Finite Group Databasebeta

← #27538950

Subgroups — #27538950

Order 13568.

556 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#20284371C211yesmaximal, characteristic
2#20283312C211yesmaximal, characteristic
3#20276926C212yesmaximal, complemented
4#20276925C212yesmaximal, complemented
5#20276919C212yesmaximal, complemented
6#20276914C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8#20276434C211yesmaximal, characteristic, complemented
9C2×#16810428C211yesmaximal, characteristic, complemented
10#20276328C211yesmaximal, characteristic
11C53×#2158C211yesmaximal, characteristic, complemented
12C22×Dic212C2211yescharacteristic
13C2×#13133223C2211yescharacteristic
14#16811199—21no—
15#16811103—21no—
16C2×#13133157C2211yescharacteristic
17#16810882—22no—
18#16810865C2212yes—
19#16810865C2212yes—
20#16810666—21no—
21#16810662—22no—
22#16810660C2212yes—
23C2×#13133151C2211yescharacteristic
24#107×D53C2212yes—
25#16810631—21no—
26#16810450C2212yes—
27#16810448C2212yes—
28#16810428C2212yes—
29#16810428C2212yes—
30#16810426C2212yes—
31#16810426C2212yes—
32#16810402C2212yes—
33#16810402C2212yes—
34C22×C16×C53C2211yescharacteristic
35C53×#427C2211yescharacteristic
36C53×#427C2211yescharacteristic
37C53×#367C2211yescharacteristic
38C53×#367C2211yescharacteristic
39C53×#366C2211yescharacteristic
40C53×#366C2211yescharacteristic
41C2×#11719945D411yescharacteristic
42C22×Dic106—21no—
43C2×Dic212—21no—
44C2×Dic212—22no—
45C2×Dic212—22no—
46C2×Dic212C2312yes—
47#13133223—22no—
48#13133223—22no—
49C2×#11719917D411yescharacteristic
50C2×#11719912D412yes—
51C2×C8×D53C2312yes—
52#13133217—41no—
53#13133205—41no—
54#13133201—42no—
55#13133197—22no—
56#13133197—22no—
57C4×Dic106—41no—
58C4×D212—41no—
59#13133177—42no—
60#13133175—22no—
61C2×C4⋊Dic53—21no—
62C4⋊C4×D53—22no—
63#13133157—22no—
64C2×#11719908C2312yes—
65#13133151—21no—
66#13133151C2312yes—
67#13133150—22no—
68#13133148—22no—
69#13133140—22no—
70#13133140—22no—
71C22×C8×C53C2311yescharacteristic
72#117×C53—21no—
73#117×C53—21no—
74C2×C16×C53—22no—
75C2×C16×C53C2311yescharacteristic
76C2×C16×C53C2311yescharacteristic
77#107×C53C2311yescharacteristic
78#107×C53C2311yescharacteristic
79#107×C53C2311yescharacteristic
80#107×C53C2311yescharacteristic
81#106×C53—21no—
82#106×C53—21no—
83C22×C4×C53C2×D411yescharacteristic
84C2×C22⋊D53D812yes—
85C22×Dic53—41no—
86C22×Dic53—41no—
87#11719945—22no—
88#11719945—22no—
89C2×D212D811yescharacteristic
90C2×C4×D53—42no—
91C2×C4×D53C2×D412yes—
92C2×Dic106—22no—
93C2×Dic106—42no—
94C2×Dic106—42no—
95C2×Dic106—41no—
96C2×Dic106D811yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53#4112yes—
99C2×C8×C53#4112yes—
100C2×C8×C53C2411yescharacteristic
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—41no—
108C42×C53—41no—
109#11719921—42no—
110#11719921—42no—
111#11719921—42no—
112C4⋊Dic53—22no—
113C4⋊Dic53—41no—
114C4⋊Dic53—42no—
115C4⋊Dic53—42no—
116#11719919—42no—
117C4×Dic53—42no—
118#11719917—22no—
119C2×#88322C2×D412yes—
120Dic212—22no—
121Dic212—22no—
122Dic212—42no—
123#11719912—24no—
124C8×D53—22no—
125C8×D53—22no—
126C16×C53—21no—
127C16×C53—21no—
128#11719908—22no—
129C23×C53C2×D811yescharacteristic
130C22×D53C2×D812yes—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53#12312yes—
136C2×C4×C53C2×D811yescharacteristic
137C2×C4×C53#13712yes—
138C2×C4×C53C22×D411yescharacteristic
139C22⋊D53—24no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—82no—
143C2×Dic53—161no—
144C2×Dic53C2×D812yes—
145D212—22no—
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—162no—
149Dic106—22no—
150Dic106—42no—
151Dic106—42no—
152Dic106—82no—
153C8×C53#12312yes—
154C8×C53C2×#4111yescharacteristic
155C8×C53C2×#4111yescharacteristic
156#88322—22no—
157#33651—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#48012yes—
161C22×C53C22×D811yescharacteristic
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53—81no—
169C4×C53#48012yes—
170C4×C53C2×#12311yescharacteristic
171C4×C53C2×#13711yescharacteristic
172Dic53—22no—
173Dic53—82no—
174Dic53—82no—
175#3059—531no—
176#2913—531no—
177#2237—532no—
178#2237—532no—
179#2236—532no—
180#2235—532no—
181#2159—531no—
