Finite Group Databasebeta

← #27297663

Subgroups — #27297663

Order 13568.

324 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 67 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#20276982C211yesmaximal, characteristic
2#20276755C211yesmaximal, characteristic, complemented
3#20274199C211yesmaximal, characteristic, complemented
4#20272758C211yesmaximal, characteristic, complemented
5#20272697C211yesmaximal, characteristic, complemented
6#20272685C211yesmaximal, characteristic, complemented
7C53×#3402C211yesmaximal, characteristic, complemented
8C2×#13133160—21no—
9C2×#13133160—21no—
10C2×#13133158—21no—
11#16811000C2211yescharacteristic
12#16810823—21no—
13C4⋊#11719908C2211yescharacteristic
14#16810423—21no—
15#16810399C2211yescharacteristic
16#16810374C2211yescharacteristic
17#16810366—21no—
18C53×#556—21no—
19C53×#497—21no—
20C53×#495C2211yescharacteristic
21C53×#464—21no—
22C53×#463—21no—
23C53×#462—21no—
24C53×#446C2211yescharacteristic
25C53×#440—21no—
26C4×Q16×C53C2211yescharacteristic
27C2×#11719924—41no—
28C2×D424—21no—
29C2×D424—21no—
30#13133214—21no—
31#13133160—42no—
32#13133160—42no—
33#13133158—42no—
34C2×#11719908—21no—
35C2×#11719908—21no—
36C4⋊#88322—21no—
37#13133137—41no—
38C2×Q16×C53—21no—
39C2×Q16×C53D411yescharacteristic
40C2×Q16×C53D411yescharacteristic
41C2×SD16×C53—21no—
42C2×SD16×C53—21no—
43C2×D8×C53D411yescharacteristic
44C4⋊Q8×C53—21no—
45#126×C53—41no—
46#126×C53—41no—
47#124×C53—21no—
48#124×C53—21no—
49C22⋊Q8×C53—41no—
50C4×Q8×C53—21no—
51C4×Q8×C53—21no—
52C4×Q8×C53D411yescharacteristic
53C4×D4×C53—21no—
54#117×C53—41no—
55#107×C53D411yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53D411yescharacteristic
58#103×C53—21no—
59#103×C53—21no—
60#102×C53—21no—
61#97×C53—21no—
62C4×C8×C53C2311yescharacteristic
63C2×Q8×C53—21no—
64C2×Q8×C53—21no—
65C2×Q8×C53D811yescharacteristic
66C2×D4×C53—21no—
67C22×C4×C53—41no—
68C2×D212—41no—
69C2×D212—41no—
70C2×D212—41no—
71Q16×C53—21no—
72Q16×C53—21no—
73Q16×C53—41no—
74Q16×C53—81no—
75SD16×C53—41no—
76SD16×C53—41no—
77SD16×C53—81no—
78D8×C53—41no—
79C2×C8×C53—21no—
80C2×C8×C53C2×D411yescharacteristic
81C2×C8×C53C2×D411yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53D811yescharacteristic
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—21no—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53C2×D411yescharacteristic
98#11719924—82no—
99C2×#88322—41no—
100D424—41no—
101D424—41no—
102D424—41no—
103D424—41no—
104#11719908—22no—
105#11719908—41no—
106C23×C53—41no—
107C22×D53—81no—
108Q8×C53—41no—
109Q8×C53—41no—
110Q8×C53—41no—
111Q8×C53—41no—
112Q8×C53—41no—
113D4×C53—41no—
114D4×C53—41no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—82no—
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53C2×D811yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124D212—81no—
125D212—81no—
126D212—81no—
127D212—81no—
128D212—161no—
129C8×C53—21no—
130C8×C53—41no—
131C8×C53#13511yescharacteristic
132C8×C53AGL(1,ℤ/8)11yescharacteristic
133#88322—81no—
134#12337—531nomaximal, Sylow
135C22×C53—81no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53#44711yescharacteristic
139C2×D53—161no—
140C2×D53—161no—
141C2×D53—161no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—81no—
147C4×C53—81no—
148C4×C53—81no—
149C4×C53#44911yescharacteristic
150C4×C53#45311yescharacteristic
151#3402D5311yescharacteristic, complemented
152#2250—531no—
153#2207—531no—
