Finite Group Databasebeta

← #27277280

Subgroups — #27277280

Order 13568.

408 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283671C211yesmaximal, characteristic
2C4×#13133152C211yesmaximal, characteristic, complemented
3#20273473C211yesmaximal, characteristic, complemented
4#20273155C211yesmaximal, characteristic
5#20273152C211yesmaximal, characteristic
6#20272606C211yesmaximal, characteristic, complemented
7C53×#1299C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810747C411yescharacteristic
10#16810747C411yescharacteristic
11#16810688C411yescharacteristic
12#16810688C411yescharacteristic
13#16810686—21no—
14#16810686—21no—
15#16810674—21no—
16C2×#13133152—21no—
17#16810647C2211yescharacteristic
18C4×#11719917C2211yescharacteristic
19C2×C4×#88322C2211yescharacteristic
20C4×C8×D53—21no—
21#16810387—21no—
22#16810375—21no—
23#16810375—21no—
24#16810355—21no—
25#16810353—21no—
26#16810352—21no—
27#16810350—21no—
28C53×#431C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C4×M16×C53C2211yescharacteristic
31C2×#11719945Q811yescharacteristic
32C2×#11719917—21no—
33C2×#11719917—21no—
34C22×#88322D411yescharacteristic
35C22×#88322D411yescharacteristic
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C4×Dic106C2×C411yescharacteristic
39C4×Dic106D411yescharacteristic
40C4×D212C2×C411yescharacteristic
41C4×D212D411yescharacteristic
42C4×C22⋊D53—21no—
43C4×C22⋊D53—21no—
44#13133176Q811yescharacteristic
45#13133174—21no—
46C42×D53—21no—
47#13133152—24no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133146—21no—
51#13133146—21no—
52#13133129—21no—
53#13133129—21no—
54#13133128—21no—
55C4×#88322—21no—
56C4×#88322C2×C411yescharacteristic
57C4×#88322C2×C411yescharacteristic
58C4×#88322D411yescharacteristic
59C4×#88322D411yescharacteristic
60C8×Dic53—21no—
61C8×Dic53—21no—
62C2×M16×C53—21no—
63C2×M16×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53C2×C411yescharacteristic
68C4⋊C8×C53C2×C411yescharacteristic
69C22⋊C8×C53—21no—
70#97×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×Q811yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C2×C22⋊D53—21no—
77#11719945—22no—
78#11719945—22no—
79C2×D212C4⋊C411yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×Dic106C4⋊C411yescharacteristic
84M16×C53—22no—
85M16×C53—41no—
86M16×C53—81no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C42×C53C22×C411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53#4111yescharacteristic
95C22⋊Dic53—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53C4⋊C411yescharacteristic
99C4⋊Dic53C4⋊C411yescharacteristic
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—22no—
106#11719917—41no—
107#11719917—81no—
108C2×#88322—22no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322C22⋊C411yescharacteristic
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322C22⋊C411yescharacteristic
115C2×#88322C22⋊C411yescharacteristic
116C8×D53—42no—
117C8×D53—42no—
118C23×C53C22⋊Q811yescharacteristic
119C22×D53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53C2×C4⋊C411yescharacteristic
128C2×C4×C53#11711yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53C22⋊Q811yescharacteristic
132C22⋊D53—24no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136D212—21no—
137D212#10812yescomplemented
138C4×D53—24no—
139C4×D53—41no—
140C4×D53—41no—
141C4×D53—41no—
142C4×D53—41no—
143Dic106—21no—
144Dic106#10812yescomplemented
145C8×C53—42no—
146C8×C53—41no—
147C8×C53—41no—
148C8×C53—81no—
149#88322—21no—
150#88322—21no—
151#88322—21no—
152#88322—42no—
153#88322—81no—
154#88322#10412yes—
155#8671—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#38011yescharacteristic
160C2×D53—22no—
161C2×D53—41no—
162C2×D53—41no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53C2×#10411yescharacteristic
167C4×C53#42111yescharacteristic
168C4×C53C2×#10811yescharacteristic
169C4×C53C2×#10811yescharacteristic
170C4×C53#47211yescharacteristic
171C4×C53#47311yescharacteristic
172Dic53—22no—
173Dic53—41no—
174Dic53—41no—
