Finite Group Databasebeta

← #28309553

Subgroups — #28309553

Order 13568.

452 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290576C211yesmaximal, characteristic
2#20289952C211yesmaximal, characteristic
3#20289710C214yesmaximal, complemented
4#20279889C211yesmaximal, characteristic
5C2×#13133276—21no—
6D53⋊(C22⋊C8)—21no—
7#16811565—21no—
8#16811501C2212yes—
9#16811468—21no—
10#16811467C2212yes—
11C2×C4×D212C2211yescharacteristic
12#16810534C412yescomplemented
13#16810534C2212yes—
14#16810524—21no—
15#16810488—21no—
16#16810480—21no—
17#16810480—21no—
18C4×C4⋊Dic53C411yescharacteristic, complemented
19C2×C4⋊Q8×C53C411yescharacteristic, complemented
20C2×D53⋊C8—41no—
21C2×C4⋊#6157—41no—
22C2×C22⋊#6157—22no—
23#13133276—42no—
24#13133274—41no—
25C22⋊#88324—22no—
26(C2×#6157)⋊C4—41no—
27C22×D212D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C4×D212—41no—
31C4×D212—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×#11719921C2×C411yescharacteristic
37C2×#11719921C2311yescharacteristic
38C2×C4×Dic53—21no—
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C22×Q8×C53C2×C412yes—
43C4⋊Q8×C53—41no—
44C4⋊Q8×C53—41no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C23×D53C2×D411yescharacteristic
50C22×#6157—42no—
51C2×Q8×C53—42no—
52C2×Q8×C53—82no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×D212—41no—
61C2×D212—41no—
62C2×D212SD1612yes—
63C2×D212Q1612yes—
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—42no—
69C2×C4×D53—42no—
70C22⋊#6157—42no—
71C22⋊#6157—42no—
72C2×#88324—42no—
73C4⋊#6157—82no—
74D53⋊C8—82no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C42×C53—21no—
79C42×C53—21no—
80#11719921—41no—
81#11719921—41no—
82#11719921—82no—
83#11719921—82no—
84C4⋊Dic53—41no—
85C4⋊Dic53—42no—
86C4×Dic53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53#10312yes—
97C22×D53C2×SD1611yescharacteristic
98C22×D53C2×Q1611yescharacteristic
99C2×#6157—82no—
100C2×#6157—82no—
101C2×#6157—82no—
102Q8×C53—84no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#10312yes—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121D212—42no—
122D212—42no—
123C4×D53—42no—
124C4×D53—42no—
125C4×D53—81no—
126C4×D53—81no—
127#88324—82no—
128#11900—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53C2×#10311yescharacteristic
133C22×C53#42111yescharacteristic
134C2×D53—21no—
135C2×D53—22no—
136C2×D53—21no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—41no—
141C2×D53—42no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—81no—
145C2×D53—81no—
146#6157—162no—
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—41no—
150C4×C53—82no—
151C4×C53#35412yes—
152Dic53—41no—
153Dic53—41no—
154Dic53—82no—
155#2563—531no—
156#1799—531no—
157#1459—531no—
158#1351—534no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53#152011yescharacteristic
162C2×C53#212211yescharacteristic
163C2×C53C2×#35411yescharacteristic
164D53—41no—
165D53—42no—
166D53—41no—
167C2×C4⋊Q8#615711yescharacteristic, complemented
168C2×C4×D4—531no—
169C2×#99—1061no—
170C2×C22⋊C8—1061no—
171#393—532no—
172#393—532no—
173#390—1061no—
174#382—1061no—
175#380—1061no—
176#380—1061no—
177#380—1061no—
178C4×C4⋊C4—531no—
179#342—532no—
180#324—1061no—
181#323—532no—
182C53#1190011yescharacteristic, complemented, Sylow
183C22×Q8C2×#615712yes—
184C22×D4—531no—
185C23×C4—531no—
186C23×C4—531no—
187C22×C8—2121no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4×D4—2121no—
191C4×D4—2121no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—2121no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C42—1061no—
203C2×C42C2×#615711yescharacteristic
204#99—2122no—
205C22⋊C8—1062no—
206C22⋊C8—2121no—
207#95—532no—
208#95—1062no—
209#95—1062no—
210#95—2121no—
211C24—531no—
212C24—531no—
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—532no—
216C2×D4—532no—
217C2×D4—2121no—
218C2×D4—2121no—
219C22×C4—22no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1062no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22⋊#615711yescharacteristic
232C22×C4C22×#615711yescharacteristic
233C2×C8—4242no—
234C2×C8—2122no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—4242no—
239C4⋊C4—2121no—
240C4⋊C4—2122no—
241C22⋊C4—4242no—
242C22⋊C4—4242no—
243C22⋊C4—2121no—
244C22⋊C4—2121no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C42—21no—
248C42—21no—
249C42—2122no—
250C23—532no—
251C23—531no—
252C23—531no—
253C23—1061no—
254C23—1061no—
255C23—1061no—
256C23—1061no—
257C23—2121no—
258C23—2121no—
259C23—2121no—
260C23—2121no—
261C23C2×C22⋊#615711yescharacteristic
262Q8—84no—
263D4—2122no—
264D4—2122no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—4241no—
276C2×C4—4241no—
277C2×C4—4242no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4#1313327511yescharacteristic
288C2×C4#1313327511yescharacteristic
289C2×C4#1313328012yes—
290C8—4242no—
291C22—21no—
292C22—21no—
293C22—1061no—
294C22—1062no—
295C22—1061no—
296C22—4241no—
297C22—4241no—
298C22—2121no—
299C22—2122no—
300C22—2121no—
301C22—2121no—
302C22—2121no—
303C22—2121no—
304C22—2121no—
305C22C2×#1313327511yescharacteristic
306C22C2×#1313328011yescharacteristic
307C22#1681162211yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—82no—
312C4—8482no—
313C4—4242no—
314C4—2121no—
315C4—2121no—
316C4#1681152912yes—
317C2—21no—
318C2—21no—
319C2—2121no—
320C2—2121no—
321C2—2122no—
322C2#2029015511yescharacteristic
323C2#2029097411yescharacteristic
324C2C2×#1681152911yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212D4
cosocleC2×#11719921C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11900
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289710C2semidirect
12. #16810534C4semidirect
18. C4×C4⋊Dic53C4semidirect
19. C2×C4⋊Q8×C53C4semidirect
167. C2×C4⋊Q8#6157semidirect
182. C53#11900semidirect

Maximal subgroups

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

subgroupindex
#202905762
#202899522
#202897102
#202798892
#1190053