Finite Group Databasebeta

← #27278960

Subgroups — #27278960

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11719917C211yesmaximal, characteristic, complemented
2#20275635C211yesmaximal, characteristic
3C4×#13133141C211yesmaximal, characteristic, complemented
4#20273166C211yesmaximal, characteristic
5#20272617C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#3103C211yesmaximal, characteristic
8#16810754—22no—
9#16810754—21no—
10Q8×#88322—21no—
11#16810647—21no—
12#16810647C2211yescharacteristic
13C2×#13133144—21no—
14C2×#13133144—21no—
15C2×#13133141—21no—
16C4×#11719917—21no—
17C4×#11719917C2211yescharacteristic
18C2×C4×#88322C2211yescharacteristic
19C4×C4⋊Dic53C2211yescharacteristic
20#16810385—21no—
21D4⋊#88322—22no—
22C53×#532C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#481—21no—
25C53×#475C411yescharacteristic, complemented
26C53×#475C411yescharacteristic, complemented
27C53×#460C411yescharacteristic, complemented
28C53×#460C411yescharacteristic, complemented
29C53×#448—21no—
30C4×SD16×C53—21no—
31C4×SD16×C53—21no—
32C53×#431C2211yescharacteristic
33C2×#11719917—21no—
34C2×#11719917D411yescharacteristic
35C2×#11719917D411yescharacteristic
36C22×#88322—21no—
37C2×C4⋊Dic53—21no—
38C2×C4×Dic53—21no—
39C4⋊#88322—22no—
40C4⋊#88322—22no—
41C4⋊#88322—21no—
42C4⋊#88322D412yes—
43#13133144—42no—
44#13133144—42no—
45#13133141—24no—
46#13133128—22no—
47#13133128—21no—
48C4×#88322—22no—
49C4×#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52#13133124—21no—
53C2×#41×C53—21no—
54C22×Q8×C53D411yescharacteristic
55C2×SD16×C53—21no—
56C2×SD16×C53—21no—
57C4⋊Q8×C53C2×C411yescharacteristic
58#127×C53C2×C411yescharacteristic
59#124×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53C2×C411yescharacteristic
65C4×Q8×C53C2×C411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67C4×Q8×C53D411yescharacteristic
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#106×C53—21no—
75#106×C53—21no—
76C4⋊C8×C53C2×C411yescharacteristic
77C4⋊C8×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84#41×C53—41no—
85#41×C53—41no—
86#41×C53—41no—
87#41×C53—41no—
88C2×Q8×C53—21no—
89C2×Q8×C53—22no—
90C2×Q8×C53—21no—
91C2×Q8×C53—41no—
92C2×Q8×C53C22⋊C411yescharacteristic
93C2×Q8×C53C22⋊C411yescharacteristic
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—41no—
105SD16×C53—42no—
106SD16×C53—42no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53C22⋊C411yescharacteristic
117C4⋊C4×C53C22⋊C411yescharacteristic
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—21no—
124C42×C53C22×C411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C4⋊Dic53—22no—
129C4⋊Dic53—41no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132#11719917—21no—
133#11719917—21no—
134#11719917—21no—
135#11719917—41no—
136#11719917—41no—
137#11719917SD1612yescomplemented
138C2×#88322—21no—
139C2×#88322—22no—
140C2×#88322—21no—
141C2×#88322—22no—
142C2×#88322—41no—
143C2×#88322—41no—
144C23×C53—41no—
145C23×C53#12111yescharacteristic
146Q8×C53—21no—
147Q8×C53—21no—
148Q8×C53—21no—
149Q8×C53—42no—
150Q8×C53—41no—
151Q8×C53—41no—
152Q8×C53#10412yescomplemented
153D4×C53—42no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—81no—
159D4×C53—81no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53—81no—
172C2×C4×C53C2×C22⋊C411yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×C4×C53C2×SD1611yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53#13711yescharacteristic
182C2×Dic53—42no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C8×C53—41no—
186C8×C53—41no—
187#88322—22no—
188#88322—42no—
189#88322—41no—
190#88322—42no—
191#8992—531nomaximal, Sylow
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#38611yescharacteristic
198C22×C53#46011yescharacteristic
199C22×C53#46411yescharacteristic
200C4×C53—21no—
201C4×C53—22no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—81no—
208C4×C53C2×#10411yescharacteristic
209C4×C53#42111yescharacteristic
210C4×C53C4×SD1611yescharacteristic
211C4×C53#44011yescharacteristic
212C4×C53#44811yescharacteristic
213C4×C53#44911yescharacteristic
214Dic53—82no—
215#3103D5311yescharacteristic
216Q8×M16—531no—
217#1980—531no—
218C4×#103—531no—
219#1492—531no—
220#1303—531no—
221#1278—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—161no—
225C2×C53#187111yescharacteristic
