Finite Group Databasebeta

← #27300101

Subgroups — #27300101

Order 13568.

432 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 101 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#20274709C212yesmaximal
4#20272841C212yesmaximal
5C2×C53×#356C211yesmaximal, characteristic
6C2×Q8×Dic53C411yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8#16810646—22no—
9C2×#13133136C2212yes—
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810509—22no—
13#16810509—22no—
14#16810489C411yescharacteristic
15#16810469C411yescharacteristic
16#16810422—22no—
17#16810391—22no—
18C2×C4⋊Q8×C53C2211yescharacteristic
19C2×#101×C53C2212yes—
20C53×#356—22no—
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#13133146—44no—
33#13133136—22no—
34#13133136—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—21no—
38#13133124—21no—
39#13133124—22no—
40C22×Q8×C53C2×C411yescharacteristic
41C22×Q8×C53C2311yescharacteristic
42C2×M16×C53—22no—
43C4⋊Q8×C53—22no—
44C4⋊Q8×C53D412yes—
45C4⋊Q8×C53Q812yes—
46C2×C4⋊C4×C53—22no—
47C2×C42×C53C2×C411yescharacteristic
48#101×C53—22no—
49#101×C53—22no—
50C2×Q8×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C4⋊C412yes—
56C2×Q8×C53C2×D411yescharacteristic
57C2×Q8×C53C2×Q811yescharacteristic
58C2×Q8×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C412yes—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68M16×C53—42no—
69M16×C53—42no—
70C2×C8×C53—42no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C42×C53—21no—
74C42×C53C4⋊C412yes—
75C4⋊Dic53—41no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C4⋊Dic53—41no—
79C4×Dic53—24no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53—41no—
83#11719917—42no—
84#11719917—42no—
85C2×#88322—42no—
86C23×C53C2×C22⋊C411yescharacteristic
87Q8×C53—41no—
88Q8×C53—41no—
89Q8×C53—42no—
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53C2×C4⋊C411yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×C4×C53#12112yes—
105C2×C4×C53C22⋊Q812yes—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53#9912yes—
117C8×C53—82no—
118#88322—82no—
119#12804—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#38011yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41011yescharacteristic
125C4×C53—22no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—22no—
129C4×C53—42no—
130C4×C53—42no—
131Dic53—22no—
132Dic53—41no—
133Dic53—42no—
134Dic53—81no—
135Dic53—81no—
136#2327—531no—
137C2×#356—531no—
138C2×#356D5311yescharacteristic
139#1801—532no—
140#1394—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53#179911yescharacteristic
144C2×C53#213311yescharacteristic
145C2×C53#213511yescharacteristic
146C2×C4⋊Q8C2×D5311yescharacteristic
147C2×C4×Q8—531no—
148C2×C4×Q8—531no—
149#423—1062no—
150C2×#101—532no—
151C2×#101C2×D5312yes—
152#391—531no—
153#391—531no—
154#387—1062no—
155#387—1062no—
156#382—531no—
157#376—531no—
158#356—22no—
159#356—1062no—
160#343—1062no—
161C53#1280411yescharacteristic, complemented, Sylow
162C22×Q8C4×D5311yescharacteristic
163C22×Q8C22×D5311yescharacteristic
164C2×M16—22no—
165C2×M16—1062no—
166C4⋊Q8—22no—
167C4⋊Q8Dic10612yes—
168C4⋊Q8C22⋊D5312yes—
169C4×Q8—2122no—
170C4×Q8—2122no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—532no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1061no—
176C2×C42—531no—
177C2×C42—532no—
178C2×C42—531no—
179C2×C42—1061no—
180C2×C42C4×D5311yescharacteristic
181C4⋊C8—2124no—
182#101—22no—
183#101—22no—
184#101—1062no—
185#101—1062no—
186#95—1061no—
187#95—1061no—
188#95—1062no—
189#95—1062no—
190#95—1062no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8#1171991912yes—
197C2×Q8C2×Dic10611yescharacteristic
198C2×Q8#1171994712yes—
199C2×Q8C2×C22⋊D5311yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—531no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4C2×C4×D5311yescharacteristic
208C22×C4D4×D5312yes—
209M16—42no—
210M16—42no—
211M16—2122no—
212M16—2122no—
213C2×C8—42no—
214C2×C8—2122no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—2121no—
218C4⋊C4—2122no—
219C4⋊C4—2121no—
220C4⋊C4—2122no—
221C42—21no—
222C42—1064no—
223C42—1062no—
224C42—2121no—
225C42—2122no—
226C42#1171991912yes—
227C23C22⋊C4×D5311yescharacteristic
228Q8—41no—
229Q8—42no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—532no—
244C2×C4—1061no—
245C2×C4—1062no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2122no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2122no—
253C2×C4—2121no—
254C2×C4C2×#1171991911yescharacteristic
255C2×C4#1313317311yescharacteristic
256C2×C4#1313318912yes—
257C2×C4#1313319412yes—
258C8—82no—
259C8—4242no—
260C22—21no—
261C22—21no—
262C22#1681048111yescharacteristic
263C22#99×D5311yescharacteristic
264C22#1681059511yescharacteristic
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—1062no—
272C4—4241no—
273C4—4241no—
274C4—2121no—
275C4—2122no—
276C2—21no—
277C2—21no—
278C2#2027470111yescharacteristic
279C2#2027640311yescharacteristic
280C2#2027640811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#356C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC2×#356D53
hypocenterC53#12804
LCSG ▷ C22×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
161. C53#12804semidirect

Maximal subgroups

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

subgroupindex
#202774352
C2×#168104222
#202747092
#202728412
C2×C53×#3562
#1280453