Finite Group Databasebeta

← #28299630

Subgroups — #28299630

Order 13568.

630 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290643C211yesmaximal, characteristic, complemented
2#20290500C211yesmaximal, characteristic, complemented
3#20289532C214yesmaximal, complemented
4#20277277C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133214C411yescharacteristic
12C2×#13133179—21no—
13C2×#13133179—21no—
14C2×#13133179C412yes—
15#16810524—21no—
16#16810524—21no—
17#16810524C2212yes—
18C42×Dic53C2211yescharacteristic
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133272—42no—
24C22×D212—21no—
25C22×D212—21no—
26C22×D212C2×C412yes—
27C22×C4×D53—22no—
28#13133214—24no—
29#13133179—44no—
30#13133179—44no—
31#13133179—42no—
32#13133179—42no—
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C42×C53C2×C411yescharacteristic
46C23×D53—22no—
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C2×D212—24no—
56C2×D212—24no—
57C2×D212—22no—
58C2×D212—42no—
59C2×D212—42no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212M1614yes—
65C2×C4×D53—44no—
66C2×C4×D53—44no—
67C2×C4×D53—44no—
68C2×#88324—24no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C42×C53—22no—
78#11719921—44no—
79#11719921—42no—
80#11719921—42no—
81#11719921—44no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53D812yes—
88C4×Dic53SD1612yes—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—24no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53#10212yes—
107C2×C4×C53#10412yes—
108C2×C4×C53C2×M1612yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53#10012yes—
118C2×Dic53#10212yes—
119C2×Dic53C2×D811yescharacteristic
120C2×Dic53C2×SD1611yescharacteristic
121C2×Dic53AGL(1,ℤ/8)12yes—
122D212—44no—
123D212—44no—
124D212—48no—
125D212—48no—
126C4×D53—88no—
127#88324—44no—
128#88324—82no—
129#7621—531nomaximal, Sylow
130C22×C53#40711yescharacteristic
131C22×C53#41011yescharacteristic
132C22×C53C2×#10011yescharacteristic
133C22×C53C2×#10211yescharacteristic
134C22×C53#41811yescharacteristic
135C22×C53C2×#10411yescharacteristic
136C22×C53#42111yescharacteristic
137C2×D53—44no—
138C2×D53—44no—
139C2×D53—44no—
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—82no—
143C2×D53—82no—
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—82no—
148Dic53—22no—
149Dic53—21no—
150Dic53—21no—
151Dic53—22no—
152Dic53—22no—
153C4×#127—531no—
154#1839—531no—
155#1767—531no—
156#1270—534no—
157C2×C53#146611yescharacteristic
158C2×C53#148611yescharacteristic
159C2×C53#149011yescharacteristic
160C2×C53#150211yescharacteristic
161C2×C53#151411yescharacteristic
162C2×C53#153011yescharacteristic
163C2×C53#154511yescharacteristic
164D53—84no—
165C2×#127—531no—
166C2×C4×D4—532no—
167C2×C4×D4—1061no—
168C2×C4×D4—1061no—
169C2×C4⋊C8—1061no—
170C2×#97—1061no—
171#390—532no—
172#390—1061no—
173#390—1061no—
174#384#615711yescharacteristic
175C43—531no—
176#336—532no—
177#336—532no—
178#336—1061no—
179#336—1061no—
180C53#762111yescharacteristic, complemented, Sylow
181C22×D4—532no—
182C22×D4—1061no—
183C22×D4—1061no—
184C23×C4—1062no—
185C22×C8—1062no—
186C22×C8—1062no—
187#127—1064no—
188C4×D4—2122no—
189C4×D4—2122no—
190C4×D4—2124no—
191C4×D4—2124no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×#615712yes—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42C2×#615711yescharacteristic
205C4⋊C8—1064no—
206#97—2122no—
207C24—1062no—
208C2×D4—534no—
209C2×D4—1064no—
210C2×D4—1064no—
211C2×D4—1062no—
212C2×D4—2122no—
213C2×D4—2122no—
214C2×D4—2122no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C22×C4—22no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—2124no—
224C22×C4—2124no—
225C22×C4—2124no—
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—1064no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C22⋊C4—2122no—
239C22⋊C4—2122no—
240C22⋊C4—2124no—
241C22⋊C4—2124no—
242C42—22no—
243C42—532no—
244C42—532no—
245C42—1062no—
246C42—1062no—
247C42—1062no—
248C42—2121no—
249C42—2121no—
250C23—1064no—
251C23—2122no—
252C23—2122no—
253C23—2122no—
254C23—2122no—
255C23—2122no—
256C23—2122no—
257C23C2×C22⋊#615711yescharacteristic
258D4—2124no—
259D4—2128no—
260D4—2124no—
261D4—2128no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—532no—
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—4248no—
285C2×C4#1313327912yes—
286C2×C4#1313328212yes—
287C2×C4D53⋊M1612yes—
288C8—4242no—
289C8—2124no—
290C22—4242no—
291C22—4242no—
292C22—4242no—
293C22—4242no—
294C22—2124no—
295C22—2124no—
296C22—2124no—
297C22#1681159511yescharacteristic
298C22#1681160411yescharacteristic
299C22D53⋊#10011yescharacteristic
300C22C2×#1313327911yescharacteristic
301C22#1681161511yescharacteristic
302C22C2×#1313328211yescharacteristic
303C22#1681162211yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—82no—
308C4—1061no—
309C4—1061no—
310C4—1062no—
311C4—1062no—
312C4—1062no—
313C2—4244no—
314C2#2028997611yescharacteristic
315C2#2029004211yescharacteristic
316C2#2029005611yescharacteristic
317C2#2029009611yescharacteristic
318C2#2029013611yescharacteristic
319C2#2029018611yescharacteristic
320C2#2029022311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7621
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20290500C2semidirect
3. #20289532C2semidirect
180. C53#7621semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202905002
#202895322
#202772772
#762153