Finite Group Databasebeta

← #27278205

Subgroups — #27278205

Order 13568.

464 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283763C211yesmaximal, characteristic
2#20275018C211yesmaximal, characteristic
3#20274922C211yesmaximal, characteristic, complemented
4#20274900C211yesmaximal, characteristic, complemented
5#20272601C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1501C211yesmaximal, characteristic
8C22×#11719917C2211yescharacteristic
9#16810701—21no—
10#16810695—21no—
11#16810695—21no—
12D4×#88322—21no—
13#16810684—21no—
14C2×#13133153—21no—
15C2×C4⋊#88322—21no—
16C2×#13133144—21no—
17#16810590—21no—
18#16810587C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20C4×C22⋊Dic53C2211yescharacteristic
21#16810388—21no—
22#16810382—21no—
23#16810378—21no—
24C53×#525C2211yescharacteristic
25C2×C22⋊C8×C53C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C53×#328C411yescharacteristic, complemented
28C53×#328C411yescharacteristic, complemented
29C53×#327C411yescharacteristic, complemented
30C53×#327C411yescharacteristic, complemented
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719917—42no—
35C2×#11719917D411yescharacteristic
36C2×#11719917D411yescharacteristic
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322D411yescharacteristic
40C22×#88322D411yescharacteristic
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—21no—
43#13133153—42no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—42no—
47#13133144—42no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133128—21no—
55C4×#88322—21no—
56#13133124—21no—
57C2×#41×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53—21no—
60#123×C53—21no—
61C22⋊Q8×C53C2×C411yescharacteristic
62C22⋊Q8×C53Q811yescharacteristic
63#121×C53C2×C411yescharacteristic
64#121×C53Q811yescharacteristic
65#120×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68#117×C53D411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C42×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53C2×C411yescharacteristic
73C22⋊C8×C53C2×C411yescharacteristic
74#95×C53C2×C411yescharacteristic
75#95×C53C2×C411yescharacteristic
76C24×C53#4111yescharacteristic
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53C4⋊C411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53C4⋊C411yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×Q811yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—41no—
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53C4⋊C411yescharacteristic
103C4⋊C4×C53C4⋊C411yescharacteristic
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—42no—
110C42×C53—21no—
111C42×C53—41no—
112C42×C53—41no—
113C22⋊Dic53—22no—
114C22⋊Dic53—41no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117#11719917—22no—
118#11719917—22no—
119#11719917—41no—
120#11719917—41no—
121#11719917—41no—
122#11719917—41no—
123#11719917—81no—
124C2×#88322—22no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—81no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53C4×D411yescharacteristic
140C23×C53C22⋊Q811yescharacteristic
141C23×C53#12311yescharacteristic
142Q8×C53—41no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—22no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53C2×C4⋊C411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53C22⋊Q811yescharacteristic
172C2×Dic53—42no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C8×C53—41no—
176C8×C53—41no—
177#88322—42no—
178#88322—41no—
179#88322—41no—
180#88322—41no—
181#88322—41no—
182#8875—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53#38511yescharacteristic
197C22×C53#44311yescharacteristic
198C22×C53#44411yescharacteristic
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—42no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—81no—
207C4×C53B(4,𝔽2)11yescharacteristic
208C4×C53#45811yescharacteristic
209Dic53—82no—
210#2976—531no—
211#1871—531no—
212#1855—531no—
213#1851—531no—
214#1501D5311yescharacteristic
215#1297—531no—
216#1282—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—41no—
220C2×C53—41no—
221C2×C53—81no—
222C2×C53#186111yescharacteristic
223C2×C53#186611yescharacteristic
224C2×C53#186711yescharacteristic
225C22×M16—531no—
226#525C2×D5311yescharacteristic
227#436—1061no—
