Finite Group Databasebeta

← #27284601

Subgroups — #27284601

Order 13568.

568 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic
2#20279194C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic
4#20274806C211yesmaximal, characteristic
5#20273287C211yesmaximal, characteristic
6#20272562C211yesmaximal, characteristic
7C53×#3189C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810956—22no—
10#16810956—22no—
11C2×#13133150—21no—
12C2×#13133141—21no—
13C2×C22⋊#88322C2211yescharacteristic
14#16810553—22no—
15#16810530—21no—
16#16810495—22no—
17#16810492—21no—
18#16810483C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810390—22no—
21#16810386—22no—
22#16810360—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊Q8×C53C2211yescharacteristic
25C53×#484—21no—
26C53×#484—21no—
27C53×#467—22no—
28C53×#451—21no—
29C53×#451—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C23×Dic53—21no—
32C22×#88322—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D412yes—
40C2×#11719919—24no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53D412yes—
43C2×C4×Dic53—21no—
44#13133150—24no—
45#13133141—24no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48#13133124—24no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53D411yescharacteristic
52C22×Q8×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×Q16×C53—22no—
55C22×C8×C53D412yes—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70#107×C53—22no—
71#103×C53—22no—
72#103×C53—22no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75C24×C53C2×D411yescharacteristic
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—42no—
81C2×Q8×C53—42no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—42no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53—82no—
99Q16×C53—82no—
100C2×C8×C53—22no—
101C2×C8×C53—42no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊Dic53—22no—
115C22⋊Dic53—42no—
116C22⋊Dic53Q1614yescomplemented
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—42no—
122#11719919—84no—
123C4×Dic53—42no—
124C2×#88322—22no—
125C2×#88322—42no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#12011yescharacteristic
129C23×C53#12111yescharacteristic
130C23×C53#12711yescharacteristic
131C23×C53C2×Q1612yes—
132C23×C53#13712yes—
133Q8×C53—42no—
134Q8×C53—42no—
135Q8×C53—42no—
136Q8×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×Dic53—24no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—44no—
159C2×Dic53—42no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C2×Dic53—81no—
165C8×C53—42no—
166#88322—42no—
167#9959—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53#39211yescharacteristic
177C22×C53#45112yes—
178C22×C53#46212yes—
179C22×C53#49411yescharacteristic
180C22×C53#49711yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53#45812yes—
189Dic53—44no—
190Dic53—82no—
191Dic53—161no—
192#3189D5311yescharacteristic
193C2×#484—531no—
194#2502—531no—
195#1996—531no—
196#1824—531no—
197#1520—531no—
198#1282—531no—
199C2×C53—22no—
200C2×C53—21no—
201C2×C53—22no—
202C2×C53—21no—
203C2×C53#200811yescharacteristic
204C2×C53#201611yescharacteristic
205C2×C53#202211yescharacteristic
206C2×C22⋊Q8C2×D5311yescharacteristic
207C2×C22⋊Q8C2×D5311yescharacteristic
208C22×C4⋊C4—531no—
209#484—21no—
210#484—21no—
211#484—1062no—
212#484—1062no—
213#467—22no—
214#451—21no—
215#451—21no—
216C2×#107—1061no—
217C2×#103—1061no—
218C2×C22⋊C8—531no—
219C2×C22⋊C8C2×D5311yescharacteristic
220#398—1062no—
221#391—1061no—
222#385—1062no—
223#384—1061no—
224#380—531no—
225C4×C22⋊C4—531no—
226#342—1062no—
227#340—1062no—
228#328—1062no—
229C53#995911yescharacteristic, complemented, Sylow
230C22×Q8C22⋊D5311yescharacteristic
231C22×Q8C22⋊D5311yescharacteristic
232C23×C4—1061no—
233C23×C4C22×D5311yescharacteristic
234C2×Q16—22no—
235C22×C8—1061no—
236C22×C8C22⋊D5312yes—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—532no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1064no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4C22⋊D5311yescharacteristic
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—532no—
260C2×C22⋊C4—1061no—
261C2×C42—1061no—
262#107—22no—
263#107—1064no—
264#103—22no—
265#103—22no—
266#103—1064no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—532no—
270C22⋊C8—1061no—
271#95—1064no—
272#95—1061no—
273#95—1061no—
274C24D4×D5311yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—4242no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4D4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4#1171994711yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297C22×C4C2×C22⋊D5311yescharacteristic
298Q16—82no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—1062no—
302C2×C8—2122no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1062no—
312C4⋊C4—1062no—
313C4⋊C4—4244no—
314C4⋊C4—2122no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—534no—
322C22⋊C4—1062no—
323C22⋊C4—2122no—
324C42—2122no—
325C23—41no—
326C23—41no—
327C23#1313318511yescharacteristic
328C23#1313318911yescharacteristic
329C23#1313321311yescharacteristic
330C23Q16×D5312yes—
331C23#1313324912yes—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—1064no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2124no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313318511yescharacteristic
361C2×C4#1313318611yescharacteristic
362C2×C4#1313318811yescharacteristic
363C2×C4#1313318911yescharacteristic
364C8—42no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1681053111yescharacteristic
375C22#1681077712yes—
376C22#1681083112yes—
377C22#1681100111yescharacteristic
378C22#1681101611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—8481no—
387C4—4242no—
388C4—2124no—
389C4#1681081312yes—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2#2027576511yescharacteristic
395C2#2027581311yescharacteristic
396C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3189C2
socleC22×C53#392
prime radicalC24×C53C2×D4
hypercenter#3189D53
hypocenterC53#9959
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
116. C22⋊Dic53Q16semidirect
229. C53#9959semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
#202791942
#202757012
#202748062
#202732872
#202725622
C53×#31892
#995953