226C2×C53#188011yescharacteristic
227C2×C53#188111yescharacteristic
228#532C2×D5311yescharacteristic
229C2×C4×Q8C2×D5311yescharacteristic
230#481—21no—
231#475Dic5311yescharacteristic, complemented
232#475Dic5311yescharacteristic, complemented
233#460Dic5311yescharacteristic, complemented
234#460Dic5311yescharacteristic, complemented
235#448—21no—
236#446—1062no—
237#446—1061no—
238C8×Q8—1061no—
239C4×SD16—21no—
240C4×SD16—21no—
241#431C2×D5311yescharacteristic
242#423—531no—
243#423—1061no—
244C2×#104—1061no—
245C2×#104—1061no—
246C2×#103—1061no—
247C4×M16—531no—
248C4×M16—1061no—
249C2×C4×C8—531no—
250C4×C4⋊C4—531no—
251#340—1061no—
252D4⋊C8—1062no—
253C53#899211yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8C22⋊D5311yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—531no—
259C2×M16—531no—
260C2×M16—1061no—
261C22×C8—1061no—
262C4⋊Q8C2×Dic5311yescharacteristic
263#127C2×Dic5311yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×Dic5311yescharacteristic
270C4×Q8C2×Dic5311yescharacteristic
271C4×Q8C22⋊D5311yescharacteristic
272C4×Q8C22⋊D5311yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C42—21no—
279C2×C42—1061no—
280C2×C42C22×D5311yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—532no—
284C4⋊C8—1062no—
285C4⋊C8—1062no—
286C4⋊C8—1061no—
287C4⋊C8C2×Dic5311yescharacteristic
288C4⋊C8C2×Dic5311yescharacteristic
289#104—2122no—
290#104—2122no—
291#103—21no—
292#103—21no—
293#103—1064no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—1062no—
298#97—1061no—
299C4×C8—21no—
300C4×C8—1062no—
301C4×C8—1061no—
302C4×C8—1061no—
303C4×C8—1061no—
304#95—1061no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊Dic5311yescharacteristic
314C2×Q8C22⋊Dic5311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—2121no—
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—532no—
329M16—1061no—
330M16—1061no—
331M16—1061no—
332M16—2121no—
333M16—2121no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—1061no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—1061no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—1062no—
350C4⋊C4—2121no—
351C4⋊C4C22⋊Dic5311yescharacteristic
352C4⋊C4C22⋊Dic5311yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—2121no—
360C42—2121no—
361C42D4×D5311yescharacteristic
362C42#1171994711yescharacteristic
363C42C22×Dic5311yescharacteristic
364C42C2×C22⋊D5311yescharacteristic
365C23—41no—
366C23#1313318911yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#1313314412yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—2122no—
396C2×C4C2×C22⋊Dic5311yescharacteristic
397C2×C4D4×Dic5311yescharacteristic
398C2×C4D4×Dic5311yescharacteristic
399C2×C4#1313318511yescharacteristic
400C2×C4#1313318911yescharacteristic
401C2×C4#1313319811yescharacteristic
402C2×C4SD16×D5311yescharacteristic
403C2×C4#1313323811yescharacteristic
404C2×C4#1313324411yescharacteristic
405C2×C4#1313324611yescharacteristic
406C8—41no—
407C8—41no—
408C8—1062no—
409C8—2122no—
410C8—2121no—
411C8—2122no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#1681050311yescharacteristic
418C22#1681082111yescharacteristic
419C22#1681084311yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—4242no—
429C4C2×#1313314411yescharacteristic
430C4#1681064011yescharacteristic
431C4SD16×Dic5311yescharacteristic
432C4#1681072411yescharacteristic
433C4#1681076011yescharacteristic
434C4#1681076711yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2#2027501811yescharacteristic
439C2#2027507211yescharacteristic
440C2#2027507911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3103C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3103D53
hypocenterC53#8992
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. Q8×#11719917C2semidirect
3. C4×#13133141C2semidirect
5. #20272617C2semidirect
6. #20272551C2semidirect
25. C53×#475C4semidirect
26. C53×#475C4semidirect
27. C53×#460C4semidirect
28. C53×#460C4semidirect
137. #11719917SD16semidirect
152. Q8×C53#104semidirect
231. #475Dic53semidirect
232. #475Dic53semidirect
233. #460Dic53semidirect
234. #460Dic53semidirect
253. C53#8992semidirect
373. Q8#13133144semidirect

Maximal subgroups

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

subgroupindex
Q8×#117199172
#202756352
C4×#131331412
#202731662
#202726172
#202725512
C53×#31032
#899253