Finite Group Databasebeta

← #27299273

Subgroups — #27299273

Order 13568.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277284C211yesmaximal, characteristic
2C2×#16810421C212yesmaximal
3#101×Dic53C212yesmaximal
4#20272842C211yesmaximal, characteristic
5C53×#1394C211yesmaximal, characteristic
6C2×#13133215C2211yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133182C411yescharacteristic
9C2×#13133136C2212yes—
10M16×Dic53—22no—
11#16810529C411yescharacteristic
12#16810520—22no—
13#16810492C411yescharacteristic
14#16810474—22no—
15C42×Dic53C411yescharacteristic
16#16810421C418yescomplemented
17#16810391—22no—
18C2×#101×C53C2212yes—
19C53×#391C2211yescharacteristic
20C22×Dic106C2×C411yescharacteristic
21C2×#11719917—22no—
22#13133215—21no—
23#13133215—21no—
24#13133215C2×C414yes—
25Q8×Dic53—41no—
26Q8×Dic53—41no—
27#13133182—41no—
28#13133182—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53D412yes—
31C2×#11719919—22no—
32C2×#11719919—21no—
33C2×#11719919—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D412yes—
40#13133136—22no—
41#13133136C2×C414yes—
42#13133127—42no—
43C8×Dic53—42no—
44#13133124—22no—
45C22×Q8×C53C2311yescharacteristic
46C2×M16×C53—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49#101×C53—22no—
50#101×C53C2×C414yes—
51#95×C53—22no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53C22×C412yes—
55C2×Q8×C53#4111yescharacteristic
56C2×Q8×C53#4111yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D412yes—
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C412yes—
65C22×Dic53C22⋊C411yescharacteristic
66C2×Dic106—21no—
67C2×Dic106—41no—
68C2×Dic106—41no—
69C2×Dic106C4212yes—
70M16×C53—24no—
71M16×C53—42no—
72C2×C8×C53—42no—
73C4⋊C4×C53—41no—
74C42×C53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—42no—
77#11719919—24no—
78#11719919—24no—
79#11719919—44no—
80#11719919—42no—
81C4×Dic53—24no—
82C4×Dic53—21no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—42no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53C4212yes—
91#11719917—24no—
92#11719917—42no—
93C2×#88322—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—41no—
97Q8×C53—41no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53C2×C4211yescharacteristic
108C2×C4×C53#11711yescharacteristic
109C2×C4×C53C4×D412yes—
110C2×C4×C53C4×D412yes—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—42no—
122C2×Dic53#9911yescharacteristic
123C2×Dic53#9911yescharacteristic
124Dic106—42no—
125Dic106—42no—
126C8×C53—44no—
127#88322—44no—
128C4×#356—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C4×C22⋊C411yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C22×C53#41011yescharacteristic
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53—81no—
140Dic53—24no—
141Dic53—21no—
142Dic53—41no—
143Dic53—42no—
144Dic53—42no—
145Dic53#35612yescomplemented
146C4×C4⋊Q8—531no—
147C2×#356—532no—
148C4×#101—532no—
149#1394—531no—
150#1394D5311yescharacteristic
151C2×C53—21no—
152C2×C53—21no—
153C2×C53C4×#9911yescharacteristic
154C2×C53C2×#35611yescharacteristic
155C2×C53#213211yescharacteristic
156C2×C4⋊Q8—531no—
157C2×C4×Q8—531no—
158C2×C4×Q8—531no—
159C2×#101—532no—
160C2×#101C2×D5312yes—
161C4×M16—1062no—
162#391—531no—
163#391C2×D5311yescharacteristic
164#389—1062no—
165#384—531no—
166C4×C4⋊C4—1062no—
167C43—531no—
168#356—538no—
169#343—1062no—
170C53C4×#35611yescharacteristic, complemented, Sylow
171C22×Q8—531no—
172C22×Q8C22×D5311yescharacteristic
173C2×M16—22no—
174C2×M16—1062no—
175C4⋊Q8—534no—
176C4⋊Q8—1061no—
177C4⋊Q8—1061no—
178C4×Q8—2121no—
179C4×Q8—2121no—
180C4×Q8—2121no—
181C4×Q8—2121no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4C4×D5311yescharacteristic
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—532no—
191C2×C42—1061no—
192C2×C42—1061no—
193C2×C42—1061no—
194C2×C42C4×D5311yescharacteristic
195#101—22no—
196#101—534no—
197#101—1062no—
198#101C4×D5314yes—
199#97—2122no—
200C4×C8—2122no—
201#95—22no—
202#95—1062no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—532no—
206C2×Q8—1061no—
207C2×Q8—2121no—
208C2×Q8—2121no—
209C2×Q8C2×C4×D5312yes—
210C2×Q8#1171994711yescharacteristic
211C2×Q8#1171994911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—531no—
215C22×C4—532no—
216C22×C4—531no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4D4×D5312yes—
221M16—24no—
222M16—42no—
223M16—1064no—
224M16—2122no—
225C2×C8—42no—
226C2×C8—2122no—
227C4⋊C4—41no—
228C4⋊C4—1064no—
229C4⋊C4—1064no—
230C4⋊C4—2121no—
231C4⋊C4—2122no—
232C4⋊C4—2122no—
233C4⋊C4—2124no—
234C42—41no—
235C42—532no—
236C42—1062no—
237C42—1062no—
238C42—1061no—
239C42—1061no—
240C42—1064no—
241C42—1061no—
242C42—2121no—
243C42—2121no—
244C42—2122no—
245C23C22⋊C4×D5311yescharacteristic
246Q8—42no—
247Q8—41no—
248Q8—41no—
249Q8—2122no—
250Q8—2122no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—531no—
261C2×C4—531no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1061no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2122no—
273C2×C4C42×D5311yescharacteristic
274C2×C4#1313317511yescharacteristic
275C2×C4#1313317712yes—
276C2×C4#1313317912yes—
277C8—44no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22#1681047011yescharacteristic
282C22#99×D5311yescharacteristic
283C22#1681059511yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—81no—
290C4—532no—
291C4—1064no—
292C4—1061no—
293C4—2122no—
294C4—2121no—
295C4—2122no—
296C2—21no—
297C2—21no—
298C2#2027450211yescharacteristic
299C2#356×D5311yescharacteristic
300C2#2027639711yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#1394D53
hypocenterC53C4×#356
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #16810421C4semidirect
145. Dic53#356semidirect
170. C53C4×#356semidirect

Maximal subgroups

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

subgroupindex
#202772842
C2×#168104212
#101×Dic532
#202728422
C53×#13942
C4×#35653