Finite Group Databasebeta

← #27369434

Subgroups — #27369434

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285843C211yesmaximal, characteristic
2C2×#16810915C211yesmaximal, characteristic, complemented
3#20284189C211yesmaximal, characteristic
4#20283795C211yesmaximal, characteristic, complemented
5#20274134C214yesmaximal, complemented
6#20274100C214yesmaximal, complemented
7#20273487C211yesmaximal, characteristic, complemented
8#20273362C211yesmaximal, characteristic
9C53×#1548C211yesmaximal, characteristic, complemented
10C2×#13133250—21no—
11#16811173C2211yescharacteristic
12C2×C4×Dic106C2211yescharacteristic
13#16811019C2212yes—
14#16810971C2212yes—
15#16810941—22no—
16#16810915—22no—
17#16810915C2214yes—
18#16810910—22no—
19#16810846—22no—
20#16810834C2212yes—
21#16810824C2212yes—
22C8×Dic106—22no—
23#16810747C2214yes—
24#16810676C2211yescharacteristic
25#16810673—22no—
26C2×#13133150—21no—
27C2×C4⋊#88322C2211yescharacteristic
28#16810625—21no—
29C2×#13133140—21no—
30#16810373C2214yes—
31#16810368C2212yes—
32#16810366C2212yes—
33C2×C4⋊Q8×C53C2211yescharacteristic
34C53×#431C2211yescharacteristic
35C53×#423C2211yescharacteristic
36C53×#333C2214yes—
37C2×#11719945—21no—
38C22×Dic106D411yescharacteristic
39#13133250—44no—
40C2×Dic212—22no—
41C2×#11719913—22no—
42C22×#88322—21no—
43#13133217C2311yescharacteristic
44#13133214C2311yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×Dic106—21no—
49C4×Dic106—21no—
50C4×Dic106D412yes—
51C4×Dic106C2312yes—
52C4×D212—21no—
53C4×D212—21no—
54C2×C4⋊Dic53D411yescharacteristic
55C2×#11719919—21no—
56C2×C4×Dic53—21no—
57#13133150—24no—
58#13133148—24no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61C4⋊#88322C2312yes—
62#13133146—24no—
63#13133142—22no—
64#13133140—24no—
65#13133139—22no—
66#13133137—24no—
67C22⋊#88322D412yes—
68C4×#88322C2312yes—
69C8×Dic53—24no—
70C22×Q8×C53—21no—
71C2×M16×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53C2312yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C2×Q8×C53—22no—
85C2×Q8×C53—41no—
86C2×Q8×C53—82no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C2×C22⋊D53—41no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94#11719945—42no—
95#11719945—42no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×Dic106—21no—
101C2×Dic106—22no—
102C2×Dic106—21no—
103C2×Dic106SD1614yes—
104C2×Dic106C2×D412yes—
105M16×C53—42no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53C2411yescharacteristic
116#11719921—41no—
117#11719921—41no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53C2×D412yes—
122#11719919—22no—
123#11719919—22no—
124C4×Dic53—22no—
125C4×Dic53—22no—
126C2×#88322—22no—
127C2×#88322—42no—
128C2×#88322C2×D414yes—
129Dic212—44no—
130#11719913—44no—
131C23×C53#12111yescharacteristic
132C22×D53—41no—
133Q8×C53—82no—
134Q8×C53—82no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—82no—
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53C2×SD1612yes—
143C2×C4×C53#13512yes—
144C2×C4×C53C22×D411yescharacteristic
145C2×C4×C53C22×D411yescharacteristic
146C2×C4×C53C2×#4111yescharacteristic
147C22⋊D53—82no—
148C2×Dic53—22no—
149C2×Dic53—24no—
150C2×Dic53—22no—
151C2×Dic53—41no—
152C2×Dic53—81no—
153D212—41no—
154D212—41no—
155D212—42no—
156D212—81no—
157C4×D53—82no—
158Dic106—22no—
159Dic106—24no—
160Dic106—41no—
161Dic106—41no—
162Dic106C2×SD1614yes—
163C8×C53—42no—
164C8×C53—42no—
165#88322—24no—
166#88322—42no—
167#19403—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53#46512yes—
171C22×C53C2×#12111yescharacteristic
172C2×D53—81no—
173C2×D53—81no—
174C2×D53—81no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53#46512yes—
180C4×C53C22×SD1611yescharacteristic
181C4×C53C2×#13511yescharacteristic
182C4×C53#57812yes—
183Dic53—24no—
184Dic53—42no—
185Dic53—81no—
186#3227—531no—
187C2×#477—531no—
188#3037—531no—
189#2982—531no—
190#1678—534no—
191#1672—534no—
192#1570—531no—
193#1548D5311yescharacteristic, complemented
194#1540—531no—
195C2×C53—22no—
