Finite Group Databasebeta

← #28309135

Subgroups — #28309135

Order 13568.

482 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13133278C211yesmaximal, characteristic, complemented
2#20289977C211yesmaximal, characteristic, complemented
3#20289698C214yesmaximal, complemented
4#20279717C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133278—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133214C411yescharacteristic
12C2×#13133205—21no—
13C2×C4×Dic106C2211yescharacteristic
14#16810541C412yes—
15#16810541C2212yes—
16#16810525—21no—
17#16810524—21no—
18#16810524—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C4⋊#88324—41no—
22#13133278—24no—
23C22⋊#88324—22no—
24#13133272—41no—
25#13133272—41no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212C2×C412yes—
31#13133214—41no—
32#13133214—41no—
33#13133205—44no—
34C4×Dic106—41no—
35C4×Dic106—41no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919C2×C411yescharacteristic
38C2×#11719919C2311yescharacteristic
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48#95×C53C2×C412yes—
49C23×D53—22no—
50C22×C4×C53—22no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C2×D212—42no—
59C2×D212—42no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212—42no—
66C2×D212—82no—
67C2×Dic106—22no—
68C2×Dic106—22no—
69C2×Dic106—41no—
70#11719940—24no—
71#11719940—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—42no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—42no—
78#11719921—44no—
79#11719921—44no—
80#11719921—82no—
81#11719921—82no—
82C4⋊Dic53—41no—
83#11719919—41no—
84#11719919—41no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—42no—
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—22no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#10412yes—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53AGL(1,ℤ/8)12yes—
118D212—82no—
119D212—84no—
120D212—84no—
121D212—82no—
122Dic106—44no—
123#88324—44no—
124#88324—82no—
125#11781—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53C2×#9911yescharacteristic
129C22×C53#41811yescharacteristic
130C22×C53C2×#10411yescharacteristic
131C2×D53—82no—
132C2×D53—82no—
133C2×D53—82no—
134C2×D53—82no—
135C2×D53—82no—
136C2×D53—82no—
137C2×D53—82no—
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—41no—
141C4×C53—82no—
142C4×C53#35512yes—
143Dic53—22no—
144Dic53—41no—
145Dic53—41no—
146Dic53—41no—
147#2548—531no—
148C4×#101—531no—
149#1467—531no—
150#1345—534no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#150711yescharacteristic
154C2×C53#212211yescharacteristic
155C2×C53C2×#35511yescharacteristic
156D53—162no—
157C2×#127—531no—
158C2×#124—1061no—
159C2×C4×Q8—531no—
160#432—1061no—
161C2×#101—1061no—
162C4×M16—1061no—
163#394—532no—
164#394—532no—
165#390—1061no—
166#390—1061no—
167#390—1061no—
168C4×C4⋊C4#615711yescharacteristic
169#343—532no—
170#324—532no—
171#324—1061no—
172C53#1178111yescharacteristic, complemented, Sylow
173C22×Q8—531no—
174C22×D4—532no—
175C22×D4—1061no—
176C22×D4—1061no—
177C2×M16—1062no—
178#127—2121no—
179#127—2121no—
180#124—2124no—
181C4×Q8—2121no—
182C4×Q8—2121no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—531no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C42—21no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42C2×#615711yescharacteristic
195C4⋊C8—2121no—
196#101—1064no—
197C22⋊C8—1062no—
198#97—2121no—
199#97—2121no—
200C4×C8—2121no—
201#95C2×#615712yes—
202C24—1062no—
203C2×Q8—1062no—
204C2×Q8—1062no—
205C2×Q8—2121no—
206C2×D4—4242no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2122no—
210C2×D4—2122no—
211C2×D4—2122no—
212C2×D4—2121no—
213C2×D4—2121no—
214C2×D4—2122no—
215C22×C4—22no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223M16—1064no—
224M16—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—42no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C22⋊C4—4242no—
232C22⋊C4—4242no—
233C22⋊C4—2124no—
234C22⋊C4—2124no—
235C42—21no—
236C42—21no—
237C42—42no—
238C42—1061no—
239C42—1061no—
240C42—1061no—
241C42—1061no—
242C23—2122no—
243C23—2122no—
244C23—2122no—
245C23—2122no—
246C23—2122no—
247C23—2122no—
248C23—2122no—
249C23C2×C22⋊#615711yescharacteristic
250Q8—2124no—
251D4—4244no—
252D4—4242no—
253D4—4244no—
254D4—4242no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—532no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4#1313327611yescharacteristic
274C2×C4#1313327611yescharacteristic
275C2×C4#1313328212yes—
276C8—4242no—
277C8—2124no—
278C22—21no—
279C22—21no—
280C22—4242no—
281C22—4242no—
282C22—4242no—
283C22—4242no—
284C22—4242no—
285C22—4242no—
286C22—4242no—
287C22C2×#1313327611yescharacteristic
288C22#1681161511yescharacteristic
289C22C2×#1313328211yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—82no—
294C4—1062no—
295C4—2121no—
296C4—2121no—
297C4—2121no—
298C4#1681153212yes—
299C2—21no—
300C2—21no—
301C2—8482no—
302C2#2029011211yescharacteristic
303C2#2029097611yescharacteristic
304C2C2×#1681153211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×#11719919C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#11781
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4×#13133278C2semidirect
2. #20289977C2semidirect
3. #20289698C2semidirect
172. C53#11781semidirect

Maximal subgroups

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

subgroupindex
C4×#131332782
#202899772
#202896982
#202797172
#1178153