Finite Group Databasebeta

← #27279091

Subgroups — #27279091

Order 13568.

800 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 147 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#20274649C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2316C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C22×#11719919—21no—
8C2×#13133140—24no—
9C2×C22⋊#88322C2212yes—
10#16810543—22no—
11#16810543C414yescomplemented
12C2×#13133124—21no—
13C2×#13133124C2211yescharacteristic
14#16810349—22no—
15C2×#117×C53—21no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#389—24no—
18C53×#380—21no—
19C53×#380C2212yes—
20C22×Dic106—22no—
21C22×Dic106—22no—
22C22×Dic106—21no—
23C22×Dic106—44no—
24C22×Dic106C2×C412yes—
25C23×Dic53—21no—
26C23×Dic53—21no—
27C22×#88322—22no—
28C2×#11719919—24no—
29C2×#11719919—44no—
30#13133140—48no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—24no—
34#13133124—21no—
35#13133124C2×C412yes—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39#117×C53—42no—
40#117×C53—42no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D414yes—
47C2×C4⋊C4×C53D412yes—
48C2×C22⋊C4×C53—22no—
49C2×C42×C53—22no—
50#95×C53—22no—
51#95×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—24no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53SD1612yes—
65C22×C4×C53Q1612yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70C22×Dic53—24no—
71C22×Dic53—24no—
72C22×Dic53—22no—
73C22×Dic53—22no—
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—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—44no—
90C4⋊C4×C53—44no—
91C4⋊C4×C53—82no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C42×C53—44no—
95#11719919—48no—
96C2×#88322—44no—
97C2×#88322—42no—
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—44no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—81no—
124C2×C4×C53#10314yescomplemented
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12112yes—
127C2×Dic53—24no—
128C2×Dic53—24no—
129C2×Dic53—42no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—44no—
137Dic106—28no—
138Dic106—48no—
139Dic106—44no—
140#88322—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—
163C4×C53—84no—
164Dic53—44no—
165Dic53—44no—
166C2×#395—531no—
167#2316D5311yescharacteristic
168#1787—532no—
169C2×#324—531no—
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—21no—
182C22×C4⋊C4—1061no—
183C22×C4⋊C4C2×D5311yescharacteristic
184C2×#103—1064no—
185C2×C22⋊C8—532no—
186#395—534no—
187#395—1062no—
188#389—24no—
189#380—21no—
190#380C2×D5312yes—
191C2×#95—531no—
192C2×#95—1061no—
193#324—1062no—
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×C4C22⋊D5311yescharacteristic
203C23×C4C22⋊D5311yescharacteristic
204C23×C4C22×D5311yescharacteristic
205C22×C8—1062no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4—2124no—
214C2×C4⋊C4C4×D5312yes—
215C2×C4⋊C4D21212yes—
216C2×C4⋊C4C22⋊D5314yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219#103—2128no—
220C22⋊C8—1062no—
221C22⋊C8—1062no—
222#95—22no—
223#95—22no—
224#95—532no—
225#95—1061no—
226#95—1064no—
227C24C2×C22⋊D5311yescharacteristic
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—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1064no—
245C22×C4—1064no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4#1171992112yes—
255C22×C4#1171992112yes—
256C22×C4#1171992312yes—
257C22×C4#1171992512yes—
258C22×C4C2×C4×D5311yescharacteristic
259C22×C4C2×D21211yescharacteristic
260C22×C4#1171994512yes—
261C22×C4C2×C22⋊D5312yes—
262C2×C8—2124no—
263C2×C8—2122no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—82no—
269C4⋊C4—2128no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—44no—
273C23—22no—
274C23—22no—
275C23C2×#1171992111yescharacteristic
276C23#1313318611yescharacteristic
277C23#1313320011yescharacteristic
278C23C2×#1171992311yescharacteristic
279C23C2×#1171992511yescharacteristic
280C23#1313324311yescharacteristic
281C23#1313324811yescharacteristic
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—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4—1064no—
303C2×C4—1064no—
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#1313314014yescomplemented
313C2×C4C4×C22⋊D5312yes—
314C2×C4#1313319112yes—
315C8—4242no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22#1681050211yescharacteristic
325C22C2×#1313314011yescharacteristic
326C22#1681062211yescharacteristic
327C22#1681077411yescharacteristic
328C22#1681077911yescharacteristic
329C22#1681080612yes—
330C22#1681081012yes—
331C22#1681095311yescharacteristic
332C22#1681095811yescharacteristic
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—84no—
338C4—2124no—
339C4—2124no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2027368412yes—
345C2#2027368912yes—
346C2#2027502211yescharacteristic
347C2#2027510811yescharacteristic
348C2#2027512311yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2316C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2316D53
hypocenterC53#9015
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
11. #16810543C4semidirect
124. C2×C4×C53#103semidirect
194. C53#9015semidirect
312. C2×C4#13133140semidirect

Maximal subgroups

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

subgroupindex
C2×#168105432
#202746492
C2×#168103492
#202725932
C53×#23162
#901553