228#435—1061no—
229#435—1061no—
230C8×D4—1061no—
231#433—1061no—
232C2×#108—1061no—
233C2×C4⋊C8—1061no—
234C2×#104—1061no—
235#408—1061no—
236#407—531no—
237C2×C22⋊C8—531no—
238C2×C22⋊C8C2×D5311yescharacteristic
239C4×C22⋊C4—531no—
240C4×C22⋊C4C2×D5311yescharacteristic
241#341—1061no—
242#339—1061no—
243#336—1061no—
244#328Dic5311yescharacteristic, complemented
245#328Dic5311yescharacteristic, complemented
246#327Dic5311yescharacteristic, complemented
247#327Dic5311yescharacteristic, complemented
248C53#887511yescharacteristic, complemented, Sylow
249C2×#41D21211yescharacteristic
250C23×C4C22×D5311yescharacteristic
251C2×M16—531no—
252C2×M16—531no—
253C2×M16—1061no—
254C2×M16—1061no—
255C2×M16—1061no—
256C2×M16—2122no—
257C22×C8—21no—
258C22×C8—531no—
259C22×C8—531no—
260C22×C8—1061no—
261C22×C8—1061no—
262#123—21no—
263C22⋊Q8Dic10611yescharacteristic
264C22⋊Q8C2×Dic5311yescharacteristic
265#121Dic10611yescharacteristic
266#121C2×Dic5311yescharacteristic
267#120—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117D21211yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—1061no—
273C2×C42—21no—
274C2×C42—1061no—
275#108—2122no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8—2122no—
279#104—2122no—
280C22⋊C8—21no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284C22⋊C8—1061no—
285C22⋊C8—1061no—
286C22⋊C8—1061no—
287C22⋊C8C2×Dic5311yescharacteristic
288C22⋊C8C2×Dic5311yescharacteristic
289#97—1061no—
290C4×C8—1061no—
291#95—1061no—
292#95C2×Dic5311yescharacteristic
293#95C2×Dic5311yescharacteristic
294C24#1171994711yescharacteristic
295#41—41no—
296#41—41no—
297C2×Q8C4⋊Dic5311yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C4⋊Dic5311yescharacteristic
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—2121no—
308C22×C4C2×Dic10611yescharacteristic
309C22×C4C2×D21211yescharacteristic
310C22×C4D4×D5311yescharacteristic
311C22×C4D4×D5311yescharacteristic
312C22×C4#1171994711yescharacteristic
313C22×C4C22×Dic5311yescharacteristic
314M16—1062no—
315M16—1062no—
316M16—4241no—
317M16—2121no—
318M16—2121no—
319M16—2121no—
320M16—2121no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—1062no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—1061no—
329C2×C8—1061no—
330C2×C8—4241no—
331C2×C8—2121no—
332C2×C8—2121no—
333C2×C8—2121no—
334C2×C8—2121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C4⋊Dic5311yescharacteristic
338C4⋊C4C4⋊Dic5311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—42no—
345C22⋊C4—1062no—
346C22⋊C4—2121no—
347C42—21no—
348C42—41no—
349C42—41no—
350C42—2121no—
351C42—2121no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23D4×Dic5311yescharacteristic
358C23#1313319411yescharacteristic
359C23#1313320111yescharacteristic
360Q8—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—4241no—
387C2×C4—4241no—
388C2×C4—2122no—
389C2×C4C2×C4⋊Dic5311yescharacteristic
390C2×C4D4×Dic5311yescharacteristic
391C2×C4#1313318711yescharacteristic
392C2×C4#1313319411yescharacteristic
393C8—41no—
394C8—41no—
395C8—2122no—
396C8—2121no—
397C8—2121no—
398C8—2121no—
399C8—2121no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22#1681049811yescharacteristic
414C22#1681074011yescharacteristic
415C22#1681074511yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—81no—
424C4—4242no—
425C4#1681081111yescharacteristic
426C4#1681081411yescharacteristic
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—81no—
432C2#2027495711yescharacteristic
433C2#2027499211yescharacteristic
434C2#2027499811yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1501C2
socleC22×C53#385
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#1501D53
hypocenterC53#8875
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
3. #20274922C2semidirect
4. #20274900C2semidirect
5. #20272601C2semidirect
6. #20272561C2semidirect
27. C53×#328C4semidirect
28. C53×#328C4semidirect
29. C53×#327C4semidirect
30. C53×#327C4semidirect
244. #328Dic53semidirect
245. #328Dic53semidirect
246. #327Dic53semidirect
247. #327Dic53semidirect
248. C53#8875semidirect

Maximal subgroups

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

subgroupindex
#202837632
#202750182
#202749222
#202749002
#202726012
#202725612
C53×#15012
#887553