Finite Group Databasebeta

← #27299280

Subgroups — #27299280

Order 13568.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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#20277435C211yesmaximal, characteristic
2C2×#16810422C211yesmaximal, characteristic
3#101×Dic53C212yesmaximal
4#20272841C212yesmaximal
5C2×C53×#356C211yesmaximal, characteristic
6C2×Q8×Dic53C411yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133136C2212yes—
9M16×Dic53—22no—
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810509—22no—
13#16810509—22no—
14#16810489C411yescharacteristic
15#16810469C411yescharacteristic
16#16810422C414yescomplemented
17#16810391—22no—
18C2×C4⋊Q8×C53C2211yescharacteristic
19C2×#101×C53C2212yes—
20C53×#356C414yescomplemented
21C2×#11719917—22no—
22Q8×Dic53—42no—
23Q8×Dic53—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53D412yes—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32#13133136—22no—
33#13133136C2×C414yes—
34#13133127—42no—
35C8×Dic53—42no—
36#13133124—22no—
37#13133124—22no—
38#13133124—21no—
39#13133124—21no—
40#13133124—22no—
41C22×Q8×C53C2×C411yescharacteristic
42C22×Q8×C53C2311yescharacteristic
43C2×M16×C53—22no—
44C4⋊Q8×C53—22no—
45C4⋊Q8×C53C2×C412yes—
46C4⋊Q8×C53C2×C412yes—
47C2×C4⋊C4×C53—22no—
48C2×C42×C53C2×C411yescharacteristic
49#101×C53—22no—
50#101×C53C2×C414yes—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53C4212yes—
57C2×Q8×C53C22×C411yescharacteristic
58C2×Q8×C53C22×C411yescharacteristic
59C2×Q8×C53#4112yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D412yes—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C412yes—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69M16×C53—24no—
70M16×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C42×C53—21no—
75C42×C53C4212yes—
76C4⋊Dic53—41no—
77C4⋊Dic53—42no—
78C4⋊Dic53—42no—
79C4⋊Dic53—41no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—42no—
84C4×Dic53—41no—
85#11719917—24no—
86#11719917—42no—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—41no—
90Q8×C53—41no—
91Q8×C53—42no—
92Q8×C53—42no—
93Q8×C53—42no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53C2×C4211yescharacteristic
105C2×C4×C53#11711yescharacteristic
106C2×C4×C53C4×D412yes—
107C2×C4×C53C4×D412yes—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53#9912yes—
119C8×C53—44no—
120#88322—44no—
121#12631—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53C4×C22⋊C411yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53#41011yescharacteristic
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—42no—
132C4×C53—42no—
133Dic53—22no—
134Dic53—41no—
135Dic53—42no—
136Dic53—81no—
137Dic53—81no—
138#2327—531no—
139C2×#356—531no—
140C2×#356D5311yescharacteristic
141C4×#101—532no—
142#1394—532no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53C4×#9911yescharacteristic
146C2×C53#213311yescharacteristic
147C2×C53#213511yescharacteristic
148C2×C4⋊Q8C2×D5311yescharacteristic
149C2×C4×Q8—531no—
150C2×C4×Q8—531no—
151C2×#101—532no—
152C2×#101C2×D5312yes—
153C4×M16—1062no—
154#391—531no—
155#391—531no—
156#387—1062no—
157#387—1062no—
158#382—531no—
159#376—531no—
160#356—534no—
161#356Dic5314yescomplemented
162#343—1062no—
163C53#1263111yescharacteristic, complemented, Sylow
164C22×Q8C4×D5311yescharacteristic
165C22×Q8C22×D5311yescharacteristic
166C2×M16—22no—
167C2×M16—1062no—
168C4⋊Q8—22no—
169C4⋊Q8C4×D5312yes—
170C4⋊Q8C2×Dic5312yes—
171C4×Q8—2122no—
172C4×Q8—2122no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1061no—
178C2×C42—531no—
179C2×C42—532no—
180C2×C42—531no—
181C2×C42—1061no—
182C2×C42C4×D5311yescharacteristic
183#101—22no—
184#101—534no—
185#101—1062no—
186#101C2×Dic5314yes—
187#97—2122no—
188C4×C8—2122no—
189#95—1061no—
190#95—1061no—
191#95—1062no—
192#95—1062no—
193#95—1062no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C4×Dic5312yes—
200C2×Q8C2×C4×D5311yescharacteristic
201C2×Q8#1171994712yes—
202C2×Q8C22×Dic5311yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—531no—
206C22×C4—532no—
207C22×C4—531no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4C2×C4×D5311yescharacteristic
211C22×C4D4×D5312yes—
212M16—24no—
213M16—42no—
214M16—1064no—
215M16—2122no—
216C2×C8—42no—
217C2×C8—2122no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—2121no—
221C4⋊C4—2122no—
222C4⋊C4—2121no—
223C4⋊C4—2122no—
224C42—21no—
225C42—1062no—
226C42—1062no—
227C42—1062no—
228C42—2121no—
229C42—2122no—
230C42C4×Dic5312yes—
231C23C22⋊C4×D5311yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235Q8—42no—
236Q8—42no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—532no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2122no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2121no—
258C2×C4C2×C4×Dic5311yescharacteristic
259C2×C4#1313317311yescharacteristic
260C2×C4#1313317712yes—
261C2×C4D4×Dic5312yes—
262C8—44no—
263C8—2124no—
264C22—21no—
265C22—21no—
266C22C22⋊C4×Dic5311yescharacteristic
267C22#99×D5311yescharacteristic
268C22#1681059511yescharacteristic
269C4—22no—
270C4—22no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1062no—
276C4—4241no—
277C4—4241no—
278C4—2121no—
279C4—2122no—
280C2—21no—
281C2—21no—
282C2#99×Dic5311yescharacteristic
283C2#2027640311yescharacteristic
284C2#2027640811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC2×C53×#356C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC2×#356D53
hypocenterC53#12631
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
16. #16810422C4semidirect
20. C53×#356C4semidirect
161. #356Dic53semidirect
163. C53#12631semidirect

Maximal subgroups

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

subgroupindex
#202774352
C2×#168104222
#101×Dic532
#202728412
C2×C53×#3562
#1263153