Finite Group Databasebeta

← #27361990

Subgroups — #27361990

Order 13568.

868 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#20286106C212yesmaximal, complemented
2#20285037C211yesmaximal, characteristic
3#20273936C214yesmaximal, complemented
4#20273901C214yesmaximal, complemented
5#20273450C211yesmaximal, characteristic
6#20273170C212yesmaximal, complemented
7C53×#3550C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810977—22no—
10#16810973C2212yes—
11#16810969C2212yes—
12#16810908C2214yes—
13#16810905—22no—
14#16810879—24no—
15#16810833C2214yes—
16#16810830—22no—
17#16810647C2212yes—
18#16810619—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810372C2212yes—
21#16810367C2212yes—
22Q8⋊#88322C2218yes—
23C53×#563—22no—
24C53×#559C2212yes—
25Q82×C53C2212yes—
26C53×#556—22no—
27D4×Q8×C53—22no—
28C53×#542—24no—
29C53×#536—22no—
30C53×#533C2212yes—
31C53×#532C2211yescharacteristic
32C2×#11719925—24no—
33C2×#11719917—22no—
34C22×#88322—21no—
35#13133217C2312yes—
36#13133209—21no—
37#13133196—22no—
38C4×Dic106—22no—
39#13133176—21no—
40#13133148—24no—
41C4⋊#88322—22no—
42C4⋊#88322C2314yes—
43#13133141—24no—
44#13133141—24no—
45#13133140—24no—
46#13133138—24no—
47C4×#88322—21no—
48C4×#88322C2312yes—
49C2×#41×C53D412yes—
50C22×Q8×C53—22no—
51C4⋊Q8×C53—24no—
52C4⋊Q8×C53—24no—
53C4⋊Q8×C53D412yes—
54C4⋊Q8×C53D412yes—
55C4⋊Q8×C53C2311yescharacteristic
56#127×C53C2311yescharacteristic
57#126×C53—28no—
58#125×C53—22no—
59#124×C53—24no—
60#124×C53—24no—
61#124×C53D412yes—
62#123×C53—24no—
63#123×C53—24no—
64C22⋊Q8×C53—24no—
65C22⋊Q8×C53—28no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53D414yes—
68#121×C53—22no—
69C4×Q8×C53—22no—
70C4×Q8×C53—24no—
71C4×Q8×C53C2314yes—
72C4×D4×C53—24no—
73C4×D4×C53—22no—
74#117×C53—22no—
75#117×C53—22no—
76C2×C42×C53C2311yescharacteristic
77#41×C53—44no—
78C2×Q8×C53—24no—
79C2×Q8×C53—22no—
80C2×Q8×C53—84no—
81C2×Q8×C53C2×D414yes—
82C2×Q8×C53C2×D412yes—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53C2×D412yes—
86C22×C4×C53—24no—
87C22×C4×C53—22no—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D411yescharacteristic
90C2×Dic106—22no—
91C4⋊C4×C53—28no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53C2×D414yes—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—28no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C42×C53—24no—
104C42×C53—24no—
105C42×C53C2×D412yes—
106C42×C53C2×D411yescharacteristic
107C42×C53C2411yescharacteristic
108C22⋊Dic53—41no—
109#11719925—84no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112#11719919—42no—
113C4×Dic53—41no—
114#11719917—82no—
115C2×#88322—22no—
116C2×#88322—24no—
117C2×#88322—42no—
118C23×C53—22no—
119C23×C53#12011yescharacteristic
120Q8×C53—24no—
121Q8×C53—24no—
122Q8×C53—42no—
123Q8×C53—44no—
124Q8×C53—84no—
125D4×C53—42no—
126D4×C53—42no—
127D4×C53—42no—
128D4×C53—82no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—24no—
133C2×C4×C53—22no—
134C2×C4×C53—24no—
135C2×C4×C53—44no—
136C2×C4×C53—84no—
137C2×C4×C53#12012yes—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53C22×D411yescharacteristic
140C2×C4×C53C22×D412yes—
141C2×Dic53—42no—
142Dic106—42no—
143Dic106—42no—
144#88322—42no—
145#88322—44no—
146#18693—531nomaximal, Sylow
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—44no—
150C22×C53—42no—
151C22×C53C2×#12011yescharacteristic
152C4×C53—22no—
153C4×C53—21no—
154C4×C53—44no—
155C4×C53—42no—
156C4×C53—44no—
157C4×C53#45512yes—
158C4×C53#57712yes—
