Finite Group Databasebeta

← #27302748

Subgroups — #27302748

Order 13568.

488 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 103 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#20279032C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20272834C214yesmaximal
4C53×#1800C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic, complemented
6C2×#13133132C2212yes—
7#16810562—21no—
8#16810559—21no—
9#16810521—21no—
10#16810503C2212yes—
11#16810495C412yescomplemented
12#16810483—21no—
13#16810483C411yescharacteristic, complemented
14C4×C22⋊Dic53—21no—
15#16810390—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#423—21no—
18C2×#100×C53C2212yes—
19C53×#343—21no—
20C23×Dic53C2×C412yes—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×#11719919—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53C2×C412yes—
30C2×C4×Dic53—41no—
31#13133132—22no—
32#13133132Q814yes—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36#13133124—41no—
37C22×D4×C53C2311yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C2×M16×C53—22no—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C4⋊C8×C53—41no—
48C4⋊C8×C53—41no—
49#100×C53—24no—
50C24×C53C22×C411yescharacteristic
51C24×C53C2×D411yescharacteristic
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C2×Q812yes—
55C2×D4×C53#4112yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—82no—
72M16×C53—42no—
73M16×C53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—21no—
79C42×C53—21no—
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—82no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719919—82no—
89C4×Dic53—82no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53C4×Q812yes—
100C23×C53C22⋊Q812yes—
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—81no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#12411yescharacteristic
119C2×C4×C53C4⋊Q811yescharacteristic
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—82no—
128C2×Dic53—82no—
129C2×Dic53—84no—
130C8×C53—82no—
131#13295—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53#39111yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53#41011yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53#35212yes—
154Dic53—82no—
155Dic53—84no—
156Dic53—82no—
157#2487—531no—
158#1800D5311yescharacteristic
159C4×#99—531no—
160#1390—534no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—42no—
164C2×C53—42no—
165C2×C53#199611yescharacteristic
166C2×C53C2×#35211yescharacteristic
167C2×C53#212211yescharacteristic
168C2×C4×D4C2×D5311yescharacteristic
169C22×C4⋊C4—531no—
170#423—21no—
171C2×#100C2×D5312yes—
172C2×#99—532no—
173#400—1061no—
174#399—1061no—
175#389—1061no—
176#386—532no—
177#385—532no—
178#380—531no—
179#380—1061no—
180C4×C22⋊C4—1061no—
181#343—21no—
182#342—1061no—
183C53#1329511yescharacteristic, complemented, Sylow
184C22×D4C22×D5311yescharacteristic
185C23×C4—532no—
186C23×C4C4×D5311yescharacteristic
187C23×C4D21211yescharacteristic
188C2×M16—22no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—1062no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—2121no—
196C2×C4⋊C4—2121no—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4—532no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4C4×D5311yescharacteristic
202C2×C22⋊C4D21211yescharacteristic
203C2×C42—2121no—
204C2×C42C22⋊D5311yescharacteristic
205C4⋊C8—41no—
206C4⋊C8—41no—
207#100—24no—
208#99—534no—
209#99—1062no—
210#95—1062no—
211#95—1061no—
212#95—1062no—
213#95—2121no—
214C24C2×C4×D5311yescharacteristic
215C24C2×D21211yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4#1171994712yes—
219C2×D4Q8×D5312yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—4242no—
229C22×C4—2122no—
230C22×C4—2122no—
231C22×C4#1171992111yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4#1171992111yescharacteristic
234C22×C4#1171992111yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—42no—
237M16—42no—
238C2×C8—42no—
239C4⋊C4—41no—
240C4⋊C4—4242no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—1064no—
248C22⋊C4—4242no—
249C22⋊C4—2122no—
250C22⋊C4—2122no—
251C42—21no—
252C42—21no—
253C42—4242no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×#1171992111yescharacteristic
263C23#1313318212yes—
264C23#1313319712yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—4242no—
281C2×C4—4242no—
282C2×C4—4244no—
283C2×C4—2124no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4#1313313111yescharacteristic
291C2×C4#1313313111yescharacteristic
292C2×C4#1313320411yescharacteristic
293C2×C4#1313321511yescharacteristic
294C8—82no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1681052811yescharacteristic
308C22C2×#1313313111yescharacteristic
309C22#1681059611yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—4242no—
317C4—4244no—
318C4—4242no—
319C4#1681041412yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2027569911yescharacteristic
325C2C2×#1681041411yescharacteristic
326C2#2027636611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1800D53
hypocenterC53#13295
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic53C4semidirect
11. #16810495C4semidirect
13. #16810483C4semidirect
183. C53#13295semidirect

Maximal subgroups

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

subgroupindex
#202790322
C4×#131331322
#202728342
C53×#18002
#1329553