Finite Group Databasebeta

← #27297614

Subgroups — #27297614

Order 13568.

322 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 75 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#20276972C211yesmaximal, characteristic
2#20276755C211yesmaximal, characteristic
3#20274075C211yesmaximal, characteristic
4#20272764C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272679C211yesmaximal, characteristic, complemented
7C53×#3395C211yesmaximal, characteristic, complemented
8C2×#13133160—21no—
9C2×#13133160—22no—
10#16811000C2211yescharacteristic
11#16810823—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810423—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810366—21no—
17C53×#557—21no—
18C53×#494—21no—
19C53×#494C2211yescharacteristic
20C53×#462—22no—
21C53×#462—21no—
22C8×Q8×C53C2211yescharacteristic
23C4×Q16×C53—21no—
24C4×Q16×C53C2211yescharacteristic
25C2×#11719924—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133214—21no—
29#13133160—44no—
30#13133160—42no—
31C2×#11719908—21no—
32C2×#11719908—21no—
33C4⋊#88322—21no—
34#13133137—41no—
35C2×Q16×C53—22no—
36C2×Q16×C53—21no—
37C2×Q16×C53D411yescharacteristic
38C2×Q16×C53D412yes—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53—21no—
41#125×C53—42no—
42#125×C53—41no—
43C4×Q8×C53—22no—
44C4×Q8×C53—21no—
45C4×Q8×C53—41no—
46C4×Q8×C53D411yescharacteristic
47#107×C53D411yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53D411yescharacteristic
50#103×C53—22no—
51#103×C53—21no—
52C4×C8×C53—21no—
53C4×C8×C53C2311yescharacteristic
54C2×Q8×C53—21no—
55C2×Q8×C53—22no—
56C2×Q8×C53D811yescharacteristic
57C2×D212—41no—
58C2×D212—41no—
59C2×D212—41no—
60Q16×C53—21no—
61Q16×C53—21no—
62Q16×C53—42no—
63Q16×C53—41no—
64Q16×C53—41no—
65Q16×C53—82no—
66C2×C8×C53—21no—
67C2×C8×C53C2×D411yescharacteristic
68C2×C8×C53C2×D411yescharacteristic
69C4⋊C4×C53—21no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53D811yescharacteristic
78C42×C53—21no—
79C42×C53—42no—
80C42×C53—41no—
81C42×C53C2×D411yescharacteristic
82#11719924—82no—
83C2×#88322—41no—
84D424—42no—
85D424—41no—
86D424—41no—
87#11719908—22no—
88#11719908—41no—
89C22×D53—81no—
90Q8×C53—21no—
91Q8×C53—41no—
92Q8×C53—42no—
93Q8×C53—42no—
94Q8×C53—41no—
95Q8×C53SD3212yescomplemented
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53#12011yescharacteristic
102C2×C4×C53C2×D811yescharacteristic
103C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
104D212—81no—
105D212—81no—
106D212—81no—
107D212—81no—
108D212—161no—
109C8×C53—21no—
110C8×C53—41no—
111C8×C53#13511yescharacteristic
112C8×C53AGL(1,ℤ/8)11yescharacteristic
113#88322—81no—
114#12330—531nomaximal, Sylow
115C22×C53#44711yescharacteristic
116C2×D53—161no—
117C2×D53—161no—
118C2×D53—161no—
119C4×C53—22no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53—81no—
123C4×C53—82no—
124C4×C53—81no—
125C4×C53#44911yescharacteristic
126C4×C53#45311yescharacteristic
127C4×C53C2×SD3211yescharacteristic
128C4×C53#51011yescharacteristic
129#3395D5311yescharacteristic, complemented
130#2248—531no—
131#2207—531no—
132#1668—531no—
133#1362—531no—
134Q8⋊C16—531no—
135#1332—531no—
136C2×C53#162111yescharacteristic
137C2×C53#218511yescharacteristic
138C2×C53#219311yescharacteristic
139D53—321no—
140#557—21no—
141C2×SD32—1061no—
142C2×SD32—1062no—
143#494—21no—
144#494C2×D5311yescharacteristic
145#493—531no—
146#462—22no—
147#462—21no—
148#460—1061no—
149C8×Q8C2×D5311yescharacteristic
150C4×Q16—21no—
151C4×Q16C2×D5311yescharacteristic
152C4⋊C16—531no—
153#357—1061no—
154C4×C16—531no—
155#335—531no—
156#331—1061no—
157C53#1233011yescharacteristic, complemented, Sylow
158C2×Q16—22no—
159C2×Q16—21no—
160C2×Q16C22⋊D5311yescharacteristic
161C2×Q16C22⋊D5312yes—
162C2×SD16—2121no—
163C2×D8—1061no—
164C2×D8—1061no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167#127—1061no—
168#125—42no—
169#125—41no—
170C4×Q8—22no—
171C4×Q8—21no—
172C4×Q8—41no—
173C4×Q8C22⋊D5311yescharacteristic
174SD32—2124no—
175SD32—2122no—
176C2×C16—1061no—
177C2×C16—1061no—
178#107C22⋊D5311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—1061no—
181C4⋊C8C22⋊D5311yescharacteristic
182#103—22no—
183#103—21no—
184#102—2121no—
185C4×C8—21no—
186C4×C8C22×D5311yescharacteristic
187C2×Q8—21no—
188C2×Q8—22no—
189C2×Q8#1171992211yescharacteristic
190C2×D4—2121no—
191C2×D4—2121no—
192C2×D4—2121no—
193Q16—21no—
194Q16—21no—
195Q16—42no—
196Q16—41no—
197Q16—41no—
198Q16—82no—
199SD16—4242no—
200D8—2122no—
201D8—2121no—
202D8—2121no—
203C2×C8—21no—
204C2×C8—2121no—
205C2×C8C2×C22⋊D5311yescharacteristic
206C2×C8C2×C22⋊D5311yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4#1171992211yescharacteristic
216C42—21no—
217C42—42no—
218C42—41no—
219C42C2×C22⋊D5311yescharacteristic
220C16—1062no—
221C16—2121no—
222C23—4241no—
223Q8—21no—
224Q8—41no—
225Q8—42no—
226Q8—42no—
227Q8—41no—
228Q8#1313316012yescomplemented
229D4—8481no—
230D4—4241no—
231D4—4241no—
232D4—4241no—
233D4—4241no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4#1313318611yescharacteristic
240C2×C4C2×#1171992211yescharacteristic
241C2×C4#1313324311yescharacteristic
242C8—21no—
243C8—41no—
244C8—4241no—
245C8#1313323711yescharacteristic
246C8#1313324211yescharacteristic
247C22—8481no—
248C22—8481no—
249C22—8481no—
250C22#1681075711yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4#1681076811yescharacteristic
258C4#1681079111yescharacteristic
259C4C2×#1313316011yescharacteristic
260C4#1681106411yescharacteristic
261C2—16961no—
262C2#2027378011yescharacteristic
263C2#2027663211yescharacteristic
264C2#2027667911yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3395C2
socleC22×C53#447
prime radical#16811000C22
hypercenter#3395D53
hypocenterC53#12330
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
4. #20272764C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272679C2semidirect
7. C53×#3395C2semidirect
95. Q8×C53SD32semidirect
129. #3395D53semidirect
157. C53#12330semidirect
228. Q8#13133160semidirect

Maximal subgroups

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

subgroupindex
#202769722
#202767552
#202740752
#202727642
Q8⋊#117199082
#202726792
C53×#33952
#1233053