175C8×#41—531no—
176C4×#108—531no—
177#1567—531no—
178#1489—531no—
179#1489—531no—
180#1299—531no—
181#1299D5311yescharacteristic, complemented
182C2×C53—41no—
183C2×C53#180611yescharacteristic
184C2×C53#181011yescharacteristic
185C2×C53#181611yescharacteristic
186D53—42no—
187C4×#41—531no—
188C8×Q8—531no—
189C8×Q8—531no—
190C8×D4—531no—
191C8×D4—531no—
192C8×D4—1061no—
193C8×D4—1061no—
194#431—1061no—
195#431C2×D5311yescharacteristic
196C2×#108—1061no—
197#423—531no—
198#423C2×D5311yescharacteristic
199C4×M16—531no—
200C4×M16C2×D5311yescharacteristic
201C2×C4×C8—531no—
202C2×C4×C8—1061no—
203#341—1061no—
204#335—1061no—
205#335—1061no—
206Q8⋊C8—1061no—
207Q8⋊C8—1061no—
208D4⋊C8—1061no—
209D4⋊C8—1061no—
210C53#867111yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—1061no—
215C2×M16—1061no—
216C22×C8—531no—
217C22×C8—531no—
218C22×C8—1061no—
219C22×C8—1061no—
220C4×Q8—531no—
221C4×Q8—531no—
222C4×D4—531no—
223C4×D4—531no—
224C4×D4—1061no—
225C4×D4—1061no—
226#117—531no—
227#117—1061no—
228C2×C42—1061no—
229C2×C42C22×D5311yescharacteristic
230#108—1064no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—1061no—
234C4⋊C8—1061no—
235C4⋊C8—1061no—
236C4⋊C8—1061no—
237C4⋊C8C4×D5311yescharacteristic
238C4⋊C8C4×D5311yescharacteristic
239C22⋊C8—21no—
240C22⋊C8—1061no—
241C22⋊C8—1061no—
242#97—21no—
243#97—1061no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—531no—
247C4×C8—531no—
248C4×C8—531no—
249C4×C8—531no—
250C4×C8—1061no—
251C4×C8—1061no—
252C4×C8—1061no—
253#41—1062no—
254#41—1062no—
255C2×Q8—531no—
256C2×D4—531no—
257C2×D4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4#1171994511yescharacteristic
262C22×C4D4×D5311yescharacteristic
263C22×C4Q8×D5311yescharacteristic
264M16—22no—
265M16—41no—
266M16—81no—
267M16—1062no—
268M16—4241no—
269M16—2121no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—531no—
278C2×C8—1062no—
279C2×C8—2122no—
280C2×C8—2122no—
281C2×C8—2121no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—531no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C42—1061no—
292C42—1061no—
293C42—1061no—
294C42C2×C4×D5311yescharacteristic
295C42#1171994511yescharacteristic
296C42D4×D5311yescharacteristic
297C42D4×D5311yescharacteristic
298C23—1061no—
299C23#1313319511yescharacteristic
300Q8—532no—
301Q8—1061no—
302D4—532no—
303D4—1064no—
304D4—1061no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1064no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4C22⋊C4×D5311yescharacteristic
320C2×C4C4⋊C4×D5311yescharacteristic
321C2×C4#1313317411yescharacteristic
322C2×C4#1313319011yescharacteristic
323C2×C4#1313319011yescharacteristic
324C2×C4#1313319511yescharacteristic
325C8—42no—
326C8—41no—
327C8—41no—
328C8—81no—
329C8—532no—
330C8—1061no—
331C8—1061no—
332C8—1061no—
333C8—4241no—
334C8—2122no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—1062no—
339C22—2121no—
340C22—2121no—
341C22#1681048211yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—1062no—
346C4—2121no—
347C4—2121no—
348C4#104×D5311yescharacteristic
349C4#1681063811yescharacteristic
350C4#108×D5311yescharacteristic
351C4#108×D5311yescharacteristic
352C4#1681089211yescharacteristic
353C4#1681089711yescharacteristic
354C2—41no—
355C2—2122no—
356C2#2027473411yescharacteristic
357C2#2027475211yescharacteristic
358C2#2027477811yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1299C2
socleC22×C53#380
prime radicalC2×#11719945Q8
cosocleC2×C42×C53C23
hypercenter#1299D53
hypocenterC53#8671
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1299

Complements

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

NH ≅ G/Ntype
2. C4×#13133152C2semidirect
3. #20273473C2semidirect
6. #20272606C2semidirect
7. C53×#1299C2semidirect
137. D212#108semidirect
144. Dic106#108semidirect
181. #1299D53semidirect
210. C53#8671semidirect

Maximal subgroups

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

subgroupindex
#202836712
C4×#131331522
#202734732
#202731552
#202731522
#202726062
C53×#12992
#867153