Finite Group Databasebeta

← #27278754

Subgroups — #27278754

Order 13568.

558 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
1#20283755C211yesmaximal, characteristic
2C4×#13133206C211yesmaximal, characteristic
3#20273110C211yesmaximal, characteristic, complemented
4#20273035C211yesmaximal, characteristic
5#20272547C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#1479C211yesmaximal, characteristic, complemented
8C2×#13133206C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810702—22no—
11#16810700—22no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810585—22no—
15#16810530C411yescharacteristic
16#16810524C411yescharacteristic
17#16810513—22no—
18#16810488C411yescharacteristic
19C4×#11719921—22no—
20C42×Dic53C2211yescharacteristic
21Q8⋊#88322—22no—
22#16810346—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53D412yes—
29C2×#11719912—24no—
30#13133206—22no—
31#13133206—22no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921D412yes—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133129—24no—
49C4×#88322—21no—
50C4×#88322—21no—
51#13133124—22no—
52#13133124—22no—
53C22×Q8×C53C2×C411yescharacteristic
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53D412yes—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53C22⋊C412yes—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53M1612yescomplemented
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53C22⋊C412yes—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—44no—
92C2×C8×C53—22no—
93C2×C8×C53—41no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53—41no—
99C42×C53—41no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104#11719921—28no—
105#11719921—22no—
106#11719921—22no—
107#11719921—44no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53—42no—
114C4×Dic53—42no—
115C4×Dic53M1612yescomplemented
116C2×#88322—22no—
117C2×#88322—41no—
118#11719912—84no—
119C23×C53C2×C22⋊C411yescharacteristic
120C22×D53—22no—
121C22×D53—22no—
122C22×D53—22no—
123C22×D53—42no—
124C22×D53—42no—
125Q8×C53—22no—
126Q8×C53—22no—
127Q8×C53—41no—
128Q8×C53—41no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13111yescharacteristic
145C2×C4×C53#13111yescharacteristic
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154D212—41no—
155D212—41no—
156D212—41no—
157D212—41no—
158C4×D53—44no—
159C4×D53—82no—
160C8×C53—42no—
161C8×C53—81no—
162#88322—42no—
163#88322—81no—
164#8961—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#38611yescharacteristic
168C22×C53#40711yescharacteristic
169C22×C53#40811yescharacteristic
170C2×D53—44no—
171C2×D53—42no—
172C2×D53—42no—
173C2×D53—42no—
174C2×D53—82no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—42no—
179C4×C53—81no—
180C4×C53#43511yescharacteristic
181C4×C53#43511yescharacteristic
182C4×C53#43611yescharacteristic
183C4×C53#43611yescharacteristic
184C4×C53#45512yes—
185Dic53—42no—
186Dic53—42no—
187Dic53—42no—
188#2975—531no—
189C4×#124—531no—
190#1479—531no—
191#1479D5311yescharacteristic, complemented
192#1460—531no—
193#1276—531no—
194#1276—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#187011yescharacteristic
198C2×C53#188411yescharacteristic
199C2×C53#188511yescharacteristic
200D53—82no—
201C2×#124—531no—
202C2×C4×Q8C2×D5311yescharacteristic
203C2×C4×D4—531no—
204#436—1062no—
205#436—1062no—
206#431—531no—
207#431—531no—
208#431C2×D5311yescharacteristic
209#431C2×D5311yescharacteristic
210#407—1062no—
211#391—531no—
212#390—531no—
213#388—1062no—
214#382—531no—
215C4×C22⋊C4—1062no—
216C43—531no—
217Q8⋊C8—22no—
218Q8⋊C8—1062no—
219#323—1062no—
220C53#896111yescharacteristic, complemented, Sylow
221C22×Q8C4×D5311yescharacteristic
222C22×D4—531no—
223C23×C4—532no—
224C2×M16—1064no—
225#124—1062no—
226#124—1062no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×D4—1061no—
232C4×D4—1061no—
233C4×D4—1061no—
234C4×D4—1061no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—532no—
239C2×C22⋊C4—1062no—
240C2×C22⋊C4—1062no—
241C2×C42—21no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—1061no—
245C2×C42—1061no—
246C2×C42C22×D5311yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—532no—
249C4⋊C8—1061no—
250C4⋊C8C22⋊D5312yes—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1064no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—1061no—
259C4×C8—1061no—
260#95—1062no—
261#95—1062no—
262C24—532no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8#1171991212yescomplemented
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—532no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—2124no—
283C22×C4C2×C4×D5311yescharacteristic
284C22×C4D4×D5311yescharacteristic
285C22×C4D4×D5311yescharacteristic
286M16—4244no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—1062no—
290C2×C8—2121no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C22⋊C4—1068no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—2124no—
301C42—41no—
302C42—41no—
303C42—532no—
304C42—1061no—
305C42—1061no—
306C42—1061no—
307C42—2122no—
308C42—2122no—
309C42#1171994511yescharacteristic
310C42D4×D5311yescharacteristic
311C42#1171994711yescharacteristic
312C42C2×C22⋊D5311yescharacteristic
313C23—1062no—
314C23—1062no—
315C23—1062no—
316C23—2122no—
317C23—2122no—
318C23C22⋊C4×D5311yescharacteristic
319Q8—22no—
320Q8—22no—
321Q8—41no—
322Q8—41no—
323D4—2121no—
324D4—2121no—
325D4—2121no—
326D4—2121no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—4242no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2124no—
343C2×C4—2122no—
344C2×C4#1313317711yescharacteristic
345C2×C4C4×C22⋊D5311yescharacteristic
346C2×C4#1313318511yescharacteristic
347C2×C4#1313318811yescharacteristic
348C2×C4#1313319011yescharacteristic
349C2×C4#1313319911yescharacteristic
350C2×C4M16×D5311yescharacteristic
351C2×C4C2×#1171991211yescharacteristic
352C2×C4#1313322311yescharacteristic
353C2×C4#1313322511yescharacteristic
354C8—42no—
355C8—81no—
356C8—4241no—
357C8—2122no—
358C22—21no—
359C22—21no—
360C22—4242no—
361C22—2124no—
362C22—2122no—
363C22—2122no—
364C22—2122no—
365C22#1681049911yescharacteristic
366C22#1681058511yescharacteristic
367C22#1681058811yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—81no—
373C4—2122no—
374C4—2122no—
375C4—2122no—
376C4#1681069111yescharacteristic
377C4#1681069411yescharacteristic
378C4#1681069811yescharacteristic
379C4#1681069911yescharacteristic
380C4#1681079912yes—
381C2—21no—
382C2—21no—
383C2—4242no—
384C2#2027500911yescharacteristic
385C2#2027509511yescharacteristic
386C2#2027510011yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1479C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1479D53
hypocenterC53#8961
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. #20273110C2semidirect
5. #20272547C2semidirect
6. #20272547C2semidirect
7. C53×#1479C2semidirect
74. C2×Q8×C53M16semidirect
115. C4×Dic53M16semidirect
191. #1479D53semidirect
220. C53#8961semidirect
268. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837552
C4×#131332062
#202731102
#202730352
#202725472
C53×#14792
#896153