Finite Group Databasebeta

← #27294683

Subgroups — #27294683

Order 13568.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133136C211yesmaximal, characteristic, complemented
2#20273059C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2548C211yesmaximal, characteristic
5#16810680—21no—
6C2×#13133136—21no—
7C4×#11719917—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C2×#127×C53C411yescharacteristic
12C2×#124×C53—21no—
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#394C412yes—
15C53×#394C2212yes—
16C53×#390—21no—
17C53×#390—21no—
18C53×#390—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22#13133136—24no—
23C22⋊#88322—22no—
24#13133128—41no—
25#13133128—41no—
26C4×#88322—41no—
27C22×Q8×C53D411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C412yes—
31#127×C53—41no—
32#127×C53—41no—
33#124×C53—44no—
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48#95×C53C2×C412yes—
49C24×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—41no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—42no—
61C2×D4×C53—82no—
62C22×C4×C53—22no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—82no—
77C22⋊C4×C53—82no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—42no—
85#11719917—24no—
86#11719917—42no—
87C2×#88322—42no—
88C2×#88322—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97Q8×C53—44no—
98D4×C53—82no—
99D4×C53—84no—
100D4×C53—84no—
101D4×C53—82no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53#9911yescharacteristic
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#10412yes—
122C2×C4×C53AGL(1,ℤ/8)12yes—
123#88322—44no—
124#88322—82no—
125#11781—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—82no—
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53C2×#9911yescharacteristic
136C22×C53#41811yescharacteristic
137C22×C53C2×#10411yescharacteristic
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—82no—
146C4×C53#35512yes—
147#2548D5311yescharacteristic
148C4×#101—531no—
149#1467—531no—
150#1345—534no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—162no—
154C2×C53#150711yescharacteristic
155C2×C53#212211yescharacteristic
156C2×C53C2×#35511yescharacteristic
157C2×#127Dic5311yescharacteristic
158C2×#124—21no—
159C2×C4×Q8C2×D5311yescharacteristic
160#432—1061no—
161C2×#101—1061no—
162C4×M16—1061no—
163#394Dic5312yes—
164#394C2×D5312yes—
165#390—21no—
166#390—21no—
167#390—21no—
168C4×C4⋊C4Dic5311yescharacteristic
169#343—532no—
170#324—532no—
171#324—1061no—
172C53#1178111yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D5311yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×Dic5312yes—
177C2×M16—1062no—
178#127—41no—
179#127—41no—
180#124—44no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×Dic5311yescharacteristic
185C2×C4⋊C4C22⋊D5311yescharacteristic
186C2×C4⋊C4C22×D5311yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—21no—
192C2×C42C2×Dic5311yescharacteristic
193C2×C42C22⋊D5311yescharacteristic
194C2×C42C22⋊D5311yescharacteristic
195C4⋊C8—2121no—
196#101—1064no—
197C22⋊C8—1062no—
198#97—2121no—
199#97—2121no—
200C4×C8—2121no—
201#95C2×Dic5312yes—
202C24—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C22×C4—22no—
216C22×C4C22⋊Dic5311yescharacteristic
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22⋊Dic5311yescharacteristic
219C22×C4C22⋊Dic5311yescharacteristic
220C22×C4C22×Dic5311yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222C22×C4C2×C22⋊D5311yescharacteristic
223M16—1064no—
224M16—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊Dic5311yescharacteristic
250Q8—44no—
251D4—84no—
252D4—82no—
253D4—84no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#1313313211yescharacteristic
273C2×C4#1313313211yescharacteristic
274C2×C4#1313314412yes—
275C2×C4#1313324312yes—
276C8—4242no—
277C8—2124no—
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#1313313211yescharacteristic
288C22#1681062811yescharacteristic
289C22C2×#1313314411yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4#1681042012yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2#2027323111yescharacteristic
303C2#2027636511yescharacteristic
304C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2548C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2548D53
hypocenterC53#11781
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C4×#13133136C2semidirect
2. #20273059C2semidirect
3. #20272718C2semidirect
172. C53#11781semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730592
#202727182
C53×#25482
#1178153