Finite Group Databasebeta

← #28308078

Subgroups — #28308078

Order 13568.

480 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 92 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#20289951C212yesmaximal
2#20289656C212yesmaximal, complemented
3#20289646C212yesmaximal, complemented
4#20277262C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×#13133204C411yescharacteristic
9C2×#13133182C411yescharacteristic
10C2×D4×Dic53C2211yescharacteristic
11#16810529C2211yescharacteristic
12#16810511—22no—
13#16810488C2211yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic
16C53×#390C411yescharacteristic
17C22×#88324—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C22×Dic106C2×C411yescharacteristic
21C23×Dic53D412yes—
22#13133204—22no—
23#13133204C814yescomplemented
24#13133182—41no—
25#13133182—41no—
26D4×Dic53—41no—
27D4×Dic53—41no—
28C2×C4⋊Dic53C2311yescharacteristic
29C2×#11719919—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53D412yes—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37#13133124—22no—
38#13133124—22no—
39C22×D4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C24×C53C22⋊C412yes—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53C2×C812yes—
47C2×D4×C53M1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53—44no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D412yes—
59C2×Dic106—21no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106C2×C812yescomplemented
63C2×#88324—44no—
64C2×#88324—84no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—44no—
67C42×C53—41no—
68C22⋊Dic53—28no—
69C22⋊Dic53—42no—
70C4⋊Dic53—41no—
71#11719919—42no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—21no—
75C4×Dic53—42no—
76C4×Dic53—42no—
77C4×Dic53—41no—
78C4×Dic53—41no—
79C4×Dic53C2×C812yescomplemented
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—42no—
83C23×C53—42no—
84C23×C53C22⋊C814yes—
85C23×C53C2×C22⋊C411yescharacteristic
86D4×C53—42no—
87D4×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53C22×C811yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—44no—
107C2×Dic53—42no—
108C2×Dic53—41no—
109C2×Dic53—82no—
110C2×Dic53#9911yescharacteristic
111C2×Dic53#9911yescharacteristic
112Dic106—42no—
113Dic106—42no—
114#88324—84no—
115#11477—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—24no—
119C22×C53—44no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—82no—
123C22×C53C2×C22⋊C811yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53#41311yescharacteristic
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—42no—
129C4×C53—81no—
130Dic53—21no—
131Dic53—42no—
132Dic53—42no—
133Dic53—41no—
134Dic53—42no—
135Dic53#35412yes—
136C4×#124—531no—
137#1459—532no—
138#1324—532no—
139#1319—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—44no—
143C2×C53#145911yescharacteristic
144C2×C53C2×#35411yescharacteristic
145C2×C53#213211yescharacteristic
146C2×#124—531no—
147C2×C4×Q8—531no—
148C2×C4×D4—531no—
149C2×C22⋊C8—1062no—
150#391—531no—
151#390#615711yescharacteristic
152#388—1062no—
153#382—531no—
154C4×C22⋊C4—1062no—
155C43—531no—
156#324—534no—
157#323—1062no—
158C53#1147711yescharacteristic, complemented, Sylow
159C22×Q8—531no—
160C22×D4C2×#615711yescharacteristic
161C23×C4—532no—
162C22×C8—2122no—
163#124—534no—
164#124—1062no—
165C4×Q8—2121no—
166C4×Q8—2121no—
167C4×D4—2121no—
168C4×D4—2121no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4C2×#615711yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—1062no—
175C2×C42—531no—
176C2×C42—531no—
177C2×C42—1061no—
178C2×C42—1061no—
179C2×C42—1061no—
180C2×C42C2×#615711yescharacteristic
181C22⋊C8—1064no—
182C22⋊C8—2122no—
183#95—1062no—
184#95—1062no—
185C24C22⋊#615712yes—
186C2×Q8—532no—
187C2×Q8—1061no—
188C2×Q8—2121no—
189C2×Q8—2121no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4D53⋊C812yes—
193C2×D4#1171993612yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—532no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—2124no—
204C22×C4C22×#615711yescharacteristic
205C2×C8—4244no—
206C2×C8—2124no—
207C4⋊C4—41no—
208C4⋊C4—2121no—
209C4⋊C4—2122no—
210C22⋊C4—44no—
211C22⋊C4—1068no—
212C22⋊C4—2122no—
213C42—41no—
214C42—532no—
215C42—1061no—
216C42—1061no—
217C42—1061no—
218C42—2122no—
219C42—2121no—
220C42—2121no—
221C42—2122no—
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23#1313327414yes—
227C23C2×C22⋊#615711yescharacteristic
228Q8—2122no—
229Q8—2122no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—531no—
240C2×C4—531no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1061no—
245C2×C4—1062no—
246C2×C4—4242no—
247C2×C4—2124no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2122no—
254C2×C4C2×D53⋊C811yescharacteristic
255C2×C4C2×#1171993611yescharacteristic
256C8—4244no—
257C22—21no—
258C22—21no—
259C22—24no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22C2×#1313327411yescharacteristic
265C22D53⋊#9911yescharacteristic
266C22#1681161011yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—81no—
271C4—532no—
272C4—1061no—
273C4—2122no—
274C4—2122no—
275C4—2122no—
276C4—2121no—
277C2—21no—
278C2—21no—
279C2—44no—
280C2#2028995311yescharacteristic
281C2D53⋊#35411yescharacteristic
282C2#2029100711yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenter#390#6157
hypocenterC53#11477
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20289656C2semidirect
3. #20289646C2semidirect
23. #13133204C8semidirect
62. C2×Dic106C2×C8semidirect
79. C4×Dic53C2×C8semidirect
158. C53#11477semidirect

Maximal subgroups

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

subgroupindex
#202899512
#202896562
#202896462
#202772622
#1147753