Finite Group Databasebeta

← #28303308

Subgroups — #28303308

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290570C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3#20289551C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5C4×C4⋊C4×D53C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×#13133274—21no—
8C2×#13133274—21no—
9C2×C4×#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12C2×C4⋊C4×D53C411yescharacteristic
13C2×C4⋊C4×D53C411yescharacteristic
14C2×C42×D53—21no—
15C2×C42×D53C2211yescharacteristic
16C4⋊C4×Dic53C2211yescharacteristic
17C4⋊C4×Dic53C2211yescharacteristic
18C4×#11719919—21no—
19C4×C4⋊Dic53C411yescharacteristic
20#16810474—21no—
21#95×D53—21no—
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—22no—
26C4⋊#88324—22no—
27#13133274—24no—
28#13133274—42no—
29C4×#88324—22no—
30C4×#88324—22no—
31C22×C4×D53—21no—
32C22×C4×D53—21no—
33C22×C4×D53C2×C411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×C4×D53D411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C4⋊C4×D53—22no—
41C4⋊C4×D53—22no—
42C4⋊C4×D53—22no—
43C4⋊C4×D53C814yescomplemented
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50C42×D53—22no—
51C42×D53—22no—
52C42×D53—42no—
53C42×D53—42no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53C2×C411yescharacteristic
61#95×C53—21no—
62C23×D53C22⋊C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—24no—
80C2×C4×D53—24no—
81C2×C4×D53—22no—
82C2×C4×D53—22no—
83C2×C4×D53—41no—
84C2×C4×D53—42no—
85C2×C4×D53—42no—
86C2×C4×D53—42no—
87C2×C4×D53—42no—
88C2×C4×D53C2×C812yes—
89C2×C4×D53M1612yes—
90C2×C4×D53D812yes—
91C2×C4×D53SD1612yes—
92C2×#88324—24no—
93C2×#88324—42no—
94C2×#88324—42no—
95C2×#88324—42no—
96C2×#88324—42no—
97C2×#88324—42no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53C2×C812yes—
101C4⋊C4×C53M1612yes—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—42no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53C2×C812yes—
108C4⋊Dic53M1612yes—
109#11719919—24no—
110#11719919—24no—
111C4×Dic53—22no—
112C4×Dic53—22no—
113C4×Dic53—42no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C4×Dic53#4112yes—
119C4×Dic53#4112yes—
120C23×C53C2×C22⋊C411yescharacteristic
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53C22⋊C812yes—
126C22×D53#10212yes—
127C22×D53#10412yes—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53#10212yes—
140C2×C4×C53#11711yescharacteristic
141C2×C4×C53#11711yescharacteristic
142C2×C4×C53C22×C811yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53C2×M1611yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53C22⋊C812yes—
157C2×Dic53#12312yes—
158C2×Dic53#12312yes—
159C2×Dic53C2×D811yescharacteristic
160C2×Dic53C2×SD1611yescharacteristic
161C2×Dic53#13712yes—
162C4×D53—22no—
163C4×D53—22no—
164C4×D53—22no—
165C4×D53—22no—
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—44no—
169C4×D53—48no—
170C4×D53—48no—
171#88324—44no—
172#88324—82no—
173#8706—531nomaximal, Sylow
174C22×C53#38111yescharacteristic
175C22×C53C2×C22⋊C811yescharacteristic
176C22×C53#40711yescharacteristic
177C22×C53C2×#10211yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53C2×#10411yescharacteristic
180C22×C53#42111yescharacteristic
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53D4⋊C814yes—
188C4×C53—22no—
189C4×C53—41no—
190C4×C53—42no—
191C4×C53—42no—
192C4×C53#43111yescharacteristic
193C4×C53#43111yescharacteristic
194C4×C53#43211yescharacteristic
195C4×C53#43211yescharacteristic
196Dic53—22no—
197Dic53—22no—
198Dic53—42no—
199Dic53—42no—
200Dic53#48012yes—
201Dic53#48312yes—
202C2×C4×C4⋊C4—531no—
203#1797—531no—
204C4×C22⋊C8—531no—
205#1279—532no—
206#1279—532no—
207C2×C53C2×D4⋊C811yescharacteristic
208C2×C53#147911yescharacteristic
209C2×C53#148611yescharacteristic
210C2×C53#149111yescharacteristic
211C2×C53#181711yescharacteristic
212C2×C53#182311yescharacteristic
213C2×C53#182611yescharacteristic
214D53—24no—
215C22×C4⋊C4—531no—
216C22×C4⋊C4—531no—
