Finite Group Databasebeta

← #27272158

Subgroups — #27272158

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133196C211yesmaximal, characteristic
2C2×#16810378C211yesmaximal, characteristic
3#20272540C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×#13133196C411yescharacteristic
8C2×C4×Dic106C411yescharacteristic
9#16810529C2211yescharacteristic
10#16810521C2211yescharacteristic
11#16810507—21no—
12#16810496—21no—
13#16810483C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810378—21no—
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—24no—
25C22×#88322—22no—
26C22×#88322—42no—
27C22×#88322—42no—
28#13133196—22no—
29#13133196C814yescomplemented
30C4×Dic106—42no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—22no—
50C2×C42×C53—42no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55C4⋊C8×C53—42no—
56C22⋊C8×C53—42no—
57C24×C53C22⋊C411yescharacteristic
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×C812yes—
70C22×C4×C53M1612yes—
71C22×C4×C53SD1612yes—
72C22×C4×C53Q1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×Dic106—21no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C2×Dic106C2×C812yescomplemented
84C2×C8×C53—42no—
85C2×C8×C53—42no—
86C2×C8×C53—42no—
87C42×C53—22no—
88C42×C53—22no—
89C42×C53—41no—
90C42×C53—41no—
91C22⋊Dic53—24no—
92C22⋊Dic53—42no—
93C4⋊Dic53—21no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53C2×C812yescomplemented
97#11719919—24no—
98#11719919—42no—
99#11719919—41no—
100#11719919—41no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103C2×#88322—24no—
104C2×#88322—48no—
105C2×#88322—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53C22⋊C812yes—
112C23×C53#10312yes—
113C23×C53#10412yes—
114C23×C53C2×C22⋊C411yescharacteristic
115C2×C4×C53—24no—
116C2×C4×C53—24no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53C22⋊C812yes—
134C2×C4×C53#9912yes—
135C2×C4×C53#10312yes—
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53C2×Q1611yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154Dic106—42no—
155Dic106—42no—
156C8×C53—82no—
157#88322—44no—
158#7589—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53Q8⋊C814yes—
166C22×C53C2×C22⋊C811yescharacteristic
167C22×C53C2×#9911yescharacteristic
168C22×C53#41311yescharacteristic
169C22×C53C2×#10311yescharacteristic
170C22×C53#41711yescharacteristic
171C22×C53C2×#10411yescharacteristic
172C22×C53#42111yescharacteristic
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—21no—
177C4×C53—22no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—22no—
181C4×C53—21no—
182Dic53—42no—
183Dic53—42no—
184Dic53—81no—
185Dic53—81no—
186C4×C22⋊Q8—531no—
187#1797D5311yescharacteristic
188C2×#336—531no—
189#1272—532no—
190#1271—532no—
191C2×C53—24no—
192C2×C53#145911yescharacteristic
193C2×C53C2×Q8⋊C811yescharacteristic
194C2×C53#147711yescharacteristic
195C2×C53#150411yescharacteristic
196C2×C53#152011yescharacteristic
197C2×C53#153211yescharacteristic
198C2×C53#155011yescharacteristic
199C23×C8—1061no—
200C2×C22⋊Q8—531no—
201C2×C4×Q8—531no—
202C22×C42C2×D5311yescharacteristic
203C2×C4⋊C8—21no—
204C2×C22⋊C8—21no—
205#391—531no—
206#389—531no—
207#387—1061no—
208#385—1061no—
209#380—531no—
210C4×C4⋊C4—531no—
211C4×C4⋊C4—1061no—
212C4×C22⋊C4—1061no—
213#336—532no—
214#336—1061no—
215#336C2×D5312yes—
216C53#758911yescharacteristic, complemented, Sylow
217C22×Q8—531no—
218C23×C4C4×D5311yescharacteristic
219C23×C4D21211yescharacteristic
220C23×C4C22⋊D5311yescharacteristic
221C22×C8—22no—
222C22×C8—1064no—
223C22×C8—1062no—
224C22×C8—2122no—
225C22×C8—2122no—
226C22⋊Q8—534no—
227C22⋊Q8—1062no—
228C4×Q8—2122no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C42—42no—
238C2×C42—1061no—
239C2×C42—1061no—
240C2×C42C4×D5311yescharacteristic
241C2×C42D21211yescharacteristic
242C2×C42C22⋊D5311yescharacteristic
243C2×C42C22×D5311yescharacteristic
244C4⋊C8—42no—
245C22⋊C8—42no—
246#95—1061no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251C24#1171992111yescharacteristic
252C2×Q8—532no—
253C2×Q8—1061no—
254C2×Q8—2121no—
255C2×Q8—2121no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4C8×D5312yes—
269C22×C4#1171991212yes—
270C22×C4#1171992111yescharacteristic
271C22×C4#1171992111yescharacteristic
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992312yes—
274C22×C4#1171992512yes—
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4C2×D21211yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—1064no—
282C2×C8—2128no—
283C2×C8—2122no—
284C2×C8—2122no—
285C4⋊C4—532no—
286C4⋊C4—1064no—
287C4⋊C4—1061no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C4⋊C4—2122no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C22⋊C4—1064no—
294C22⋊C4—2122no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—2122no—
300C42—2122no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23#1313312912yes—
306C23#1313314012yes—
307C23#1313314312yes—
308C23C2×#1171992111yescharacteristic
309Q8—2122no—
310Q8—2122no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—1062no—
330C2×C4—1062no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4#1313312912yes—
342C2×C4#1313313112yes—
343C2×C4#1313314012yes—
344C2×C4C2×C8×D5311yescharacteristic
345C2×C4C2×#1171991211yescharacteristic
346C2×C4C2×#1171992311yescharacteristic
347C2×C4C2×#1171992511yescharacteristic
348C2×C4#1313324511yescharacteristic
349C2×C4#1313325011yescharacteristic
350C8—82no—
351C8—2124no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22#1681035314yes—
359C22C2×#1313312911yescharacteristic
360C22C2×#1313313111yescharacteristic
361C22#1681060511yescharacteristic
362C22C2×#1313314011yescharacteristic
363C22#1681062511yescharacteristic
364C22C2×#1313314311yescharacteristic
365C22#1681064111yescharacteristic
366C4—21no—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—4241no—
376C4—4241no—
377C4—2122no—
378C4—2122no—
379C2—24no—
380C2#2027303111yescharacteristic
381C2C2×#1681035311yescharacteristic
382C2#2027310311yescharacteristic
383C2#2027322011yescharacteristic
384C2#2027328611yescharacteristic
385C2#2027333611yescharacteristic
386C2#2027340611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#7589
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272537C2semidirect
29. #13133196C8semidirect
83. C2×Dic106C2×C8semidirect
96. C4⋊Dic53C2×C8semidirect
216. C53#7589semidirect

Maximal subgroups

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

subgroupindex
C4×#131331962
C2×#168103782
#202725402
#202725372
C53×#17972
#758953