154#1688—531no—
155#1360—531no—
156#1338—531no—
157#1334—531no—
158C2×C53—161no—
159C2×C53#162111yescharacteristic
160C2×C53#219011yescharacteristic
161C2×C53#219311yescharacteristic
162D53—321no—
163#556—21no—
164C2×SD32—1061no—
165C2×SD32—1061no—
166C2×D16—1061no—
167#497—21no—
168#495C2×D5311yescharacteristic
169#493—531no—
170#464—21no—
171#463—21no—
172#462—21no—
173#460—1061no—
174#446C2×D5311yescharacteristic
175#440—21no—
176C4×Q16C2×D5311yescharacteristic
177C4⋊C16—531no—
178#357—1061no—
179#346—531no—
180#334—531no—
181#331—1061no—
182C53#1233711yescharacteristic, complemented, Sylow
183C2×Q16—21no—
184C2×Q16C22⋊D5311yescharacteristic
185C2×Q16C22⋊D5311yescharacteristic
186C2×SD16—21no—
187C2×SD16—21no—
188C2×SD16—2121no—
189C2×D8—1061no—
190C2×D8—1061no—
191C2×D8C22⋊D5311yescharacteristic
192C4⋊Q8—21no—
193#127—1061no—
194#126—41no—
195#126—41no—
196#124—21no—
197#124—21no—
198C22⋊Q8—41no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8C22⋊D5311yescharacteristic
202C4×D4—21no—
203#117—41no—
204SD32—2122no—
205SD32—2122no—
206D16—2122no—
207C2×C16—1061no—
208C2×C16—1061no—
209#107C22⋊D5311yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—1061no—
212C4⋊C8C22⋊D5311yescharacteristic
213#103—21no—
214#103—21no—
215#102—21no—
216#102—2121no—
217#97—21no—
218C4×C8C22×D5311yescharacteristic
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8#1171992211yescharacteristic
222C2×D4—21no—
223C2×D4—2121no—
224C2×D4—2121no—
225C2×D4—2121no—
226C22×C4—41no—
227Q16—21no—
228Q16—21no—
229Q16—41no—
230Q16—81no—
231SD16—41no—
232SD16—41no—
233SD16—81no—
234SD16—4242no—
235D8—41no—
236D8—2121no—
237D8—2121no—
238D8—2121no—
239D8—2121no—
240C2×C8—21no—
241C2×C8—2121no—
242C2×C8C2×C22⋊D5311yescharacteristic
243C2×C8C2×C22⋊D5311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4#1171992211yescharacteristic
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42C2×C22⋊D5311yescharacteristic
260C16—1062no—
261C16—2121no—
262C23—41no—
263C23—4241no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269D4—41no—
270D4—41no—
271D4—8481no—
272D4—4241no—
273D4—4241no—
274D4—4241no—
275D4—4241no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4#1313318611yescharacteristic
283C2×C4C2×#1171992211yescharacteristic
284C2×C4#1313324311yescharacteristic
285C8—21no—
286C8—41no—
287C8—4241no—
288C8#1313323711yescharacteristic
289C8#1313324211yescharacteristic
290C22—81no—
291C22—81no—
292C22—81no—
293C22—8481no—
294C22—8481no—
295C22—8481no—
296C22#1681075711yescharacteristic
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#1681076811yescharacteristic
305C4#1681079111yescharacteristic
306C2—161no—
307C2—16961no—
308C2#2027378011yescharacteristic
309C2#2027666111yescharacteristic
310C2#2027667911yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3402C2
socleC22×C53#447
prime radical#20276982C2
hypercenter#3402D53
hypocenterC53#12337
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3402

Complements

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

NH ≅ G/Ntype
2. #20276755C2semidirect
3. #20274199C2semidirect
4. #20272758C2semidirect
5. #20272697C2semidirect
6. #20272685C2semidirect
7. C53×#3402C2semidirect
151. #3402D53semidirect
182. C53#12337semidirect

Maximal subgroups

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

subgroupindex
#202769822
#202767552
#202741992
#202727582
#202726972
#202726852
C53×#34022
#1233753