217C22×C42—531no—
218C22×C42—1061no—
219C2×C4⋊C8—532no—
220C2×C22⋊C8—1061no—
221C2×C22⋊C8—1061no—
222C2×C4×C8—532no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4—531no—
225C4×C4⋊C4—531no—
226C4×C4⋊C4—1061no—
227C4×C4⋊C4—1061no—
228C4×C4⋊C4#615711yescharacteristic
229C2×#95—1061no—
230#336—1061no—
231#336—1061no—
232C53#870611yescharacteristic, complemented, Sylow
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—1061no—
237C23×C4—1061no—
238C22×C8—1062no—
239C22×C8—1062no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—534no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1062no—
248C2×C4⋊C4C2×#615711yescharacteristic
249C2×C4⋊C4C2×#615711yescharacteristic
250C2×C42—21no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—1062no—
255C2×C42—1062no—
256C2×C42—1061no—
257C2×C42—1061no—
258C2×C42—1061no—
259C2×C42—2122no—
260C2×C42—2122no—
261C2×C42C2×#615711yescharacteristic
262C4⋊C8—1062no—
263C4⋊C8—1062no—
264C22⋊C8—1064no—
265C22⋊C8—2122no—
266C4×C8—1062no—
267C4×C8—1062no—
268#95—21no—
269#95—1061no—
270#95—1061no—
271#95—1061no—
272C24—531no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1064no—
287C22×C4—1064no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—2121no—
295C22×C4—2122no—
296C22×C4—2122no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4C22⋊#615711yescharacteristic
300C22×C4C22⋊#615711yescharacteristic
301C22×C4C22×#615711yescharacteristic
302C2×C8—1064no—
303C2×C8—2122no—
304C2×C8—2122no—
305C2×C8—2122no—
306C2×C8—2122no—
307C2×C8—2122no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—532no—
311C4⋊C4—532no—
312C4⋊C4—1064no—
313C4⋊C4—1064no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4D53⋊C812yes—
317C4⋊C4#1171993612yes—
318C42—21no—
319C42—21no—
320C42—42no—
321C42—532no—
322C42—532no—
323C42—1062no—
324C42—1062no—
325C42—2121no—
326C42—2121no—
327C42—2121no—
328C42—2121no—
329C42—2122no—
330C23—532no—
331C23—532no—
332C23—532no—
333C23—1061no—
334C23—1061no—
335C23—1061no—
336C23—1061no—
337C23C2×C22⋊#615711yescharacteristic
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—532no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—532no—
353C2×C4—532no—
354C2×C4—532no—
355C2×C4—1064no—
356C2×C4—1064no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1062no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1062no—
367C2×C4—2124no—
368C2×C4—2128no—
369C2×C4—2128no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4#1313327912yes—
375C2×C4#1313329611yescharacteristic
376C2×C4#1313329611yescharacteristic
377C2×C4C2×D53⋊C811yescharacteristic
378C2×C4D53⋊M1611yescharacteristic
379C2×C4D53⋊M1611yescharacteristic
380C2×C4C2×#1171993611yescharacteristic
381C8—4242no—
382C8—2124no—
383C22—534no—
384C22—1062no—
385C22—1062no—
386C22—1062no—
387C22—1062no—
388C22—1062no—
389C22—1062no—
390C22#1681156611yescharacteristic
391C22D53⋊(C22⋊C8)11yescharacteristic
392C22#1681159411yescharacteristic
393C22C2×#1313327911yescharacteristic
394C22#1681161611yescharacteristic
395C22D53⋊#10411yescharacteristic
396C22#1681162011yescharacteristic
397C4—22no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—532no—
402C4—532no—
403C4—1062no—
404C4—1062no—
405C4—2122no—
406C4—2122no—
407C4#1681164111yescharacteristic
408C4#1681164211yescharacteristic
409C4#1681164511yescharacteristic
410C4#1681164611yescharacteristic
411C2—1064no—
412C2D53⋊(D4⋊C8)11yescharacteristic
413C2#2029001811yescharacteristic
414C2#2029004111yescharacteristic
415C2#2029005811yescharacteristic
416C2#2029060711yescharacteristic
417C2#2029061811yescharacteristic
418C2#2029062311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#8706
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289551C2semidirect
4. #20289551C2semidirect
43. C4⋊C4×D53C8semidirect
232. C53#8706semidirect

Maximal subgroups

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

subgroupindex
#202905702
C4×#131332742
#202895512
C4×C4⋊C4×D532
#870653