Finite Group Databasebeta

← #27279167

Subgroups — #27279167

Order 13568.

484 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283800C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic
3#20273162C212yesmaximal
4#20272608C212yesmaximal
5C53×#2399C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—21no—
8#16810680C2212yes—
9#16810676—21no—
10C2×#13133136—21no—
11C4×#11719917C2212yes—
12#16810392—21no—
13Q8⋊#88322—24no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#398—21no—
17C53×#398C411yescharacteristic
18C53×#398C411yescharacteristic
19C53×#395—21no—
20C53×#395C411yescharacteristic
21C53×#395C411yescharacteristic
22C4×C4⋊C4×C53—21no—
23C4×C4⋊C4×C53—21no—
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322D414yes—
29#13133136—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33#13133128—22no—
34C4×#88322—22no—
35C4×#88322—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×Q8×C53D411yescharacteristic
38C4×Q8×C53—22no—
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41C4×Q8×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C42×C53—21no—
52C2×C42×C53—21no—
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C2×Q8×C53—22no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—41no—
67C2×Q8×C53—42no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C42×C53—24no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53C2×D412yes—
96C42×C53#4112yes—
97#11719917—24no—
98#11719917—42no—
99C2×#88322—24no—
100C2×#88322—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102Q8×C53—24no—
103Q8×C53—44no—
104Q8×C53—42no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53#10412yes—
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12112yes—
134C2×C4×C53#12311yescharacteristic
135#88322—44no—
136#88322—44no—
137#9030—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53#38611yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—22no—
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—42no—
148C4×C53—42no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53#45612yes—
155#2983—531no—
156#2399D5311yescharacteristic
157C4×#101—531no—
158#1491—532no—
159#1300—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53#187111yescharacteristic
163C2×C53#188611yescharacteristic
164C2×C53#189111yescharacteristic
165C2×C4×Q8C2×D5311yescharacteristic
166C2×C4×Q8C2×D5311yescharacteristic
167#446—1064no—
168#432—532no—
169#432—1061no—
170#431—1061no—
171C2×#101—1061no—
172C4×M16—532no—
173#398—21no—
174#398Dic5311yescharacteristic
175#398Dic5311yescharacteristic
176#395—21no—
177#395Dic5311yescharacteristic
178#395Dic5311yescharacteristic
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181#376C2×D5311yescharacteristic
182#343—1061no—
183Q8⋊C8—1064no—
184C53#903011yescharacteristic, complemented, Sylow
185C22×Q8C2×Dic5311yescharacteristic
186C22×Q8C22⋊D5311yescharacteristic
187C2×M16—1062no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C4⋊C4C2×Dic5311yescharacteristic
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊D5311yescharacteristic
204C2×C42C22⋊D5311yescharacteristic
205C2×C42C22×D5311yescharacteristic
206C4⋊C8—534no—
207C4⋊C8—1062no—
208C4⋊C8—1062no—
209#101—1064no—
210C22⋊C8—1062no—
211#97—1062no—
212#97—1062no—
213#97—1062no—
214C4×C8—1062no—
215C4×C8—1062no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×Dic5311yescharacteristic
222#95C2×Dic5311yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C22⋊Dic5311yescharacteristic
233C22×C4C22⋊Dic5311yescharacteristic
234C22×C4C22⋊Dic5311yescharacteristic
235C22×C4C22⋊Dic5311yescharacteristic
236C22×C4C22×Dic5311yescharacteristic
237C22×C4C2×C22⋊D5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239M16—1064no—
240M16—2122no—
241C2×C8—1064no—
242C2×C8—2122no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C42—24no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42D4×D5312yes—
260C42#1171994712yes—
261C23C2×C22⋊Dic5311yescharacteristic
262Q8—24no—
263Q8—44no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#1313314412yes—
291C2×C4D4×Dic5312yes—
292C2×C4#1313318511yescharacteristic
293C2×C4#1313318912yes—
294C2×C4#1313319811yescharacteristic
295C8—2124no—
296C8—2124no—
297C22—21no—
298C22—21no—
299C22#1681050311yescharacteristic
300C22C2×#1313314411yescharacteristic
301C22#1681064011yescharacteristic
302C4—22no—
303C4—22no—
304C4—21no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4#1681080512yes—
314C2—21no—
315C2—21no—
316C2#2027501811yescharacteristic
317C2#2027510911yescharacteristic
318C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2399C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2399D53
hypocenterC53#9030
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
184. C53#9030semidirect

Maximal subgroups

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

subgroupindex
#202838002
C4×#131331362
#202731622
#202726082
C53×#23992
#903053