182#2158D5311yescharacteristic, complemented
183#2141—531no—
184C2×#358—531no—
185#2113—531no—
186C2×C53—22no—
187C2×C53C2×#48011yescharacteristic
188C2×C53#341611yescharacteristic
189C2×C53#341711yescharacteristic
190D53—42no—
191C22×Q16—531no—
192C2×#131—531no—
193#536—1061no—
194#520—1061no—
195C2×M32—531no—
196C22×C16C2×D5311yescharacteristic
197#470—1062no—
198#467—532no—
199#467—532no—
200#428—1062no—
201#428—1061no—
202#427—532no—
203#427C2×D5311yescharacteristic
204#427C2×D5311yescharacteristic
205C2×#107—531no—
206C2×#107—532no—
207#419—1061no—
208#367—532no—
209#367C2×D5311yescharacteristic
210#367C2×D5311yescharacteristic
211#366—532no—
212#366C2×D5311yescharacteristic
213#366C2×D5311yescharacteristic
214#358—532no—
215#358—532no—
216#358—532no—
217#358—532no—
218C22⋊C16—532no—
219C22⋊C16—532no—
220C53#3365111yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C22×Q8—1061no—
223C2×Q16—532no—
224C2×Q16—1062no—
225C2×Q16—1061no—
226C2×Q16—1062no—
227#131—1062no—
228#131—1062no—
229C2×M16—531no—
230C2×M16—532no—
231C22×C8—532no—
232C22×C8C22×D5311yescharacteristic
233C4⋊Q8—2121no—
234#124—2121no—
235#123—2122no—
236C22⋊Q8—1062no—
237C22⋊Q8—1062no—
238C4×Q8—2121no—
239C4×D4—2121no—
240C4×D4—2122no—
241#117—21no—
242#117—21no—
243#117—1062no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1061no—
246M32—1062no—
247C2×C16—22no—
248C2×C16—532no—
249C2×C16C22×D5311yescharacteristic
250C2×C16C22×D5311yescharacteristic
251#107—532no—
252#107—1062no—
253#107—1061no—
254#107C22×D5311yescharacteristic
255#107C22×D5311yescharacteristic
256#107C22×D5311yescharacteristic
257#107C22×D5311yescharacteristic
258#106—21no—
259#106—21no—
260#106—1062no—
261#103—1062no—
262#103—1062no—
263#41—1062no—
264#41—1062no—
265C2×Q8—531no—
266C2×Q8—1062no—
267C2×Q8—2122no—
268C2×Q8—2122no—
269C2×Q8—2121no—
270C2×D4—532no—
271C2×D4—531no—
272C22×C4—532no—
273C22×C4—2122no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4D4×D5311yescharacteristic
277Q16—1062no—
278Q16—1062no—
279Q16—2122no—
280M16—1062no—
281M16—1064no—
282C2×C8—532no—
283C2×C8—1062no—
284C2×C8—1062no—
285C2×C8#1171994512yes—
286C2×C8D4×D5311yescharacteristic
287C2×C8#1171994912yes—
288C2×C8C23×D5311yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—1062no—
294C4⋊C4—2122no—
295C4⋊C4—2122no—
296C4⋊C4—2122no—
297C4⋊C4—2121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—2122no—
301C22⋊C4—2122no—
302C22⋊C4—2122no—
303C42—41no—
304C42—41no—
305C42—2122no—
306C16—21no—
307C16—21no—
308C16—1062no—
309C23—532no—
310C23D8×D5311yescharacteristic
311Q8—1062no—
312Q8—4242no—
313Q8—2122no—
314Q8—2122no—
315D4—1064no—
316D4—1062no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—532no—
322C2×C4—8482no—
323C2×C4—8481no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—4242no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4#1313320212yes—
330C2×C4D8×D5311yescharacteristic
331C2×C4#1313324912yes—
332C2×C4C2×D4×D5311yescharacteristic
333C8—1062no—
334C8#1313320212yes—
335C8C2×#1171994511yescharacteristic
336C8C2×#1171994911yescharacteristic
337C22—21no—
338C22—21no—
339C22—1062no—
340C22—1062no—
341C22—1062no—
342C22#1681093812yes—
343C22C2×D8×D5311yescharacteristic
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—1062no—
349C4—4242no—
350C4—4242no—
351C4#1681093812yes—
352C4C2×#1313320211yescharacteristic
353C4C2×#1313324911yescharacteristic
354C2—22no—
355C2—2122no—
356C2C2×#1681093811yescharacteristic
357C2#2028785811yescharacteristic
358C2#2028786511yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2158C2
socleC22×C53C22×D8
prime radicalC2×#11719945D4
hypercenter#2158D53
hypocenterC53#33651
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
3. #20276926C2semidirect
4. #20276925C2semidirect
5. #20276919C2semidirect
6. #20276914C2semidirect
7. #20276504C2semidirect
8. #20276434C2semidirect
9. C2×#16810428C2semidirect
11. C53×#2158C2semidirect
182. #2158D53semidirect
220. C53#33651semidirect

Maximal subgroups

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

subgroupindex
#202843712
#202833122
#202769262
#202769252
#202769192
#202769142
#202765042
#202764342
C2×#168104282
#202763282
C53×#21582
#3365153