Finite Group Databasebeta

← #27299277

Subgroups — #27299277

Order 13568.

464 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 113 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#20277433C211yesmaximal, 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#16810510—22no—
13#16810505—22no—
14#16810488C411yescharacteristic
15#16810469C411yescharacteristic
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—42no—
26#13133182—42no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×#11719919—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35#13133136—22no—
36#13133136C2×C414yes—
37#13133127—42no—
38C8×Dic53—42no—
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42#13133124—21no—
43C22×Q8×C53C2311yescharacteristic
44C2×M16×C53—22no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47#101×C53—22no—
48#101×C53C2×C414yes—
49#95×C53—22no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C22×C412yes—
53C2×Q8×C53#4111yescharacteristic
54C2×Q8×C53#4111yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D412yes—
59C22×Dic53—21no—
60C22×Dic53—21no—
61C22×Dic53C22⋊C412yes—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C2×Dic106—21no—
65C2×Dic106—41no—
66C2×Dic106—41no—
67C2×Dic106C4212yes—
68M16×C53—24no—
69M16×C53—42no—
70C2×C8×C53—42no—
71C4⋊C4×C53—41no—
72C42×C53—41no—
73C4⋊Dic53—41no—
74C4⋊Dic53—42no—
75#11719919—24no—
76#11719919—24no—
77#11719919—42no—
78C4×Dic53—21no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—42no—
83C4×Dic53C4212yes—
84#11719917—24no—
85#11719917—42no—
86C2×#88322—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—42no—
89Q8×C53—41no—
90Q8×C53—41no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53C2×C4211yescharacteristic
101C2×C4×C53#11711yescharacteristic
102C2×C4×C53C4×D412yes—
103C2×C4×C53C4×D412yes—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53#9912yes—
116Dic106—42no—
117Dic106—42no—
118C8×C53—44no—
119#88322—44no—
120#12631—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53C4×C22⋊C411yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53#41011yescharacteristic
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—81no—
131C4×C53—81no—
132Dic53—22no—
133Dic53—42no—
134Dic53—41no—
135Dic53—42no—
136Dic53—42no—
137#2327—531no—
138C2×#356—532no—
139C4×#101—532no—
140#1394—531no—
141#1394D5311yescharacteristic
142C2×C53—21no—
143C2×C53—21no—
144C2×C53C4×#9911yescharacteristic
145C2×C53#213311yescharacteristic
146C2×C53#213511yescharacteristic
147C2×C4⋊Q8—531no—
148C2×C4×Q8—531no—
149C2×C4×Q8—531no—
150C2×#101—532no—
151C2×#101C2×D5312yes—
152C4×M16—1062no—
153#391—531no—
154#391C2×D5311yescharacteristic
155#387—1062no—
156#387—1062no—
157#382—531no—
158#376—531no—
159#356—538no—
160#343—1062no—
161C53#1263111yescharacteristic, complemented, Sylow
162C22×Q8—531no—
163C22×Q8C22×D5311yescharacteristic
164C2×M16—22no—
165C2×M16—1062no—
166C4⋊Q8—534no—
167C4⋊Q8—1061no—
168C4⋊Q8—1061no—
169C4×Q8—2122no—
170C4×Q8—2122no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C42—531no—
177C2×C42—531no—
178C2×C42—532no—
179C2×C42—1061no—
180C2×C42C4×D5311yescharacteristic
181#101—22no—
182#101—534no—
183#101—1062no—
184#101C4×D5314yes—
185#97—2122no—
186C4×C8—2122no—
187#95—22no—
188#95—1061no—
189#95—1061no—
190#95—1062no—
191#95—1062no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—532no—
195C2×Q8—1061no—
196C2×Q8—2121no—
197C2×Q8—2121no—
198C2×Q8C2×C4×D5312yes—
199C2×Q8#1171994711yescharacteristic
200C2×Q8#1171994911yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—531no—
204C22×C4—532no—
205C22×C4—531no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4D4×D5312yes—
210M16—24no—
211M16—42no—
212M16—1064no—
213M16—2122no—
214C2×C8—42no—
215C2×C8—2122no—
216C4⋊C4—41no—
217C4⋊C4—1064no—
218C4⋊C4—1064no—
219C4⋊C4—2121no—
220C4⋊C4—2122no—
221C4⋊C4—2122no—
222C42—41no—
223C42—532no—
224C42—1062no—
225C42—1062no—
226C42—1062no—
227C42—1061no—
228C42—2122no—
229C23C22⋊C4×D5311yescharacteristic
230Q8—42no—
231Q8—41no—
232Q8—41no—
233Q8—2122no—
234Q8—2122no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—532no—
245C2×C4—1061no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2122no—
255C2×C4—2121no—
256C2×C4C42×D5311yescharacteristic
257C2×C4#1313317511yescharacteristic
258C2×C4#1313317712yes—
259C2×C4#1313317912yes—
260C8—44no—
261C8—2124no—
262C22—21no—
263C22—21no—
264C22#1681047011yescharacteristic
265C22#99×D5311yescharacteristic
266C22#1681059511yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—81no—
272C4—81no—
273C4—1062no—
274C4—2122no—
275C4—2122no—
276C4—2121no—
277C4—2122no—
278C2—21no—
279C2—21no—
280C2#2027450211yescharacteristic
281C2#2027640111yescharacteristic
282C2#2027640811yescharacteristic
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
hypocenterC53#12631
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
161. C53#12631semidirect

Maximal subgroups

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

subgroupindex
#202774332
C2×#168104212
#101×Dic532
#202728422
C53×#13942
#1263153