Finite Group Databasebeta

← #27275513

Subgroups — #27275513

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811100C211yesmaximal, characteristic
2#20273111C212yesmaximal, complemented
3#20273109C212yesmaximal, complemented
4#20273029C211yesmaximal, characteristic
5C53×#1458C211yesmaximal, characteristic
6C23×Dic106C411yescharacteristic
7#16811100—24no—
8#16811100—24no—
9C2×C4×Dic106C411yescharacteristic
10C2×C4×Dic106C411yescharacteristic
11C2×C4×Dic106C2211yescharacteristic
12C2×C4×Dic106C2211yescharacteristic
13C2×#13133182—21no—
14C2×#13133182—21no—
15C2×#13133176C411yescharacteristic
16C2×#13133173—21no—
17C2×#13133173—21no—
18#16810676C2212yes—
19#16810355—24no—
20#16810353—24no—
21#16810349—21no—
22#16810347—21no—
23C2×#117×C53C2211yescharacteristic
24C53×#431C2212yes—
25C53×#324—21no—
26C53×#323—21no—
27C22×Dic106—21no—
28C22×Dic106—22no—
29C22×Dic106—22no—
30C22×Dic106—22no—
31C22×Dic106—22no—
32C22×Dic106—22no—
33C22×Dic106C814yescomplemented
34C22×Dic106C2×C411yescharacteristic
35C22×Dic106C2×C411yescharacteristic
36C23×Dic53—21no—
37C4×Dic106—22no—
38C4×Dic106—22no—
39C4×Dic106—22no—
40C4×Dic106—22no—
41C4×Dic106—22no—
42C4×Dic106—22no—
43C4×Dic106D414yes—
44C4×Dic106D414yes—
45#13133182—28no—
46#13133182—28no—
47#13133176—22no—
48#13133176—22no—
49#13133173—28no—
50#13133173—28no—
51C2×C4⋊Dic53C2×C411yescharacteristic
52C2×C4⋊Dic53C2×C411yescharacteristic
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×C22⋊Dic53—21no—
58C2×C4×Dic53—21no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61C2×C4×Dic53—21no—
62C2×C4×Dic53—21no—
63C4⋊#88322—22no—
64C22⋊#88322—22no—
65C4×#88322—22no—
66C23×C4×C53C2×C411yescharacteristic
67#117×C53—22no—
68#117×C53—22no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53D411yescharacteristic
73C2×C42×C53C2×C411yescharacteristic
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—22no—
76C22⋊C8×C53—22no—
77C4×C8×C53—22no—
78C24×C53C22⋊C411yescharacteristic
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C2×C812yes—
87C22×C4×C53M1612yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—24no—
92C22×Dic53—24no—
93C22×Dic53—24no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C2×Dic106—24no—
99C2×Dic106—24no—
100C2×Dic106—28no—
101C2×Dic106—28no—
102C2×Dic106—24no—
103C2×Dic106—21no—
104C2×Dic106—21no—
105C2×Dic106—21no—
106C2×Dic106—21no—
107C2×Dic106—21no—
108C2×Dic106—21no—
109C2×Dic106—28no—
110C2×Dic106—28no—
111C2×Dic106C22⋊C412yes—
112C2×Dic106C22⋊C412yes—
113C2×Dic106C2×C812yes—
114C2×Dic106C2×C812yes—
115C2×Dic106M1612yes—
116C2×Dic106M1612yes—
117C2×C8×C53—42no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—22no—
122C42×C53—21no—
123C42×C53—21no—
124C42×C53C2×D412yes—
125C42×C53C2×D412yes—
126C22⋊Dic53—24no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53C22⋊C412yes—
130C4⋊Dic53C22⋊C412yes—
131#11719919—24no—
132#11719919—24no—
133#11719919—24no—
134#11719919—24no—
135C4×Dic53—24no—
136C4×Dic53—24no—
137C4×Dic53—24no—
138C4×Dic53—24no—
139C4×Dic53—24no—
140C2×#88322—42no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C22⋊C812yes—
146C23×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53C22⋊C812yes—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53#12012yes—
163C2×C4×C53#12012yes—
164C2×C4×C53C22×C811yescharacteristic
165C2×C4×C53C2×M1611yescharacteristic
166C2×C4×C53C2×M1611yescharacteristic
167C2×C4×C53C2×M1611yescharacteristic
168C2×Dic53—216no—
169C2×Dic53—22no—
170C2×Dic53—22no—
171C2×Dic53—22no—
172C2×Dic53—22no—
173C2×Dic53—22no—
174C2×Dic53—22no—
175C2×Dic53—22no—
176C2×Dic53—22no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179Dic106—216no—
180Dic106—22no—
181Dic106—22no—
182Dic106—22no—
183Dic106—22no—
184Dic106—22no—
185Dic106—22no—
186Dic106C22⋊C814yes—
187Dic106C22⋊C814yes—
188C8×C53—82no—
189#88322—82no—
190#8323—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—22no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53#37911yescharacteristic
199C22×C53C2×C22⋊C811yescharacteristic
200C22×C53#40711yescharacteristic
201C4×C53—22no—
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53C2×C22⋊C811yescharacteristic
207C4×C53C2×C22⋊C811yescharacteristic
208C4×C53#40711yescharacteristic
209C4×C53#40711yescharacteristic
210Dic53—24no—
211Dic53—24no—
212Dic53—42no—
213Dic53—42no—
214C2×#519—531no—
215#1479—532no—
216#1479—532no—
217#1458—531no—
218#1458D5311yescharacteristic
219C2×C53—21no—
220C2×C53—21no—
