Finite Group Databasebeta

← #27279084

Subgroups — #27279084

Order 13568.

756 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16810543C211yesmaximal, characteristic
2#20277318C211yesmaximal, characteristic
3#20274651C212yesmaximal
4#20272595C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C22×#11719919—21no—
8C2×#13133142—24no—
9#16810543—22no—
10#16810543C414yescomplemented
11#16810509—24no—
12#16810486—21no—
13#16810483C2212yes—
14C4×C22⋊Dic53—21no—
15C2×#13133124—21no—
16C2×#13133124C2211yescharacteristic
17C22×C4⋊C4×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2212yes—
19C53×#324—22no—
20C22×Dic106—22no—
21C22×Dic106—22no—
22C22×Dic106—21no—
23C22×Dic106—44no—
24C22×Dic106C2×C412yes—
25C23×Dic53—21no—
26C23×Dic53—21no—
27C2×C4⋊Dic53D414yes—
28C2×#11719919—24no—
29C2×#11719919—44no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53—22no—
32#13133142—48no—
33#13133124—22no—
34#13133124—24no—
35#13133124—24no—
36#13133124—22no—
37#13133124—21no—
38#13133124C2×C412yes—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C22×C8×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D412yes—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—22no—
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53#4112yes—
64C22×Dic53—24no—
65C22×Dic53—24no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—24no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—42no—
74C2×Dic106—28no—
75C2×Dic106—24no—
76C2×Dic106—44no—
77C2×Dic106—48no—
78C2×Dic106—44no—
79C2×Dic106—44no—
80C2×Dic106—42no—
81C2×C8×C53—44no—
82C2×C8×C53—42no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—24no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—44no—
88#11719919—48no—
89C4×Dic53—82no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53#12011yescharacteristic
94C23×C53#12311yescharacteristic
95C23×C53#13512yes—
96C23×C53#13712yes—
97C2×C4×C53—24no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—81no—
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12112yes—
112C2×Dic53—24no—
113C2×Dic53—24no—
114C2×Dic53—42no—
115C2×Dic53—44no—
116C2×Dic53—48no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—44no—
121C2×Dic53—44no—
122C2×Dic53—44no—
123C2×Dic53—44no—
124Dic106—28no—
125Dic106—48no—
126Dic106—44no—
127C8×C53—82no—
128#9014—531nomaximal, Sylow
129C22×C53—24no—
130C22×C53—24no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53#38611yescharacteristic
136C22×C53#41611yescharacteristic
137C22×C53#41911yescharacteristic
138C22×C53#45212yes—
139C22×C53#45612yes—
140C22×C53B(4,𝔽2)12yes—
141C22×C53#48512yes—
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—42no—
145C4×C53—42no—
146Dic53—44no—
147Dic53—44no—
148Dic53—84no—
149C2×#395—531no—
150#2313—531no—
151#1787—532no—
152C2×#324D5311yescharacteristic
153#1294—532no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—22no—
157C2×C53#160314yes—
158C2×C53#187311yescharacteristic
159C2×C53#188611yescharacteristic
160C2×C53#188911yescharacteristic
161C23×Q8—531no—
162C22×C4⋊C4—1061no—
163C22×C4⋊C4C2×D5311yescharacteristic
164C2×#103—1064no—
165C2×C22⋊C8C2×D5312yes—
166#395—534no—
167#395—1062no—
168#387—1064no—
169#381—1061no—
170#380—532no—
171C4×C22⋊C4—1061no—
172C2×#95—531no—
173C2×#95—1061no—
174#324—22no—
175C53#901411yescharacteristic, complemented, Sylow
176C22×Q8—532no—
177C22×Q8—1061no—
178C22×Q8—1062no—
179C22×Q8—1062no—
180C22×Q8—2124no—
181C23×C4—1061no—
182C23×C4—1061no—
183C23×C4D21211yescharacteristic
184C23×C4D21211yescharacteristic
185C23×C4C22×D5311yescharacteristic
186C22×C8—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—534no—
190C2×C4⋊C4—1064no—
191C2×C4⋊C4—2124no—
192C2×C4⋊C4C4×D5312yes—
193C2×C4⋊C4C22⋊D5312yes—
194C2×C22⋊C4—1062no—
195C2×C42—1062no—
196#103—2128no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#95—532no—
200#95—1062no—
201#95—1064no—
202#95—1062no—
203#95—1061no—
204#95—1064no—
205C24C2×D21211yescharacteristic
206C2×Q8—1068no—
207C2×Q8—1064no—
208C2×Q8—2124no—
209C2×Q8—2124no—
210C2×Q8—2124no—
211C2×Q8—2128no—
212C2×Q8—2122no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—1064no—
222C22×C4—1064no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1064no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4—2121no—
229C22×C4—2121no—
230C22×C4—2122no—
231C22×C4#1171992112yes—
232C22×C4#1171992112yes—
233C22×C4C2×C4×D5311yescharacteristic
234C22×C4D4×D5312yes—
235C22×C4#1171994712yes—
236C22×C4C2×C22⋊D5311yescharacteristic
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—2124no—
242C4⋊C4—2128no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C42—4242no—
246C23—22no—
247C23—22no—
248C23C2×#1171992111yescharacteristic
249C23#1313318711yescharacteristic
250C23#1313320111yescharacteristic
251C23#1313323912yes—
252C23#1313325012yes—
253Q8—1068no—
254Q8—2128no—
255Q8—2124no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—1064no—
270C2×C4—1064no—
271C2×C4—2124no—
272C2×C4—2124no—
273C2×C4—2124no—
274C2×C4—2122no—
275C2×C4—2128no—
276C2×C4—2122no—
277C2×C4—2124no—
278C2×C4—2122no—
279C2×C4—2124no—
280C2×C4—2122no—
281C2×C4#1313317712yes—
282C2×C4#1313318912yes—
283C8—82no—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22#1681050011yescharacteristic
291C22#1681062011yescharacteristic
292C22#1681063311yescharacteristic
293C22#1681078712yes—
294C22#1681080812yes—
295C22#1681081112yes—
296C22#1681096512yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—4244no—
302C4—2124no—
303C4—2124no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2027368014yes—
308C2#2027502511yescharacteristic
309C2#2027510611yescharacteristic
310C2#2027512111yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#9014
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
10. #16810543C4semidirect
175. C53#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#168105432
#202773182
#202746512
#202725952
C2×C53×#3242
#901453