Finite Group Databasebeta

← #28304360

Subgroups — #28304360

Order 13568.

474 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 100 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#20291611C211yesmaximal, characteristic
2#20290017C212yesmaximal
3#20289958C211yesmaximal, characteristic
4#20289539C212yesmaximal
5#20277435C211yesmaximal, characteristic
6#16811674—24no—
7#16811643—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10Q8⋊#88324—24no—
11#16811468—22no—
12C2×Q8×Dic53C2211yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810530C411yescharacteristic
16#16810509—22no—
17#16810509—22no—
18#16810489C411yescharacteristic
19#16810469C2211yescharacteristic
20C2×C4⋊Q8×C53C411yescharacteristic
21#13133284—24no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26#13133272—24no—
27C4×#88324—22no—
28C4×#88324D414yes—
29Q8×Dic53—22no—
30Q8×Dic53—22no—
31Q8×Dic53—22no—
32Q8×Dic53—22no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53D412yes—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D412yes—
40C2×C4×Dic53C2311yescharacteristic
41#13133124—22no—
42#13133124—22no—
43#13133124—22no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C22×Q8×C53C2×C411yescharacteristic
48C4⋊Q8×C53—22no—
49C4⋊Q8×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52C2×Q8×C53—22no—
53C2×Q8×C53—21no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C412yes—
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×#88324—24no—
70C2×#88324—42no—
71C4⋊C4×C53—28no—
72C42×C53—21no—
73C42×C53M1612yes—
74C4⋊Dic53—22no—
75C4⋊Dic53—24no—
76C4⋊Dic53—22no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—42no—
83C4×Dic53C2×D412yes—
84C4×Dic53#4112yes—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—24no—
87Q8×C53—42no—
88Q8×C53—42no—
89Q8×C53—42no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53C4×D412yes—
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53#13112yes—
103C2×Dic53—22no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53#12011yescharacteristic
113C2×Dic53#12112yes—
114C2×Dic53#12311yescharacteristic
115#88324—44no—
116#88324—82no—
117#9104—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#38611yescharacteristic
121C22×C53#40711yescharacteristic
122C22×C53#40811yescharacteristic
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—42no—
126C4×C53#43512yes—
127C4×C53#43612yes—
128Dic53—22no—
129Dic53—21no—
130Dic53—21no—
131Dic53—42no—
132Dic53—81no—
133Dic53—81no—
134#2983—531no—
135#2327—531no—
136#1479—532no—
137#1461—531no—
138#1274—532no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#187011yescharacteristic
142C2×C53#190911yescharacteristic
143C2×C53#191211yescharacteristic
144C2×C4⋊Q8#615711yescharacteristic
145C2×C4×Q8—531no—
146C2×C4×Q8—531no—
147#446—1064no—
148#432—1062no—
149#431—532no—
150#431—532no—
151#391—531no—
152#391—531no—
153#387—1062no—
154#387—1062no—
155#382—531no—
156#376—531no—
157Q8⋊C8—1064no—
158#324—1062no—
159C53#910411yescharacteristic, complemented, Sylow
160C22×Q8C2×#615711yescharacteristic
161C22×Q8C2×#615711yescharacteristic
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4×Q8—1062no—
165C4×Q8—1062no—
166C4×Q8—1062no—
167C4×Q8—1062no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—1061no—
173C2×C42—531no—
174C2×C42—532no—
175C2×C42—531no—
176C2×C42—1061no—
177C2×C42C2×#615711yescharacteristic
178C4⋊C8—1062no—
179C4⋊C8—1062no—
180C4⋊C8—1064no—
181C22⋊C8—1062no—
182C22⋊C8—1062no—
183#97—1064no—
184C4×C8—534no—
185C4×C8—1062no—
186#95—1061no—
187#95—1061no—
188#95—1062no—
189#95—1062no—
190#95—1062no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8#1171994012yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4C22⋊#615712yes—
207C22×C4C22×#615711yescharacteristic
208C2×C8—1064no—
209C2×C8—2122no—
210C4⋊C4—28no—
211C4⋊C4—1064no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C42—21no—
217C42—532no—
218C42—532no—
219C42—1062no—
220C42—1062no—
221C42—1062no—
222C42—2122no—
223C42#1171994012yes—
224C23C2×C22⋊#615711yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—42no—
228Q8—42no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—531no—
239C2×C4—532no—
240C2×C4—531no—
241C2×C4—1062no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2122no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2122no—
250C2×C4D4×#615712yes—
251C2×C4C2×#1171994011yescharacteristic
252C2×C4C2×#1171994011yescharacteristic
253C2×C4#1313330412yes—
254C8—4242no—
255C8—2124no—
256C22—21no—
257C22—21no—
258C22#1681157111yescharacteristic
259C22#1681159311yescharacteristic
260C22#1681159611yescharacteristic
261C4—42no—
262C4—42no—
263C4—42no—
264C4—1062no—
265C4—1061no—
266C4—1061no—
267C4—4241no—
268C4—4241no—
269C4—2122no—
270C4#1681165412yes—
271C4#1681165812yes—
272C2—21no—
273C2—21no—
274C2#2029069211yescharacteristic
275C2#2029075711yescharacteristic
276C2#2029076211yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#9104
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
159. C53#9104semidirect

Maximal subgroups

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

subgroupindex
#202916112
#202900172
#202899582
#202895392
#202774352
#910453