Finite Group Databasebeta

← #27301140

Subgroups — #27301140

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811053C211yesmaximal, characteristic
2C2×#16810740C211yesmaximal, characteristic, complemented
3C2×#16810431C211yesmaximal, characteristic, complemented
4#20275601C211yesmaximal, characteristic, complemented
5#20272808C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#359C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9C2×#13133224—21no—
10#16811053—22no—
11#16811053—22no—
12C2×#13133163C411yescharacteristic, complemented
13C2×#13133161C411yescharacteristic, complemented
14C2×#13133160C411yescharacteristic, complemented
15C2×#13133158C411yescharacteristic, complemented
16C22×#11719908C2211yescharacteristic
17#16810740—22no—
18#16810740—22no—
19C2×#13133153C2211yescharacteristic
20C2×#13133144—21no—
21C2×#13133143—21no—
22#16810591—21no—
23C2×C8×Dic53C2211yescharacteristic
24#16810431—22no—
25C2×#135×C53C2211yescharacteristic
26C22×C16×C53C2211yescharacteristic
27C2×#108×C53C2211yescharacteristic
28C53×#359—22no—
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37#13133224—24no—
38#13133224—42no—
39C2×#11719917—21no—
40C2×#11719917—41no—
41C22×#88322—41no—
42C22×#88322D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133163—22no—
45#13133161—22no—
46#13133160—22no—
47#13133158—22no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50C2×#11719908C2×C411yescharacteristic
51C2×#11719908C2×C411yescharacteristic
52#13133153—21no—
53#13133153D412yes—
54#13133144—24no—
55#13133143—42no—
56#13133129—41no—
57#13133129—41no—
58C8×Dic53—21no—
59C8×Dic53—21no—
60C2×#41×C53—21no—
61#135×C53—21no—
62#135×C53—21no—
63#135×C53D412yes—
64#135×C53D412yes—
65C2×Q16×C53C2×C411yescharacteristic
66C2×SD16×C53—21no—
67C2×D8×C53C2×C411yescharacteristic
68C2×M16×C53—21no—
69C22×C8×C53C2311yescharacteristic
70C2×C16×C53—21no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73#108×C53—21no—
74#108×C53—21no—
75#41×C53—22no—
76#41×C53—22no—
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C2×C22⋊D53—41no—
85C22×Dic53—21no—
86#11719945—41no—
87#11719945—41no—
88#11719945—41no—
89#11719945—41no—
90C2×D212—21no—
91C2×C4×D53—41no—
92C2×Dic106—21no—
93Q16×C53—21no—
94Q16×C53C22⋊C412yes—
95SD16×C53—22no—
96SD16×C53—22no—
97D8×C53—21no—
98D8×C53C22⋊C412yes—
99M16×C53—41no—
100M16×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C4×Dic53—21no—
109C4×Dic53—21no—
110#11719917—22no—
111#11719917—44no—
112#11719917—41no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—44no—
116C2×#88322—41no—
117C2×#88322D811yescharacteristic
118C2×#88322D811yescharacteristic
119Dic212—21no—
120Dic212—21no—
121D424—21no—
122D424—21no—
123#11719913—41no—
124#11719913—41no—
125C16×C53—41no—
126#11719908—21no—
127#11719908—21no—
128C23×C53—41no—
129C23×C53#13511yescharacteristic
130C22×D53—41no—
131Q8×C53—22no—
132Q8×C53—41no—
133D4×C53—22no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C22⋊D53—81no—
147C22⋊D53—81no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152D212—41no—
153D212—41no—
154C4×D53—81no—
155C4×D53—81no—
156Dic106—41no—
157Dic106—41no—
158C8×C53—81no—
159C8×C53C2×C22⋊C411yescharacteristic
160C8×C53C4×D411yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12311yescharacteristic
163#88322—21no—
164#88322—21no—
165#88322—42no—
166#13011—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—42no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53C4×D811yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#46311yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—42no—
179C4×C53#38611yescharacteristic
180C4×C53#44211yescharacteristic
181C4×C53#44711yescharacteristic
182C4×C53#45911yescharacteristic
183Dic53—42no—
184Dic53—81no—
185C2×#508—531no—
186C2×#443—531no—
187C2×#359—531no—
188C2×#359D5311yescharacteristic, complemented
189#1974—531no—
190#1381—531no—
191#1374—531no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—82no—
195C2×C53#187911yescharacteristic
196C2×C53#217811yescharacteristic
197C2×C53#217911yescharacteristic
198D53—161no—
199C2×#135—531no—
200C2×#135C2×D5311yescharacteristic
201C2×#131—1061no—
