Finite Group Databasebeta

← #27301157

Subgroups — #27301157

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133162C211yesmaximal, characteristic
2C2×#16810717C211yesmaximal, characteristic, complemented
3C2×#16810426C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic
5#20275579C211yesmaximal, characteristic
6#20272792C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×#13133182—21no—
11C2×#13133162—22no—
12C2×#13133162—22no—
13C2×#13133162C414yescomplemented
14C22×#11719908C2211yescharacteristic
15#16810717—22no—
16#16810717—22no—
17C2×#13133140—21no—
18C2×#13133137—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810524—21no—
21#16810426—22no—
22#16810423—22no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×D212—21no—
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212C2×C412yes—
32C2×Dic212D412yes—
33C2×Dic212D412yes—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424—21no—
38C2×D424—21no—
39C2×D424C2×C412yes—
40C22×#88322D411yescharacteristic
41#13133182—24no—
42#13133182—42no—
43C2×#11719919—41no—
44C2×#11719921—41no—
45C2×#11719921—41no—
46C2×C4×Dic53—41no—
47C2×C4×Dic53D411yescharacteristic
48#13133162—28no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51C2×#11719908—21no—
52#13133140—24no—
53#13133137—42no—
54C8×Dic53—21no—
55C8×Dic53—21no—
56C22×C8×C53C2311yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53C2×C412yes—
62#107×C53—21no—
63#107×C53—21no—
64#107×C53—21no—
65#107×C53D412yes—
66C23×D53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—41no—
71C22×Dic53D811yescharacteristic
72C22×Dic53D811yescharacteristic
73C2×D212—22no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×C8×C53C22×C411yescharacteristic
86C2×C8×C53C2×D411yescharacteristic
87C2×C8×C53C2×D411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C4⋊C4×C53—22no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95#11719921—82no—
96#11719921—82no—
97#11719919—44no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—44no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322—21no—
104Dic212—22no—
105Dic212—22no—
106Dic212—22no—
107Dic212C22⋊C414yescomplemented
108D424—22no—
109D424—22no—
110D424—41no—
111D424—41no—
112C16×C53—22no—
113#11719908—41no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122C2×C4×C53—42no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53C4×D411yescharacteristic
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53SD3212yescomplemented
143D212—42no—
144D212—42no—
145D212—81no—
146D212—81no—
147Dic106—24no—
148Dic106—42no—
149Dic106—42no—
150Dic106—42no—
151C8×C53C2×C22⋊C411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—41no—
156#13013—531nomaximal, Sylow
157C22×C53C4×D811yescharacteristic
158C22×C53#44711yescharacteristic
159C22×C53#45911yescharacteristic
160C22×C53C2×SD3211yescharacteristic
161C22×C53#50811yescharacteristic
162C22×C53#50911yescharacteristic
163C22×C53#51011yescharacteristic
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—161no—
168C2×D53—161no—
169C2×D53—161no—
170C2×D53—161no—
171C4×C53—82no—
172C4×C53—161no—
173C4×C53#38611yescharacteristic
174C4×C53#44211yescharacteristic
175C4×C53#44911yescharacteristic
176C4×C53#46311yescharacteristic
177Dic53—22no—
178Dic53—44no—
179Dic53—41no—
180C22×SD32—531no—
181C2×C4×Q16—531no—
182C2×#358—531no—
183C2×#357—531no—
184#1970—531no—
185#1380D5311yescharacteristic, complemented
186#1374—531no—
187C2×C53#187911yescharacteristic
188C2×C53C4×SD3211yescharacteristic
189C2×C53#217511yescharacteristic
190C2×C53#218511yescharacteristic
191C2×C53#218611yescharacteristic
192C2×C53#220711yescharacteristic
193C2×C53#221111yescharacteristic
194D53—162no—
195C22×Q16—531no—
196C22×D8—531no—
197C2×C4×Q8—1061no—
198C2×SD32—534no—
199C2×SD32—1062no—
