Finite Group Databasebeta

← #27300971

Subgroups — #27300971

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811040C211yesmaximal, characteristic, complemented
2C2×#16810407C211yesmaximal, characteristic, complemented
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276329C211yesmaximal, characteristic, complemented
5#20272797C211yesmaximal, characteristic, complemented
6#20272793C211yesmaximal, characteristic, complemented
7C2×C53×#443C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—21no—
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13C22×#11719908—21no—
14C22×#11719908C2211yescharacteristic
15C22×#11719908C2211yescharacteristic
16#16810407—22no—
17#16810407—22no—
18C22⋊#11719908—21no—
19C22⋊#11719908—21no—
20C2×#135×C53C411yescharacteristic
21C2×#131×C53C2211yescharacteristic
22C2×#131×C53C2211yescharacteristic
23C53×#443—22no—
24C53×#443—22no—
25C2×#108×C53C411yescharacteristic
26C2×#104×C53C411yescharacteristic
27C2×#104×C53C411yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29#13133157—24no—
30#13133157—21no—
31#13133157—21no—
32#13133157—21no—
33#13133157D412yes—
34C2×#11719908—24no—
35C2×#11719908—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908D411yescharacteristic
43C2×#11719908D411yescharacteristic
44C2×#41×C53C2×C411yescharacteristic
45C2×#41×C53C2×C411yescharacteristic
46#135×C53—22no—
47#135×C53—22no—
48C2×Q16×C53C811yescharacteristic
49C2×SD16×C53C811yescharacteristic
50C2×SD16×C53C811yescharacteristic
51C2×D8×C53C811yescharacteristic
52#131×C53—21no—
53#131×C53—21no—
54#131×C53—21no—
55#131×C53—21no—
56#131×C53—21no—
57#131×C53—21no—
58#131×C53D412yes—
59#131×C53D412yes—
60C2×M16×C53—21no—
61C2×M16×C53—21no—
62C2×M16×C53C2×C411yescharacteristic
63C2×M16×C53C2×C411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53C2×C411yescharacteristic
67C22×C8×C53C2311yescharacteristic
68C2×C42×C53C2×C411yescharacteristic
69#108×C53—22no—
70#104×C53—22no—
71#104×C53—22no—
72C4×C8×C53—21no—
73C4×C8×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53C22⋊C412yes—
81#41×C53M1612yes—
82C2×Q8×C53C2×C811yescharacteristic
83C2×Q8×C53C2×C811yescharacteristic
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C2×C811yescharacteristic
87C2×D4×C53C2×C811yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C22×C411yescharacteristic
92Q16×C53—22no—
93SD16×C53—22no—
94SD16×C53—22no—
95D8×C53—22no—
96M16×C53—21no—
97M16×C53—22no—
98M16×C53—22no—
99M16×C53—21no—
100M16×C53—21no—
101M16×C53—41no—
102M16×C53—41no—
103M16×C53C22⋊C412yes—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—21no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53C2×C811yescharacteristic
113C2×C8×C53C2×C811yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53#4111yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C42×C53—21no—
121C42×C53—21no—
122#11719908—21no—
123#11719908—22no—
124#11719908—21no—
125#11719908—41no—
126#11719908—41no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53#13111yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53C22⋊C812yes—
134D4×C53—22no—
135D4×C53—22no—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53C22⋊C812yes—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—21no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C22×C811yescharacteristic
155C2×C4×C53C2×M1611yescharacteristic
156C2×C4×C53#13111yescharacteristic
157C8×C53—22no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—41no—
161C8×C53#12011yescharacteristic
162C8×C53#12111yescharacteristic
163C8×C53#12111yescharacteristic
164C8×C53#12311yescharacteristic
165#12980—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53#40811yescharacteristic
175C22×C53C8×D411yescharacteristic
176C22×C53#43511yescharacteristic
177C4×C53—22no—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53#38611yescharacteristic
181C4×C53C2×C22⋊C811yescharacteristic
182C4×C53C8×D411yescharacteristic
183C4×C53#43511yescharacteristic
184C2×#504—531no—
185C2×#443D5311yescharacteristic
186C2×#350—531no—
187C2×#350—531no—
188#2113—531no—
189#1376—531no—
190#1374—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—42no—
194C2×C53—81no—
195C2×C53#186911yescharacteristic
