Finite Group Databasebeta

← #27283179

Subgroups — #27283179

Order 13568.

610 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
1#20283073C211yesmaximal, characteristic
2#20274866C211yesmaximal, characteristic, complemented
3#20274570C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6C2×#16810366C211yesmaximal, characteristic
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133206—21no—
10C2×#13133206—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133146—22no—
14C2×#13133127—21no—
15#16810524—22no—
16#16810524C414yes—
17C42×Dic53C2211yescharacteristic
18#16810377—21no—
19#16810369—22no—
20#16810366—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#332—22no—
26C22×D212—21no—
27C22×D212—21no—
28C22×D212C2×C412yes—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214—21no—
35#13133206—42no—
36#13133206—42no—
37#13133206—42no—
38#13133206—42no—
39C2×#11719921—24no—
40C2×#11719921—24no—
41C2×C4×Dic53C2×C412yes—
42C2×C4×Dic53D412yes—
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133146—44no—
50#13133127—42no—
51C22×Q8×C53—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C412yes—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53D412yes—
65C23×D53—22no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C22×C4×C53—22no—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53C22⋊C412yes—
77C22×Dic53C22⋊C412yes—
78C2×D212—22no—
79C2×D212—22no—
80C2×D212—44no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—42no—
90C2×D212—42no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—42no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104#11719921—48no—
105#11719921—48no—
106C4×Dic53—22no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109C4×Dic53—22no—
110C4×Dic53—42no—
111C4×Dic53D812yes—
112C4×Dic53SD1612yes—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—42no—
118C23×C53C2×C22⋊C411yescharacteristic
119C22×D53—42no—
120C22×D53—44no—
121C22×D53—42no—
122C22×D53—44no—
123C22×D53—42no—
124Q8×C53—81no—
125Q8×C53—81no—
126Q8×C53—81no—
127Q8×C53—81no—
128C2×C4×C53—22no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12411yescharacteristic
137C2×C4×C53#12711yescharacteristic
138C2×C4×C53C2×D811yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)12yes—
143C2×C4×C53#13712yes—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53#10214yes—
150D212—82no—
151D212—82no—
152D212—81no—
153D212—81no—
154D212—81no—
155D212—81no—
156D212—82no—
157D212—81no—
158D212—81no—
159C8×C53—42no—
160C8×C53—81no—
161#88322—42no—
162#88322—81no—
163#9704—531nomaximal, Sylow
164C22×C53#39011yescharacteristic
165C22×C53#41312yes—
166C22×C53C2×#10211yescharacteristic
167C22×C53#41712yes—
168C22×C53#41811yescharacteristic
169C2×D53—84no—
170C2×D53—84no—
171C2×D53—82no—
172C2×D53—82no—
173C2×D53—82no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—82no—
177C4×C53#45911yescharacteristic
178C4×C53#45911yescharacteristic
179C4×C53#46011yescharacteristic
180C4×C53#46011yescharacteristic
181C4×C53#48912yes—
182C4×C53#49611yescharacteristic
183C4×C53#49711yescharacteristic
184Dic53—24no—
185Dic53—42no—
186#2858—531no—
187#1842—531no—
188#1767—531no—
189C2×#332—531no—
190C2×#332D5311yescharacteristic, complemented
191C2×#331—531no—
192C2×#331—531no—
193C2×C53#154712yes—
194C2×C53#155712yes—
195C2×C53#195911yescharacteristic
196C2×C53#196711yescharacteristic
197C2×C53#198211yescharacteristic
198D53—162no—
199C2×C4⋊Q8C2×D5311yescharacteristic
200C2×#127—531no—
201C2×#124—1061no—
202C2×#124—1061no—
203C2×C4⋊C8—531no—
204C2×C4⋊C8—531no—
205C2×C4⋊C8—1062no—
206C2×C4⋊C8C2×D5311yescharacteristic
207C2×C4⋊C8C2×D5311yescharacteristic