159C4×C53#57812yes—
160Dic53—82no—
161#3550D5311yescharacteristic
162#3255—532no—
163#3131—531no—
164#1645—534no—
165#1640—534no—
166#1560—531no—
167#1493—532no—
168C2×C53—41no—
169C2×C53—82no—
170C2×C53#301312yes—
171C2×C53C2×#45511yescharacteristic
172#563—22no—
173#559C2×D5312yes—
174Q82C2×D5312yes—
175#556—22no—
176D4×Q8—22no—
177#542—24no—
178#536—22no—
179#533—531no—
180#533C2×D5312yes—
181#532C2×D5311yescharacteristic
182#488—1062no—
183#487—532no—
184#486—532no—
185#475—534no—
186#475—1062no—
187#470—1064no—
188#462—534no—
189#462—1062no—
190#423—532no—
191#416—1062no—
192C2×C4×C8—531no—
193#333—532no—
194#331—532no—
195Q8⋊C8—538no—
196C53#1869311yescharacteristic, complemented, Sylow
197C2×#41C22⋊D5312yes—
198C22×Q8—22no—
199C2×Q16—1064no—
200C2×M16—1062no—
201C22×C8—1061no—
202C4⋊Q8—24no—
203C4⋊Q8—24no—
204C4⋊Q8—532no—
205C4⋊Q8C22⋊D5312yes—
206C4⋊Q8C22⋊D5312yes—
207C4⋊Q8C22×D5311yescharacteristic
208#127C22×D5311yescharacteristic
209#126—28no—
210#125—22no—
211#125—1061no—
212#124—24no—
213#124—24no—
214#124C22⋊D5312yes—
215#123—24no—
216#123—24no—
217C22⋊Q8—24no—
218C22⋊Q8—28no—
219C22⋊Q8—22no—
220C22⋊Q8—1062no—
221C22⋊Q8C22⋊D5314yes—
222#121—22no—
223C4×Q8—24no—
224C4×Q8—22no—
225C4×Q8—1062no—
226C4×Q8C22×D5314yes—
227C4×D4—24no—
228C4×D4—22no—
229#117—22no—
230#117—22no—
231#117—1061no—
232C2×C42C22×D5311yescharacteristic
233#106—1064no—
234C4⋊C8—534no—
235C4⋊C8—1062no—
236#103—1064no—
237#103—1064no—
238#103—1064no—
239#102—1064no—
240C4×C8—532no—
241C4×C8—1061no—
242#41—44no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—84no—
246C2×Q8—1062no—
247C2×Q8C2×C22⋊D5314yes—
248C2×Q8C2×C22⋊D5312yes—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4C2×C22⋊D5312yes—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4C2×C22⋊D5312yes—
255C22×C4C2×C22⋊D5311yescharacteristic
256Q16—4244no—
257M16—4242no—
258C2×C8—1062no—
259C2×C8—1064no—
260C2×C8—2122no—
261C4⋊C4—28no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—1062no—
269C4⋊C4—1062no—
270C4⋊C4—2122no—
271C4⋊C4C2×C22⋊D5314yes—
272C22⋊C4—24no—
273C22⋊C4—28no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—2121no—
277C42—24no—
278C42—24no—
279C42—2121no—
280C42C2×C22⋊D5311yescharacteristic
281C42C2×C22⋊D5312yes—
282C42C23×D5311yescharacteristic
283C23—22no—
284C23#1313318611yescharacteristic
285Q8—24no—
286Q8—24no—
287Q8—42no—
288Q8—44no—
289Q8—84no—
290Q8—2122no—
291Q8—2122no—
292D4—42no—
293D4—42no—
294D4—42no—
295D4—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—44no—
303C2×C4—84no—
304C2×C4—2122no—
305C2×C4#1313318612yes—
306C2×C4#1313318611yescharacteristic
307C2×C4C22×C22⋊D5311yescharacteristic
308C2×C4C22×C22⋊D5312yes—
309C8—2122no—
310C8—2124no—
311C22—22no—
312C22—21no—
313C22—44no—
314C22—42no—
315C22C2×#1313318611yescharacteristic
316C4—22no—
317C4—21no—
318C4—44no—
319C4—42no—
320C4—44no—
321C4—4242no—
322C4#1681080112yes—
323C4#1681143112yes—
324C4#1681143812yes—
325C2—41no—
326C2—82no—
327C2#2028403012yes—
328C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3550C2
socleC22×C53C2×#120
prime radicalC53×#532C22
hypercenter#3550D53
hypocenterC53#18693
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20286106C2semidirect
3. #20273936C2semidirect
4. #20273901C2semidirect
6. #20273170C2semidirect
196. C53#18693semidirect

Maximal subgroups

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

subgroupindex
#202861062
#202850372
#202739362
#202739012
#202734502
#202731702
C53×#35502
#1869353