Finite Group Databasebeta

← #27279089

Subgroups — #27279089

Order 13568.

800 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 143 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#20277333C211yesmaximal, characteristic
3#20274651C212yesmaximal
4#20272595C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133176—21no—
8C22×#11719919—21no—
9C2×#13133142—24no—
10#16810543—22no—
11#16810543C414yescomplemented
12#16810522—24no—
13#16810483—21no—
14#16810483C2212yes—
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—
27#13133176—42no—
28#13133176—42no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53D414yes—
32C2×#11719919—24no—
33C2×#11719919—44no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53—22no—
36#13133142—48no—
37#13133124—22no—
38#13133124—24no—
39#13133124—22no—
40#13133124—21no—
41#13133124C2×C412yes—
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—22no—
48C2×C4⋊C4×C53C2×C412yes—
49C2×C4⋊C4×C53D412yes—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53SD1612yes—
64C22×C4×C53Q1612yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—24no—
70C22×Dic53—24no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C2×Dic106—28no—
81C2×Dic106—24no—
82C2×Dic106—44no—
83C2×Dic106—48no—
84C2×Dic106—44no—
85C2×Dic106—44no—
86C2×Dic106—42no—
87C2×C8×C53—44no—
88C2×C8×C53—42no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—24no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C4⋊Dic53—44no—
94C4⋊Dic53—44no—
95C4⋊Dic53—82no—
96#11719919—48no—
97C4×Dic53—44no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#12011yescharacteristic
102C23×C53#12311yescharacteristic
103C23×C53C2×SD1611yescharacteristic
104C23×C53C2×Q1611yescharacteristic
105C23×C53AGL(1,ℤ/8)11yescharacteristic
106C23×C53#13711yescharacteristic
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—81no—
121C2×C4×C53#10314yescomplemented
122C2×C4×C53C4×D412yes—
123C2×C4×C53#12112yes—
124C2×Dic53—24no—
125C2×Dic53—24no—
126C2×Dic53—42no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—44no—
134C2×Dic53—44no—
135C2×Dic53—44no—
136C2×Dic53—44no—
137Dic106—28no—
138Dic106—48no—
139Dic106—44no—
140C8×C53—82no—
141#9015—531nomaximal, Sylow
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53#38611yescharacteristic
151C22×C53C2×#10311yescharacteristic
152C22×C53#41711yescharacteristic
153C22×C53#45011yescharacteristic
154C22×C53#45111yescharacteristic
155C22×C53#45612yes—
156C22×C53B(4,𝔽2)12yes—
157C22×C53#48311yescharacteristic
158C22×C53#48411yescharacteristic
159C4×C53—24no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—42no—
163Dic53—44no—
164Dic53—44no—
165Dic53—84no—
166C2×#395—531no—
167#2316—531no—
168#1787—532no—
169C2×#324D5311yescharacteristic
170#1294—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53#160412yes—
176C2×C53#160512yes—
177C2×C53#187211yescharacteristic
178C2×C53#188611yescharacteristic
179C2×C53#188911yescharacteristic
180C23×Q8—531no—
181C2×#117—1061no—
182C22×C4⋊C4—1061no—
183C22×C4⋊C4C2×D5311yescharacteristic
184C2×#103—1064no—
185C2×C22⋊C8C2×D5312yes—
186#395—534no—
187#395—1062no—
188#389—1064no—
189#380—532no—
190#380—1061no—
191C2×#95—531no—
192C2×#95—1061no—
193#324—22no—
194C53#901511yescharacteristic, complemented, Sylow
195C22×Q8—532no—
196C22×Q8—1061no—
197C22×Q8—1062no—
198C22×Q8—1062no—
199C22×Q8—2124no—
200C23×C4—1061no—
201C23×C4—1061no—
202C23×C4D21211yescharacteristic
203C23×C4D21211yescharacteristic
204C23×C4C22×D5311yescharacteristic
205C22×C8—22no—
206#117—2122no—
207#117—2122no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—534no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1064no—
214C2×C4⋊C4—2124no—
215C2×C4⋊C4C4×D5312yes—
216C2×C4⋊C4C22⋊D5312yes—
217C2×C22⋊C4—1062no—
218C2×C42—1062no—
219#103—2128no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222#95—532no—
223#95—1062no—
224#95—1062no—
225#95—1061no—
226#95—1064no—
227C24C2×D21211yescharacteristic
228C2×Q8—1068no—
229C2×Q8—1064no—
230C2×Q8—2124no—
231C2×Q8—2124no—
232C2×Q8—2124no—
233C2×Q8—2128no—
234C2×Q8—2122no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1064no—
244C22×C4—1064no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4—1064no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4#1171991312yes—
255C22×C4Dic21212yes—
256C22×C4#1171992112yes—
257C22×C4#1171992112yes—
258C22×C4C2×C4×D5311yescharacteristic
259C22×C4D4×D5312yes—
260C22×C4#1171994712yes—
261C22×C4C2×C22⋊D5311yescharacteristic
262C2×C8—44no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—4242no—
267C4⋊C4—2124no—
268C4⋊C4—2124no—
269C4⋊C4—2128no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C42—2124no—
273C23—22no—
274C23—22no—
275C23C2×#1171992111yescharacteristic
276C23#1313318711yescharacteristic
277C23#1313320111yescharacteristic
278C23C2×#1171991311yescharacteristic
279C23C2×Dic21211yescharacteristic
280C23#1313324511yescharacteristic
281C23#1313325011yescharacteristic
282Q8—1068no—
283Q8—2128no—
284Q8—2124no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—1064no—
300C2×C4—1064no—
301C2×C4—2124no—
302C2×C4—2124no—
303C2×C4—2124no—
304C2×C4—2122no—
305C2×C4—2124no—
306C2×C4—2124no—
307C2×C4—2122no—
308C2×C4—2124no—
309C2×C4—2122no—
310C2×C4—2124no—
311C2×C4—2122no—
312C2×C4#1313314214yescomplemented
313C2×C4#1313317712yes—
314C2×C4#1313318912yes—
315C8—82no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22#1681050011yescharacteristic
325C22C2×#1313314211yescharacteristic
326C22#1681062511yescharacteristic
327C22#1681077611yescharacteristic
328C22#1681078111yescharacteristic
329C22#1681080812yes—
330C22#1681081112yes—
331C22#1681095411yescharacteristic
332C22#1681095911yescharacteristic
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—4244no—
338C4—2124no—
339C4—2124no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2027368612yes—
345C2#2027369112yes—
346C2#2027502011yescharacteristic
347C2#2027510611yescharacteristic
348C2#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#9015
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
11. #16810543C4semidirect
121. C2×C4×C53#103semidirect
194. C53#9015semidirect
312. C2×C4#13133142semidirect

Maximal subgroups

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

subgroupindex
C2×#168105432
#202773332
#202746512
#202725952
C2×C53×#3242
#901553