208C2×#97—1061no—
209#390—534no—
210#390—1062no—
211C43—531no—
212#336—1061no—
213#332—22no—
214#332—1062no—
215#331—1062no—
216#331—1062no—
217C53#970411yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C22×D4—532no—
220C22×D4—1061no—
221C22×D4—1061no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1061no—
225C22×C8—1061no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230#127—1061no—
231#127—1061no—
232#127—1061no—
233#127—1061no—
234#124—2122no—
235#124—2122no—
236#124—2122no—
237#124—2122no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C4×D5312yes—
240C2×C22⋊C4—1064no—
241C2×C22⋊C4—1064no—
242C2×C42—531no—
243C2×C42—531no—
244C2×C42—532no—
245C2×C42—532no—
246C2×C42C22×D5311yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—532no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253C4⋊C8—1061no—
254C4⋊C8—2124no—
255C4⋊C8C22⋊D5312yes—
256#97—2122no—
257C24—1062no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—1062no—
265C2×D4—1062no—
266C2×D4—2124no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4—2121no—
275C2×D4—2122no—
276C2×D4—2122no—
277C22×C4—22no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4D4×D5311yescharacteristic
282C22×C4D4×D5311yescharacteristic
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—1062no—
289C2×C8—2121no—
290C2×C8—2122no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—2128no—
298C22⋊C4—2128no—
299C42—532no—
300C42—532no—
301C42—1061no—
302C42—1062no—
303C42—1061no—
304C42—1062no—
305C42—2122no—
306C42#1171994511yescharacteristic
307C42D4×D5311yescharacteristic
308C42#1171994711yescharacteristic
309C42C2×C22⋊D5311yescharacteristic
310C23—2122no—
311C23—2124no—
312C23—2122no—
313C23—2124no—
314C23—2122no—
315C23C22⋊C4×D5311yescharacteristic
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—81no—
320D4—4242no—
321D4—4242no—
322D4—4241no—
323D4—4241no—
324D4—4241no—
325D4—4241no—
326D4—4241no—
327D4—4241no—
328D4—4242no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—534no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4#1313317711yescharacteristic
340C2×C4C4×C22⋊D5311yescharacteristic
341C2×C4#1313318911yescharacteristic
342C2×C4#1313319011yescharacteristic
343C2×C4#1313320311yescharacteristic
344C2×C4#1313321311yescharacteristic
345C2×C4D8×D5311yescharacteristic
346C2×C4SD16×D5311yescharacteristic
347C2×C4#1313324111yescharacteristic
348C2×C4#1313324412yes—
349C2×C4#1313324411yescharacteristic
350C2×C4#1313324912yes—
351C8—42no—
352C8—81no—
353C8—4241no—
354C8—2122no—
355C22—4244no—
356C22—4242no—
357C22—4242no—
358C22—4242no—
359C22—4244no—
360C22#1681052311yescharacteristic
361C22#1681060612yes—
362C22#102×D5311yescharacteristic
363C22#1681062312yes—
364C22#1681062611yescharacteristic
365C4—21no—
366C4—21no—
367C4—82no—
368C4—1064no—
369C4—2122no—
370C4#1681081611yescharacteristic
371C4#1681081711yescharacteristic
372C4#1681082111yescharacteristic
373C4#1681082211yescharacteristic
374C4#1681098412yes—
375C4#1681100911yescharacteristic
376C4#1681101611yescharacteristic
377C2—8482no—
378C2#2027339112yes—
379C2#2027343612yes—
380C2#2027552811yescharacteristic
381C2#2027556311yescharacteristic
382C2#2027564011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9704
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20274866C2semidirect
3. #20274570C2semidirect
4. C2×#16810369C2semidirect
7. C2×C53×#332C2semidirect
190. C2×#332D53semidirect
217. C53#9704semidirect

Maximal subgroups

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

subgroupindex
#202830732
#202748662
#202745702
C2×#168103692
C2×#168103662
C2×C53×#3322
#970453