Finite Group Databasebeta

← #27297642

Subgroups — #27297642

Order 13568.

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

normal subgroups only

This group has 306 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#20276967C211yesmaximal, characteristic
2#20276765C211yesmaximal, characteristic
3#20274085C211yesmaximal, characteristic
4#20272764C211yesmaximal, characteristic
5Q8⋊#11719908C211yesmaximal, characteristic
6#20272679C211yesmaximal, characteristic
7C53×#3395C211yesmaximal, characteristic
8C2×#13133163—21no—
9C2×#13133163—22no—
10#16811003C2211yescharacteristic
11#16810908—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810426—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810370—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×Dic212—21no—
26C2×Dic212—21no—
27#13133217—21no—
28#13133163—44no—
29#13133163—42no—
30C2×#11719908—21no—
31C2×#11719908—21no—
32#13133148—41no—
33C4⋊#88322—21no—
34#13133141—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×Dic106—41no—
58Q16×C53—21no—
59Q16×C53—21no—
60Q16×C53—42no—
61Q16×C53—41no—
62Q16×C53—41no—
63Q16×C53—82no—
64C2×C8×C53—21no—
65C2×C8×C53C2×D411yescharacteristic
66C2×C8×C53C2×D411yescharacteristic
67C4⋊C4×C53—21no—
68C4⋊C4×C53—21no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53D811yescharacteristic
76C42×C53—21no—
77C42×C53—42no—
78C42×C53—41no—
79C42×C53C2×D411yescharacteristic
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C2×#88322—41no—
83Dic212—42no—
84Dic212—41no—
85Dic212—41no—
86#11719908—22no—
87#11719908—41no—
88Q8×C53—21no—
89Q8×C53—41no—
90Q8×C53—42no—
91Q8×C53—42no—
92Q8×C53—41no—
93Q8×C53Q3212yescomplemented
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53C2×D811yescharacteristic
101C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
102C2×Dic53—81no—
103Dic106—81no—
104Dic106—81no—
105C8×C53—21no—
106C8×C53—41no—
107C8×C53#13511yescharacteristic
108C8×C53AGL(1,ℤ/8)11yescharacteristic
109#88322—81no—
110#12334—531nomaximal, Sylow
111C22×C53#44711yescharacteristic
112C4×C53—22no—
113C4×C53—21no—
114C4×C53—41no—
115C4×C53—81no—
116C4×C53—82no—
117C4×C53—81no—
118C4×C53#44911yescharacteristic
119C4×C53#45311yescharacteristic
120C4×C53C2×Q3211yescharacteristic
121C4×C53#51011yescharacteristic
122Dic53—161no—
123#3395D5311yescharacteristic
124#2247—531no—
125#2209—531no—
126#1670—531no—
127#1362—531no—
128Q8⋊C16—531no—
129#1332—531no—
130C2×C53#162111yescharacteristic
131C2×C53#218811yescharacteristic
132C2×C53#219511yescharacteristic
133#557—21no—
134C2×Q32—1061no—
135C2×Q32—1062no—
136#494—21no—
137#494—531no—
138#494C2×D5311yescharacteristic
139#475—1061no—
140#462—22no—
141#462—21no—
142C8×Q8C2×D5311yescharacteristic
143C4×Q16—21no—
144C4×Q16C2×D5311yescharacteristic
145C4⋊C16—531no—
146#358—1061no—
147C4×C16—531no—
148#335—531no—
149#332—1061no—
150C53#1233411yescharacteristic, complemented, Sylow
151C2×Q16—22no—
152C2×Q16—21no—
153C2×Q16—1061no—
154C2×Q16—1061no—
155C2×Q16C22⋊D5311yescharacteristic
156C2×Q16C22⋊D5312yes—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—1061no—
160#125—42no—
161#125—41no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8—41no—
165C4×Q8C22⋊D5311yescharacteristic
166Q32—2124no—
167Q32—2122no—
168C2×C16—1061no—
169C2×C16—1061no—
170#107C22⋊D5311yescharacteristic
171#106—2121no—
172C4⋊C8—21no—
173C4⋊C8—1061no—
174C4⋊C8C22⋊D5311yescharacteristic
175#103—22no—
176#103—21no—
177#103—2121no—
178C4×C8—21no—
179C4×C8C22×D5311yescharacteristic
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—2121no—
183C2×Q8#1171992211yescharacteristic
184Q16—21no—
185Q16—21no—
186Q16—42no—
187Q16—41no—
188Q16—41no—
189Q16—82no—
190Q16—2122no—
191Q16—2121no—
192Q16—2121no—
193C2×C8—21no—
194C2×C8—2121no—
195C2×C8C2×C22⋊D5311yescharacteristic
196C2×C8C2×C22⋊D5311yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C4⋊C4#1171992211yescharacteristic
208C42—21no—
209C42—42no—
210C42—41no—
211C42C2×C22⋊D5311yescharacteristic
212C16—1062no—
213C16—2121no—
214Q8—21no—
215Q8—41no—
216Q8—42no—
217Q8—42no—
218Q8—41no—
219Q8—4241no—
220Q8—4241no—
221Q8#1313316312yescomplemented
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—4241no—
228C2×C4#1313318611yescharacteristic
229C2×C4C2×#1171992211yescharacteristic
230C2×C4#1313324311yescharacteristic
231C8—21no—
232C8—41no—
233C8—4241no—
234C8#1313323711yescharacteristic
235C8#1313324211yescharacteristic
236C22#1681075711yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—81no—
241C4—82no—
242C4—81no—
243C4—8481no—
244C4#1681076811yescharacteristic
245C4#1681079111yescharacteristic
246C4C2×#1313316311yescharacteristic
247C4#1681106411yescharacteristic
248C2#2027378011yescharacteristic
249C2#2027664911yescharacteristic
250C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3395C2
socle, prime radicalC22×C53#447
hypercenter#3395D53
hypocenterC53#12334
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
93. Q8×C53Q32semidirect
150. C53#12334semidirect
221. Q8#13133163semidirect

Maximal subgroups

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

subgroupindex
#202769672
#202767652
#202740852
#202727642
Q8⋊#117199082
#202726792
C53×#33952
#1233453