Finite Group Databasebeta

← #27275781

Subgroups — #27275781

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811330C211yesmaximal, characteristic
2#20274803C212yesmaximal, complemented
3#20273286C212yesmaximal, complemented
4#20273033C211yesmaximal, characteristic
5C53×#1799C211yesmaximal, characteristic
6#16811330—24no—
7#16811330—24no—
8C2×#13133217C411yescharacteristic, complemented
9C2×#13133208—21no—
10C2×#13133196C411yescharacteristic, complemented
11C2×#13133196C2211yescharacteristic
12C2×#13133194—21no—
13C2×#13133194—21no—
14C2×C4×Dic106C411yescharacteristic, complemented
15C2×D4×Dic53—21no—
16C22×C4⋊Dic53C411yescharacteristic, complemented
17C22×C4⋊Dic53C2211yescharacteristic
18C2×C22⋊#88322C2212yes—
19#16810385—24no—
20#16810359—24no—
21#16810349—21no—
22#16810347—21no—
23C2×C4×D4×C53C2211yescharacteristic
24C2×#99×C53—21no—
25C53×#380C2212yes—
26C53×#342—21no—
27C22×Dic106—21no—
28C22×Dic106C2×C411yescharacteristic
29C23×Dic53—21no—
30C23×Dic53—21no—
31C22×#88322—22no—
32#13133217—22no—
33#13133217—22no—
34#13133208—28no—
35#13133196—22no—
36#13133196—22no—
37#13133196—22no—
38#13133196D414yes—
39#13133194—28no—
40#13133194—28no—
41C4×Dic106—22no—
42C4×Dic106—22no—
43D4×Dic53—28no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53C2×C411yescharacteristic
51C2×C4⋊Dic53C2×C411yescharacteristic
52C2×C4⋊Dic53C2×C411yescharacteristic
53C2×C4⋊Dic53D414yes—
54C2×#11719919—21no—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—21no—
59C2×C4×Dic53—21no—
60C22⋊#88322—22no—
61C22⋊#88322—22no—
62C22×D4×C53D411yescharacteristic
63C23×C4×C53C2×C411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C4×D4×C53—22no—
66C4×D4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C42×C53C2×C411yescharacteristic
73#99×C53—42no—
74#95×C53—22no—
75C24×C53C22⋊C411yescharacteristic
76C24×C53C2×D411yescharacteristic
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53SD1612yes—
80C2×D4×C53Q1612yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D412yes—
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—24no—
98C22×Dic53—24no—
99C22×Dic53—82no—
100C22×Dic53—82no—
101C2×Dic106—21no—
102C2×Dic106—24no—
103C2×Dic106—42no—
104C2×Dic106—82no—
105C2×Dic106C22⋊C412yes—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—44no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C42×C53—21no—
113C42×C53—21no—
114C22⋊Dic53—24no—
115C22⋊Dic53—24no—
116C4⋊Dic53—24no—
117C4⋊Dic53—24no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—24no—
122C4⋊Dic53C22⋊C412yes—
123C4⋊Dic53C22⋊C412yes—
124C4⋊Dic53C22⋊C412yes—
125#11719919—24no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53—24no—
129C2×#88322—44no—
130C2×#88322—42no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53#10312yes—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53#12012yes—
140C23×C53C2×SD1611yescharacteristic
141C23×C53C2×Q1611yescharacteristic
142C23×C53#13712yes—
143D4×C53—24no—
144D4×C53—24no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—24no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53—82no—
160C2×C4×C53#10312yes—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53#12012yes—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—22no—
167C2×Dic53—22no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—84no—
173C2×Dic53—84no—
174Dic106—42no—
175Dic106—42no—
176Dic106—84no—
177#88322—82no—
178#8380—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—21no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53#37911yescharacteristic
192C22×C53C2×#10311yescharacteristic
193C22×C53#41911yescharacteristic
194C22×C53#44812yes—
195C22×C53#45112yes—
196C4×C53—22no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—162no—
202C4×C53#41012yes—
203Dic53—42no—
204Dic53—42no—
205Dic53—42no—
206Dic53—42no—
207C2×#554—531no—
208#1824—532no—
209#1799D5311yescharacteristic
210#1520—532no—
211#1459—531no—
212C2×C53—22no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—42no—
217C2×C53#178211yescharacteristic
218C2×C53#178711yescharacteristic