202#508—1062no—
203#508—1062no—
204C2×Q32—531no—
205C2×SD32—531no—
206C2×SD32—531no—
207C2×D16—531no—
208C22×C16—531no—
209C22×C16C2×D5311yescharacteristic
210#443—1062no—
211#443—1062no—
212C2×#108—531no—
213C2×#108C2×D5311yescharacteristic
214C2×#104—1061no—
215C2×#104—1061no—
216#408—1061no—
217C2×C4×C8—531no—
218#359—22no—
219#359—1062no—
220C53#1301111yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—1061no—
223#135—21no—
224#135—21no—
225#135—1061no—
226#135—1061no—
227#135—1061no—
228#135—1061no—
229#135D21212yes—
230#135C22⋊D5312yes—
231C2×Q16—531no—
232C2×Q16C4×D5311yescharacteristic
233C2×SD16—21no—
234C2×SD16—1061no—
235C2×D8—531no—
236C2×D8C4×D5311yescharacteristic
237#131—1064no—
238#131—2122no—
239C2×M16—21no—
240C2×M16—1061no—
241C2×M16—2121no—
242C22×C8—531no—
243C22×C8—2121no—
244C22×C8C22×D5311yescharacteristic
245C2×C42—531no—
246Q32—1062no—
247SD32—1062no—
248SD32—1062no—
249D16—1062no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—531no—
254C2×C16—531no—
255C2×C16—1061no—
256C2×C16—1061no—
257#108—21no—
258#108—21no—
259#108—532no—
260#108—1061no—
261#104—1064no—
262#104—2122no—
263C22⋊C8—2121no—
264C22⋊C8—2121no—
265C4×C8—1061no—
266C4×C8—1061no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271#41—2121no—
272#41—2121no—
273#41—2121no—
274#41—2121no—
275C2×Q8—21no—
276C2×Q8—1061no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—1061no—
280C2×D4—2121no—
281C22×C4—41no—
282C22×C4—1061no—
283C22×C4—2121no—
284C22×C4D4×D5311yescharacteristic
285Q16—21no—
286Q16—1061no—
287Q16—1061no—
288Q16#1171992112yes—
289SD16—22no—
290SD16—22no—
291SD16—2121no—
292SD16—2121no—
293D8—21no—
294D8—1061no—
295D8—1061no—
296D8#1171992112yes—
297M16—41no—
298M16—41no—
299M16—1062no—
300M16—2124no—
301M16—2121no—
302C2×C8—41no—
303C2×C8—531no—
304C2×C8—531no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—2124no—
308C2×C8—2121no—
309C2×C8C2×C4×D5311yescharacteristic
310C2×C8C2×D21211yescharacteristic
311C2×C8D4×D5311yescharacteristic
312C2×C8#1171994711yescharacteristic
313C2×C8#1171994711yescharacteristic
314C2×C8C2×C22⋊D5311yescharacteristic
315C42—1061no—
316C42—1061no—
317C16—41no—
318C16—1061no—
319C16—1061no—
320C23—41no—
321C23—2121no—
322C23#1313323811yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—2121no—
326Q8—2121no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—41no—
331D4—4241no—
332D4—4241no—
333D4—2121no—
334D4—2121no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—1062no—
339C2×C4—4241no—
340C2×C4—4241no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4#1313317711yescharacteristic
345C2×C4#1313318711yescharacteristic
346C2×C4#1313318911yescharacteristic
347C2×C4D8×D5311yescharacteristic
348C2×C4#1313324111yescharacteristic
349C2×C4#1313324411yescharacteristic
350C8—81no—
351C8—1061no—
352C8—1061no—
353C8—2122no—
354C8C2×#1171992111yescharacteristic
355C8#1313317711yescharacteristic
356C8#1313318911yescharacteristic
357C8#1313320111yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—4241no—
364C22—4241no—
365C22—4241no—
366C22#1681070311yescharacteristic
367C22#1681076911yescharacteristic
368C22#1681084011yescharacteristic
369C4—42no—
370C4—4241no—
371C4—2122no—
372C4#1681050011yescharacteristic
373C4#1681073211yescharacteristic
374C4#1681075811yescharacteristic
375C4#1681081611yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—8481no—
380C2#2027506111yescharacteristic
381C2#2027659311yescharacteristic
382C2#2027659811yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#359C2
socleC22×C53C4×D8
prime radicalC2×#16811053C2
cosocleC22×C8×C53C23
hypercenterC2×#359D53
hypocenterC53#13011
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C2×#16810740C2semidirect
3. C2×#16810431C2semidirect
4. #20275601C2semidirect
5. #20272808C2semidirect
6. #20272792C2semidirect
7. C2×C53×#359C2semidirect
12. C2×#13133163C4semidirect
13. C2×#13133161C4semidirect
14. C2×#13133160C4semidirect
15. C2×#13133158C4semidirect
188. C2×#359D53semidirect
220. C53#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#168110532
C2×#168107402
C2×#168104312
#202756012
#202728082
#202727922
C2×C53×#3592
#1301153