196C2×C53#217011yescharacteristic
197C2×C53#217111yescharacteristic
198C2×#135Dic5311yescharacteristic
199C2×#131C2×D5311yescharacteristic
200C2×#131C2×D5311yescharacteristic
201#504—1062no—
202#504—1062no—
203C2×M32—531no—
204C2×M32—531no—
205C2×M32—1061no—
206C22×C16—531no—
207C22×C16—531no—
208C22×C16—1061no—
209#443—22no—
210#443—22no—
211C2×#108Dic5311yescharacteristic
212C2×#104Dic5311yescharacteristic
213C2×#104Dic5311yescharacteristic
214C2×C4×C8C2×D5311yescharacteristic
215#350—1062no—
216#350—1062no—
217C22⋊C16—1061no—
218C22⋊C16—1061no—
219C53#1298011yescharacteristic, complemented, Sylow
220C2×#41C2×Dic5311yescharacteristic
221C2×#41C2×Dic5311yescharacteristic
222#135—22no—
223#135—22no—
224C2×Q16#8832211yescharacteristic
225C2×SD16#8832211yescharacteristic
226C2×SD16#8832211yescharacteristic
227C2×D8#8832211yescharacteristic
228#131—21no—
229#131—21no—
230#131—21no—
231#131—21no—
232#131—21no—
233#131—21no—
234#131C22⋊D5312yes—
235#131C22⋊D5312yes—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16C2×Dic5311yescharacteristic
239C2×M16C2×Dic5311yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C22×C8C2×Dic5311yescharacteristic
243C22×C8C22×D5311yescharacteristic
244C2×C42C2×Dic5311yescharacteristic
245M32—532no—
246M32—1064no—
247M32—1061no—
248M32—1061no—
249M32—1061no—
250C2×C16—531no—
251C2×C16—531no—
252C2×C16—1064no—
253C2×C16—1061no—
254C2×C16—1061no—
255C2×C16—1061no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16—1061no—
260#108—22no—
261#104—22no—
262#104—22no—
263C4×C8—21no—
264C4×C8—21no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—21no—
271#41#1171991712yes—
272#41C22⋊Dic5312yes—
273C2×Q8C2×#8832211yescharacteristic
274C2×Q8C2×#8832211yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C2×#8832211yescharacteristic
278C2×D4C2×#8832211yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4C22×Dic5311yescharacteristic
283Q16—22no—
284SD16—22no—
285SD16—22no—
286D8—22no—
287M16—21no—
288M16—22no—
289M16—22no—
290M16—21no—
291M16—21no—
292M16—41no—
293M16—41no—
294M16C22⋊Dic5312yes—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8C2×#8832211yescharacteristic
304C2×C8C2×#8832211yescharacteristic
305C2×C8D4×D5311yescharacteristic
306C2×C8D4×D5311yescharacteristic
307C2×C8#1171994711yescharacteristic
308C2×C8#1171994711yescharacteristic
309C2×C8C2×C22⋊D5311yescharacteristic
310C2×C8C2×C22⋊D5311yescharacteristic
311C42—21no—
312C42—21no—
313C16—1061no—
314C16—1061no—
315C16—1062no—
316C16—2121no—
317C16—2121no—
318C23—21no—
319C23—21no—
320C23#1313322411yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8C22⋊#8832212yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—41no—
331D4—41no—
332D4C22⋊#8832212yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4C2×C22⋊Dic5311yescharacteristic
343C2×C4D4×Dic5311yescharacteristic
344C2×C4D4×Dic5311yescharacteristic
345C2×C4C22×#8832211yescharacteristic
346C2×C4C2×#1171991711yescharacteristic
347C2×C4#1313322411yescharacteristic
348C8—22no—
349C8—21no—
350C8—21no—
351C8—41no—
352C8#1313318511yescharacteristic
353C8#1313318911yescharacteristic
354C8#1313318911yescharacteristic
355C8#1313319811yescharacteristic
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22#1681059011yescharacteristic
365C22D4×#8832211yescharacteristic
366C22#1681069511yescharacteristic
367C4—22no—
368C4—42no—
369C4—41no—
370C4#1681050311yescharacteristic
371C4C2×C22⋊#8832211yescharacteristic
372C4D4×#8832211yescharacteristic
373C4#1681069511yescharacteristic
374C2—21no—
375C2—21no—
376C2—42no—
377C2—81no—
378C2#2027500811yescharacteristic
379C2#2027655211yescharacteristic
380C2#2027655711yescharacteristic
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#443C2
socleC22×C53C8×D4
prime radicalC2×#135×C53C4
cosocleC22×C8×C53C23
hypercenterC2×#443D53
hypocenterC53#12980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#16811040C2semidirect
2. C2×#16810407C2semidirect
3. C2×#16810407C2semidirect
4. #20276329C2semidirect
5. #20272797C2semidirect
6. #20272793C2semidirect
219. C53#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#168110402
C2×#168104072
#202763292
#202727972
#202727932
C2×C53×#4432
#1298053