313C4—1061no—
314C4—1062no—
315C4—1061no—
316C4—1062no—
317C4—4242no—
318C4—2121no—
319C2—24no—
320C2#2028996311yescharacteristic
321C2#2028997511yescharacteristic
322C2#2028998911yescharacteristic
323C2#2028999211yescharacteristic
324C2#2029088911yescharacteristic
325C2#2029090511yescharacteristic
326C2#2029090811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#9883
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
183. C53#9883semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
#202895532
C2×#168105222
#988353