196C2×C53#304411yescharacteristic
197C2×C53C2×#46511yescharacteristic
198C2×C53#306811yescharacteristic
199D53—161no—
200C2×#137—1061no—
201#532—531no—
202C2×C4⋊Q8C2×D5311yescharacteristic
203C2×C4×Q8—531no—
204#498—532no—
205#486—532no—
206#481—1062no—
207#477—534no—
208#477—1062no—
209#476—1062no—
210#464—1062no—
211#462—532no—
212#460—532no—
213C8×Q8—534no—
214C8×Q8—1062no—
215#431—531no—
216#431—1062no—
217#431C2×D5311yescharacteristic
218C2×#107—1061no—
219#423C2×D5311yescharacteristic
220C2×C4⋊C8—531no—
221#417—1061no—
222C2×#103—1061no—
223#334—534no—
224#333C2×D5314yes—
225#332—532no—
226#331—532no—
227C53#1940311yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C22×Q8—21no—
230C22×Q8—531no—
231#137—2124no—
232C2×Q16—1062no—
233C2×SD16—1062no—
234C2×M16—21no—
235C22×C8—1061no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—531no—
240C4⋊Q8C22×D5312yes—
241#127—531no—
242#124—1061no—
243#121—1061no—
244#121—1061no—
245C4×Q8—532no—
246C4×Q8—532no—
247C4×Q8—1061no—
248C4×Q8—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—1061no—
255C2×C42—1061no—
256C2×C42C22×D5311yescharacteristic
257#107—1064no—
258#106—1064no—
259C4⋊C8—21no—
260C4⋊C8—532no—
261C4⋊C8—532no—
262C4⋊C8—1064no—
263C4⋊C8—1061no—
264C4⋊C8C22×D5312yes—
265C4⋊C8C22×D5312yes—
266#103—1064no—
267#103—1062no—
268#102—1064no—
269#102—1062no—
270C22⋊C8—21no—
271C22⋊C8—532no—
272C4×C8—21no—
273C4×C8—532no—
274C4×C8—1064no—
275#41—2122no—
276#41—2122no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8—534no—
281C2×Q8—532no—
282C2×Q8—1061no—
283C2×Q8—1062no—
284C2×Q8—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—2121no—
289C22×C4—41no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—2121no—
293C22×C4#1171994511yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C22×C4D4×D5311yescharacteristic
296Q16—2124no—
297SD16—2124no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—534no—
302C2×C8—1062no—
303C2×C8—2122no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—532no—
309C4⋊C4—1061no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C42—1062no—
317C42—1062no—
318C42#1171994511yescharacteristic
319C42D4×D5311yescharacteristic
320C42D4×D5311yescharacteristic
321C42C23×D5311yescharacteristic
322C23—2121no—
323C23#1313319011yescharacteristic
324Q8—82no—
325Q8—82no—
326Q8—534no—
327Q8—1064no—
328Q8—1062no—
329Q8—2121no—
330Q8—2121no—
331D4—4241no—
332D4—4242no—
333D4—2121no—
334D4—2121no—
335D4—2122no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—1064no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—4242no—
344C2×C4—4241no—
345C2×C4—2121no—
346C2×C4#1313319011yescharacteristic
347C2×C4#1313319011yescharacteristic
348C2×C4#1313319011yescharacteristic
349C2×C4SD16×D5312yes—
350C2×C4#1313324012yes—
351C2×C4C2×D4×D5311yescharacteristic
352C2×C4C2×D4×D5311yescharacteristic
353C2×C4C2×#1171994511yescharacteristic
354C8—42no—
355C8—42no—
356C8—1064no—
357C8—2122no—
358C22—21no—
359C22—21no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22#1681085512yes—
364C22C2×#1313319011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—1064no—
370C4—4241no—
371C4—2122no—
372C4#1681085512yes—
373C4C2×SD16×D5311yescharacteristic
374C4C2×#1313324011yescharacteristic
375C4#1681143912yes—
376C2—22no—
377C2—8481no—
378C2#2028424411yescharacteristic
379C2C2×#1681085511yescharacteristic
380C2#2028444211yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1548C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#1548D53
hypocenterC53#19403
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. C2×#16810915C2semidirect
4. #20283795C2semidirect
5. #20274134C2semidirect
6. #20274100C2semidirect
7. #20273487C2semidirect
9. C53×#1548C2semidirect
193. #1548D53semidirect
227. C53#19403semidirect

Maximal subgroups

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

subgroupindex
#202858432
C2×#168109152
#202841892
#202837952
#202741342
#202741002
#202734872
#202733622
C53×#15482
#1940353