200C2×SD32—1062no—
201C22×C16—531no—
202C22×C16C2×D5311yescharacteristic
203C4×Q16—1062no—
204C4×Q16—1062no—
205C2×#107C2×D5311yescharacteristic
206C2×#107C2×D5311yescharacteristic
207C2×#103—1061no—
208C2×#102—1061no—
209C2×C4×C8—531no—
210#390—1061no—
211#358—1062no—
212#357—1062no—
213C53#1301311yescharacteristic, complemented, Sylow
214C22×Q8—1061no—
215C22×D4—1061no—
216C2×Q16—532no—
217C2×Q16—532no—
218C2×Q16—532no—
219C2×Q16—1061no—
220C2×Q16—1061no—
221C2×Q16—1061no—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C22×C8—531no—
229C22×C8C22×D5311yescharacteristic
230C4×Q8—1064no—
231C4×Q8—2122no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—2121no—
235C2×C22⋊C4—2121no—
236C2×C22⋊C4—2121no—
237C2×C42—531no—
238C2×C42—2121no—
239SD32—1068no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245C2×C16C4×D5312yes—
246#107—21no—
247#107—21no—
248#107—21no—
249#107D21212yes—
250#103—1064no—
251#102—2122no—
252C4×C8—1061no—
253C4×C8—1061no—
254C24—2121no—
255C2×Q8—1062no—
256C2×Q8—1062no—
257C2×Q8—1062no—
258C2×Q8—2121no—
259C2×Q8—2121no—
260C2×Q8—2121no—
261C2×D4—1062no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—2121no—
272C22×C4D4×D5311yescharacteristic
273Q16—534no—
274Q16—1062no—
275Q16—1062no—
276Q16—1062no—
277D8—1062no—
278D8—1062no—
279D8—2121no—
280D8—2121no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8C2×C4×D5311yescharacteristic
285C2×C8C2×D21211yescharacteristic
286C2×C8#1171994511yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8D4×D5311yescharacteristic
289C2×C8#1171994911yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—2124no—
295C22⋊C4—4242no—
296C22⋊C4—4242no—
297C42—1061no—
298C42—1061no—
299C42—2124no—
300C16—22no—
301C16—2121no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—4241no—
308C23—2122no—
309C23D8×D5311yescharacteristic
310Q8—1064no—
311Q8—2122no—
312Q8—2122no—
313Q8—2122no—
314D4—4241no—
315D4—4241no—
316D4—2122no—
317D4—2122no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—532no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4#1313317911yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313319011yescharacteristic
336C2×C4#1313324011yescharacteristic
337C2×C4#1313324411yescharacteristic
338C2×C4#1313324411yescharacteristic
339C8—2121no—
340C8C22⋊C4×D5311yescharacteristic
341C8C4×D21211yescharacteristic
342C8#1313319211yescharacteristic
343C8#1313320211yescharacteristic
344C22—8481no—
345C22—8481no—
346C22—8481no—
347C22—8481no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22#1681070511yescharacteristic
352C22#1681075811yescharacteristic
353C22#1681081711yescharacteristic
354C22SD32×D5311yescharacteristic
355C22#1681105411yescharacteristic
356C22#1681105711yescharacteristic
357C22#1681106511yescharacteristic
358C4—82no—
359C4—161no—
360C4—1062no—
361C4—2124no—
362C4—2121no—
363C4#1681050111yescharacteristic
364C4#1681073411yescharacteristic
365C4#1681077111yescharacteristic
366C4#1681084111yescharacteristic
367C2—8482no—
368C2#2027506411yescharacteristic
369C2#2027656611yescharacteristic
370C2#2027657811yescharacteristic
371C2#2027663311yescharacteristic
372C2#2027663911yescharacteristic
373C2#2027675411yescharacteristic
374C2#2027678011yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#13013
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16810717C2semidirect
3. C2×#16810426C2semidirect
6. #20272792C2semidirect
7. C53×#1380C2semidirect
13. C2×#13133162C4semidirect
107. Dic212C22⋊C4semidirect
142. C2×Dic53SD32semidirect
185. #1380D53semidirect
213. C53#13013semidirect

Maximal subgroups

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

subgroupindex
C22×#131331622
C2×#168107172
C2×#168104262
C2×#168104232
#202755792
#202727922
C53×#13802
#1301353