219C2×C53#179511yescharacteristic
220#554—1064no—
221#554—1064no—
222C2×C4⋊Q8—531no—
223C2×#125—1061no—
224C2×C22⋊Q8—531no—
225C2×C22⋊Q8—531no—
226C2×C22⋊Q8—1061no—
227C2×C22⋊Q8—1061no—
228C2×C4×Q8—531no—
229C2×C4×D4—1061no—
230C2×C4×D4C2×D5311yescharacteristic
231C22×C4⋊C4—531no—
232C22×C4⋊C4—531no—
233C2×#99—21no—
234C2×C22⋊C8—532no—
235#380C2×D5312yes—
236#342—21no—
237#340—1064no—
238#328—1064no—
239#324—1061no—
240#323—1061no—
241C53#838011yescharacteristic, complemented, Sylow
242C22×Q8—531no—
243C22×Q8—1061no—
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—1061no—
246C23×C4—1061no—
247C23×C4C4×D5311yescharacteristic
248C23×C4C22×D5311yescharacteristic
249C22×C8—1062no—
250C4⋊Q8—1062no—
251C4⋊Q8—1062no—
252#125—1068no—
253C22⋊Q8—534no—
254C22⋊Q8—1068no—
255C22⋊Q8—1068no—
256C22⋊Q8—1062no—
257C22⋊Q8—1062no—
258C22⋊Q8—1062no—
259C4×Q8—1062no—
260C4×Q8—1062no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—1068no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—534no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4D21211yescharacteristic
279C2×C22⋊C4—22no—
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1061no—
282C2×C22⋊C4D21211yescharacteristic
283C2×C22⋊C4C22⋊D5311yescharacteristic
284C2×C42—1061no—
285C2×C42C4×D5311yescharacteristic
286#99—42no—
287C22⋊C8—1062no—
288C22⋊C8—1062no—
289#95—22no—
290C24#1171992111yescharacteristic
291C24C2×C22⋊D5311yescharacteristic
292C2×Q8—532no—
293C2×Q8—1064no—
294C2×Q8—1061no—
295C2×Q8—4242no—
296C2×Q8—2122no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4#1171992312yes—
300C2×D4#1171992512yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—1064no—
307C22×C4—1064no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—4242no—
313C22×C4—4242no—
314C22×C4#1171992111yescharacteristic
315C22×C4#1171992111yescharacteristic
316C22×C4#1171992111yescharacteristic
317C22×C4C2×C4×D5311yescharacteristic
318C22×C4C2×D21211yescharacteristic
319C22×C4D4×D5312yes—
320C22×C4D4×D5312yes—
321C2×C8—2124no—
322C2×C8—2122no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—532no—
326C4⋊C4—532no—
327C4⋊C4—532no—
328C4⋊C4—1064no—
329C4⋊C4—1064no—
330C4⋊C4—1064no—
331C4⋊C4—1064no—
332C4⋊C4—1061no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—1064no—
336C4⋊C4—1064no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—1064no—
342C22⋊C4—1064no—
343C42—21no—
344C42—21no—
345C42—1064no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23#1313314012yes—
353C23C2×#1171992111yescharacteristic
354C23#1313318512yes—
355C23C2×#1171992311yescharacteristic
356C23C2×#1171992511yescharacteristic
357C23#1313324712yes—
358Q8—4244no—
359Q8—2122no—
360Q8—2122no—
361D4—24no—
362D4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—1062no—
379C2×C4—1062no—
380C2×C4—1062no—
381C2×C4—1062no—
382C2×C4—4244no—
383C2×C4—4244no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4#1313314012yes—
389C2×C4C22⋊C4×D5311yescharacteristic
390C2×C4C22⋊C4×D5311yescharacteristic
391C2×C4#1313318712yes—
392C8—4242no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22#1681047811yescharacteristic
406C22C2×#1313314011yescharacteristic
407C22#1681063111yescharacteristic
408C22#1681076112yes—
409C22#1681077812yes—
410C4—22no—
411C4—21no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—162no—
416C4—2122no—
417C4—2122no—
418C4—2122no—
419C4—2122no—
420C4#1681059512yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—42no—
426C2#2027463111yescharacteristic
427C2#2027464911yescharacteristic
428C2#2027468311yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1799C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1799D53
hypocenterC53#8380
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20274803C2semidirect
3. #20273286C2semidirect
8. C2×#13133217C4semidirect
10. C2×#13133196C4semidirect
14. C2×C4×Dic106C4semidirect
16. C22×C4⋊Dic53C4semidirect
241. C53#8380semidirect

Maximal subgroups

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

subgroupindex
C2×#168113302
#202748032
#202732862
#202730332
C53×#17992
#838053