Finite Group Databasebeta

← #27300969

Subgroups — #27300969

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×#16810738C211yesmaximal, characteristic
2C2×#16810408C211yesmaximal, characteristic, complemented
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276328C211yesmaximal, characteristic, complemented
5#20272796C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#504C211yesmaximal, characteristic, complemented
8C2×#13133237C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810738—22no—
13#16810738—22no—
14C2×#13133152C411yescharacteristic
15C2×#13133144C411yescharacteristic
16C2×#13133143C411yescharacteristic
17C2×C8×Dic53C2211yescharacteristic
18#16810408—22no—
19#16810407—22no—
20#16810402—21no—
21#16810402—21no—
22C2×#131×C53C2211yescharacteristic
23C53×#504—22no—
24C53×#504—22no—
25C2×M32×C53—21no—
26C2×M32×C53C2211yescharacteristic
27C22×C16×C53—21no—
28C22×C16×C53C2211yescharacteristic
29C2×#11719945C2×C411yescharacteristic
30#13133237—22no—
31#13133237—22no—
32C2×#11719925C811yescharacteristic
33C2×#11719923C811yescharacteristic
34C2×#11719924C811yescharacteristic
35C2×#11719922C811yescharacteristic
36#13133223—21no—
37#13133223—21no—
38#13133223—21no—
39#13133223—21no—
40C2×#11719917C2×C411yescharacteristic
41C2×#11719912—21no—
42C22×#88322C2×C411yescharacteristic
43C2×C8×D53—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45#13133157—21no—
46#13133157—21no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51#13133152—22no—
52#13133144—22no—
53#13133143—22no—
54C8×Dic53—21no—
55C8×Dic53—21no—
56C2×#41×C53C2×C411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53D412yes—
60#131×C53D412yes—
61C2×M16×C53—21no—
62C2×M16×C53C2×C411yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53C2311yescharacteristic
65M32×C53—24no—
66M32×C53—21no—
67M32×C53D412yes—
68C2×C16×C53—24no—
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71C2×C16×C53—21no—
72C2×C16×C53D411yescharacteristic
73C2×C16×C53D411yescharacteristic
74#41×C53—21no—
75#41×C53—21no—
76#41×C53C22⋊C412yes—
77#41×C53M1612yes—
78C2×Q8×C53C2×C811yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53C2×C811yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53C22×C411yescharacteristic
83C2×C22⋊D53—21no—
84C22×Dic53—21no—
85#11719945—21no—
86#11719945—21no—
87#11719945—21no—
88#11719945—21no—
89C2×D212C2×C811yescharacteristic
90C2×C4×D53—21no—
91C2×Dic106C2×C811yescharacteristic
92M16×C53—21no—
93M16×C53—22no—
94M16×C53—22no—
95M16×C53C22⋊C412yes—
96C2×C8×C53—21no—
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C2×C8×C53#4111yescharacteristic
105#11719925—22no—
106#11719924—22no—
107#11719923—22no—
108#11719922—22no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111#11719917—21no—
112#11719917—21no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322C2×C811yescharacteristic
117C2×#88322C2×C811yescharacteristic
118#11719912—41no—
119#11719912—41no—
120C8×D53—41no—
121C8×D53—41no—
122C16×C53—21no—
123C16×C53—22no—
124C16×C53—21no—
125#11719908—41no—
126#11719908—41no—
127C23×C53—21no—
128C23×C53#13111yescharacteristic
129C22×D53—21no—
130Q8×C53—21no—
131Q8×C53C22⋊C812yes—
132D4×C53—22no—
133D4×C53—22no—
134D4×C53—21no—
135D4×C53C22⋊C812yes—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C22×C811yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13111yescharacteristic
145C22⋊D53—41no—
146C22⋊D53—41no—
147C2×Dic53—22no—
148C2×Dic53—21no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151D212—21no—
152D212—21no—
153C4×D53—41no—
154C4×D53—41no—
155Dic106—21no—
156Dic106—21no—
157C8×C53—22no—
158C8×C53#12011yescharacteristic
159C8×C53#12111yescharacteristic
160C8×C53#12111yescharacteristic
161C8×C53#12311yescharacteristic
162#88322—21no—
163#88322—21no—
164#88322—41no—
165#12980—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53#40811yescharacteristic
172C22×C53C8×D411yescharacteristic
173C22×C53#43511yescharacteristic
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—22no—
178C4×C53#38611yescharacteristic
179C4×C53C2×C22⋊C811yescharacteristic
180C4×C53C8×D411yescharacteristic
181C4×C53#43511yescharacteristic
182Dic53—42no—
183Dic53—41no—
184C2×#504D5311yescharacteristic, complemented
185C2×#443—531no—
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#186911yescharacteristic
195C2×C53#217011yescharacteristic