221C2×C53—42no—
222C2×C53#177911yescharacteristic
223C2×C53#179611yescharacteristic
224C2×C53#179611yescharacteristic
225C23×Q8—531no—
226#519—1064no—
227#519—1064no—
228C2×C4×Q8—531no—
229C2×C4×Q8—531no—
230C2×C4×Q8—531no—
231C2×C4×Q8—531no—
232C2×C4×Q8—1061no—
233C2×C4×Q8—1061no—
234C2×#117—531no—
235C2×#117—1061no—
236C2×#117—1061no—
237C2×#117C2×D5311yescharacteristic
238#431—532no—
239#431C2×D5312yes—
240Q8⋊C8—1064no—
241Q8⋊C8—1064no—
242#324—21no—
243#324—1061no—
244#323—21no—
245#323—1061no—
246C53#832311yescharacteristic, complemented, Sylow
247C22×Q8—531no—
248C22×Q8—531no—
249C22×Q8—534no—
250C22×Q8—1062no—
251C22×Q8—1062no—
252C22×Q8—1062no—
253C22×Q8—1062no—
254C22×Q8—1062no—
255C22×Q8—1061no—
256C23×C4—1061no—
257C23×C4C4×D5311yescharacteristic
258C4×Q8—534no—
259C4×Q8—534no—
260C4×Q8—1062no—
261C4×Q8—1062no—
262C4×Q8—1062no—
263C4×Q8—1062no—
264C4×Q8—1068no—
265C4×Q8—1068no—
266C4×Q8—1062no—
267C4×Q8—1062no—
268#117—22no—
269#117—22no—
270#117—1068no—
271#117—1068no—
272#117—1062no—
273#117—1062no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4D21211yescharacteristic
281C2×C4⋊C4C22⋊D5311yescharacteristic
282C2×C22⋊C4—1061no—
283C2×C22⋊C4D21211yescharacteristic
284C2×C22⋊C4C22⋊D5311yescharacteristic
285C2×C42—1061no—
286C2×C42—1061no—
287C2×C42—1061no—
288C2×C42—1061no—
289C2×C42—1061no—
290C2×C42C4×D5311yescharacteristic
291C2×C42C22×D5311yescharacteristic
292C4⋊C8—22no—
293C4⋊C8—1062no—
294C22⋊C8—22no—
295C22⋊C8—1062no—
296C4×C8—22no—
297C4×C8—1062no—
298C24#1171992111yescharacteristic
299C2×Q8—532no—
300C2×Q8—532no—
301C2×Q8—532no—
302C2×Q8—532no—
303C2×Q8—532no—
304C2×Q8—532no—
305C2×Q8—1068no—
306C2×Q8—1068no—
307C2×Q8—1064no—
308C2×Q8—1064no—
309C2×Q8—1068no—
310C2×Q8—1068no—
311C2×Q8—1064no—
312C2×Q8—1061no—
313C2×Q8—1061no—
314C2×Q8—1061no—
315C2×Q8—1061no—
316C2×Q8—1061no—
317C2×Q8—1061no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—1064no—
323C22×C4—1064no—
324C22×C4—1064no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4C8×D5312yes—
330C22×C4#1171991212yes—
331C22×C4#1171992111yescharacteristic
332C22×C4#1171992111yescharacteristic
333C22×C4#1171992111yescharacteristic
334C22×C4C2×C4×D5311yescharacteristic
335C22×C4C2×D21211yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C2×C8—42no—
338C2×C8—2122no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—532no—
342C4⋊C4—532no—
343C4⋊C4—1064no—
344C4⋊C4—1064no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—1064no—
348C4⋊C4—1064no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—1064no—
352C42—21no—
353C42—21no—
354C42—1064no—
355C42—1064no—
356C42—1064no—
357C42—1064no—
358C42—1064no—
359C42D4×D5312yes—
360C42D4×D5312yes—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23#1313312912yes—
366C23C2×#1171992111yescharacteristic
367Q8—534no—
368Q8—534no—
369Q8—10616no—
370Q8—1062no—
371Q8—1062no—
372Q8—1062no—
373Q8—1062no—
374Q8—1062no—
375Q8—1062no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—10616no—
389C2×C4—1062no—
390C2×C4—1062no—
391C2×C4—1062no—
392C2×C4—1062no—
393C2×C4—1062no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—1062no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4#1313312912yes—
400C2×C4C22⋊C4×D5311yescharacteristic
401C2×C4C22⋊C4×D5311yescharacteristic
402C2×C4#1313318512yes—
403C2×C4#1313318712yes—
404C2×C4C2×C8×D5311yescharacteristic
405C2×C4M16×D5311yescharacteristic
406C2×C4M16×D5311yescharacteristic
407C2×C4C2×#1171991211yescharacteristic
408C8—82no—
409C8—4242no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22#1681047811yescharacteristic
418C22C2×#1313312911yescharacteristic
419C22#1681058611yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—1064no—
426C4—1064no—
427C4—2122no—
428C4—2122no—
429C4C22⋊C8×D5311yescharacteristic
430C4C22⋊C8×D5311yescharacteristic
431C4#1681058511yescharacteristic
432C4#1681058511yescharacteristic
433C2—21no—
434C2—21no—
435C2—42no—
436C2#2027462411yescharacteristic
437C2#2027468811yescharacteristic
438C2#2027469111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1458C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1458D53
hypocenterC53#8323
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20273111C2semidirect
3. #20273109C2semidirect
33. C22×Dic106C8semidirect
246. C53#8323semidirect

Maximal subgroups

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

subgroupindex
C2×#168111002
#202731112
#202731092
#202730292
C53×#14582
#832353