196C2×C53#217111yescharacteristic
197D53—81no—
198C2×#135—531no—
199C2×#131—531no—
200C2×#131C2×D5311yescharacteristic
201#504—22no—
202#504—22no—
203C2×M32—21no—
204C2×M32—531no—
205C2×M32C2×D5311yescharacteristic
206C22×C16—21no—
207C22×C16—531no—
208C22×C16C2×D5311yescharacteristic
209#443—1062no—
210#443—1062no—
211C2×#108—531no—
212C2×#104—531no—
213C2×#104—531no—
214C2×C4×C8—531no—
215#350—1062no—
216#350—1062no—
217C22⋊C16—1061no—
218C22⋊C16—1061no—
219C53#1298011yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41C4×D5311yescharacteristic
222#135—1062no—
223#135—1062no—
224C2×Q16—531no—
225C2×SD16—531no—
226C2×SD16—531no—
227C2×D8—531no—
228#131—21no—
229#131—21no—
230#131—1061no—
231#131—1061no—
232#131—1061no—
233#131—1061no—
234#131D21212yes—
235#131C22⋊D5312yes—
236C2×M16—21no—
237C2×M16—531no—
238C2×M16—1061no—
239C2×M16C4×D5311yescharacteristic
240C22×C8—21no—
241C22×C8—531no—
242C22×C8—1061no—
243C22×C8C22×D5311yescharacteristic
244C2×C42—531no—
245M32—24no—
246M32—21no—
247M32—1061no—
248M32—1061no—
249M32C22⋊D5312yes—
250C2×C16—24no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16—1061no—
255C2×C16—1061no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16C22⋊D5311yescharacteristic
259C2×C16C22⋊D5311yescharacteristic
260#108—1062no—
261#104—1062no—
262#104—1062no—
263C4×C8—1061no—
264C4×C8—1061no—
265#41—21no—
266#41—21no—
267#41—1061no—
268#41—1061no—
269#41—1061no—
270#41—1061no—
271#41#1171991212yes—
272#41#1171992112yes—
273C2×Q8—531no—
274C2×Q8C8×D5311yescharacteristic
275C2×D4—21no—
276C2×D4—531no—
277C2×D4—1061no—
278C2×D4C8×D5311yescharacteristic
279C22×C4—21no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4C2×C4×D5311yescharacteristic
283Q16—1062no—
284SD16—1062no—
285SD16—1062no—
286D8—1062no—
287M16—21no—
288M16—22no—
289M16—22no—
290M16—1061no—
291M16—1061no—
292M16—2121no—
293M16—2121no—
294M16#1171992112yes—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—531no—
299C2×C8—531no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—2121no—
304C2×C8—2121no—
305C2×C8C2×D21211yescharacteristic
306C2×C8#1171994511yescharacteristic
307C2×C8#1171994511yescharacteristic
308C2×C8C2×C22⋊D5311yescharacteristic
309C2×C8C2×C22⋊D5311yescharacteristic
310C2×C8C2×C22⋊D5311yescharacteristic
311C42—1061no—
312C42—1061no—
313C16—21no—
314C16—22no—
315C16—21no—
316C16—2121no—
317C16—2121no—
318C23—21no—
319C23—1061no—
320C23#1313322311yescharacteristic
321Q8—21no—
322Q8—1061no—
323Q8—1061no—
324Q8#1313312912yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—1061no—
329D4—1061no—
330D4—2121no—
331D4—2121no—
332D4#1313312912yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—1062no—
337C2×C4—1061no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4C2×#1171992111yescharacteristic
343C2×C4C4×C22⋊D5311yescharacteristic
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4C2×C8×D5311yescharacteristic
346C2×C4C2×#1171991211yescharacteristic
347C2×C4#1313322311yescharacteristic
348C8—22no—
349C8—1061no—
350C8—1061no—
351C8—2121no—
352C8#1313318611yescharacteristic
353C8#1313319111yescharacteristic
354C8#1313319111yescharacteristic
355C8#1313320011yescharacteristic
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22#1681058911yescharacteristic
365C22C8×C22⋊D5311yescharacteristic
366C22#1681069411yescharacteristic
367C4—22no—
368C4—2122no—
369C4—2121no—
370C4#1681050211yescharacteristic
371C4C2×#1313312911yescharacteristic
372C4C8×C22⋊D5311yescharacteristic
373C4#1681069411yescharacteristic
374C2—21no—
375C2—21no—
376C2—42no—
377C2—4241no—
378C2#2027500711yescharacteristic
379C2#2027655011yescharacteristic
380C2#2027655511yescharacteristic
subgroupG/H
centerC2×C8C2×C22⋊D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#504C2
socleC22×C53C8×D4
prime radicalC2×#13133237C4
cosocleC22×C8×C53C23
hypercenterC2×#504D53
hypocenterC53#12980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C16 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. C2×#16810408C2semidirect
3. C2×#16810407C2semidirect
4. #20276328C2semidirect
5. #20272796C2semidirect
6. #20272792C2semidirect
7. C2×C53×#504C2semidirect
184. C2×#504D53semidirect
219. C53#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#168107382
C2×#168104082
C2×#168104072
#202763282
#202727962
#202727922
C2×C53×#5042
#1298053