Finite Group Databasebeta

← #27297613

Subgroups — #27297613

Order 13568.

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

normal subgroups only

This group has 322 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#20287720C211yesmaximal, characteristic, complemented
2#20276970C211yesmaximal, characteristic
3#20274072C211yesmaximal, characteristic
4#20272765C211yesmaximal, characteristic, complemented
5#20272693C211yesmaximal, characteristic, complemented
6#20272679C211yesmaximal, characteristic, complemented
7C53×#2207C211yesmaximal, characteristic
8#16811351—21no—
9C2×#13133161—22no—
10#16811003C2211yescharacteristic
11#16811001—21no—
12#16810833—21no—
13#16810832—22no—
14#16810820—21no—
15C8×Dic106C2211yescharacteristic
16#16810717—21no—
17C4×#11719908C2211yescharacteristic
18#16810376C2211yescharacteristic
19#16810367—21no—
20C2×SD32×C53—21no—
21C53×#493C2211yescharacteristic
22C4×Q16×C53C2211yescharacteristic
23C53×C4⋊C16C2211yescharacteristic
24C53×#357—21no—
25C2×Dic212—21no—
26C2×Dic212D412yes—
27C2×#11719925—22no—
28C2×#11719923—41no—
29#13133217—21no—
30#13133215—22no—
31#13133209—41no—
32#13133207—42no—
33C4×Dic106—41no—
34C4×Dic106D411yescharacteristic
35#13133182—22no—
36#13133161—44no—
37C2×#11719908—21no—
38C4⋊#88322—21no—
39C4⋊#88322D411yescharacteristic
40#13133146—21no—
41#13133140—22no—
42#13133138—41no—
43C8×Dic53—21no—
44C2×Q16×C53D411yescharacteristic
45C2×D8×C53—21no—
46C2×D8×C53—21no—
47#127×C53—21no—
48C4×Q8×C53—21no—
49SD32×C53—42no—
50C2×C16×C53—21no—
51#107×C53D411yescharacteristic
52#103×C53—21no—
53C4×C8×C53C2311yescharacteristic
54C2×Q8×C53—21no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×Dic106—22no—
59C2×Dic106D811yescharacteristic
60Q16×C53—21no—
61Q16×C53—21no—
62D8×C53—42no—
63D8×C53—41no—
64D8×C53—41no—
65C2×C8×C53C2×D411yescharacteristic
66C2×C8×C53C2×D411yescharacteristic
67C4⋊C4×C53—21no—
68C4⋊C4×C53—41no—
69C42×C53—41no—
70C42×C53C2×D411yescharacteristic
71#11719925—82no—
72#11719923—82no—
73C4⋊Dic53—41no—
74C4⋊Dic53—41no—
75C4⋊Dic53D811yescharacteristic
76#11719919—21no—
77#11719919—42no—
78#11719919—42no—
79#11719919—42no—
80C4×Dic53—21no—
81C4×Dic53—42no—
82C2×#88322—21no—
83C2×#88322—41no—
84Dic212—42no—
85Dic212—41no—
86Dic212—41no—
87C16×C53—41no—
88#11719908—22no—
89C23×C53—81no—
90Q8×C53—41no—
91Q8×C53—41no—
92D4×C53—81no—
93D4×C53—81no—
94D4×C53—81no—
95D4×C53—81no—
96D4×C53—161no—
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—21no—
103C2×Dic53—21no—
104C2×Dic53—42no—
105Dic106—21no—
106Dic106—42no—
107Dic106—42no—
108Dic106SD3212yescomplemented
109C8×C53—21no—
110C8×C53#13511yescharacteristic
111C8×C53AGL(1,ℤ/8)11yescharacteristic
112#88322—41no—
113#88322—81no—
114#12330—531nomaximal, Sylow
115C22×C53—161no—
116C22×C53—161no—
117C22×C53—161no—
118C22×C53#44711yescharacteristic
119C4×C53—21no—
120C4×C53—81no—
121C4×C53—81no—
122C4×C53#44911yescharacteristic
123C4×C53#45311yescharacteristic
124C4×C53C2×SD3211yescharacteristic
125C4×C53#51011yescharacteristic
126Dic53—22no—
127Dic53—41no—
128Dic53—82no—
129#3395—531no—
130#2248—531no—
131#2207D5311yescharacteristic
132#1668—531no—
133#1362—531no—
134Q8⋊C16—531no—
135#1332—531no—
136C2×C53—321no—
137C2×C53#162111yescharacteristic
138C2×C53#218511yescharacteristic
139C2×C53#219311yescharacteristic
140#557—1061no—
141C2×SD32—21no—
142C2×SD32—1062no—
143#494—531no—
144#494—1061no—
145#493C2×D5311yescharacteristic
146#462—1062no—
147#462—1061no—
148#460—1061no—
149C8×Q8—531no—
150C4×Q16—1061no—
151C4×Q16C2×D5311yescharacteristic
152C4⋊C16C2×D5311yescharacteristic
153#357—21no—
154C4×C16—531no—
155#335—531no—
156#331—1061no—
157C53#1233011yescharacteristic, complemented, Sylow
158C2×Q16—532no—
159C2×Q16—1062no—
160C2×Q16—1061no—
161C2×Q16C22⋊D5311yescharacteristic
162C2×SD16—2121no—
163C2×D8—21no—
164C2×D8—21no—
165C4⋊Q8—1062no—
166C4⋊Q8—1061no—
167#127—21no—
168#125—2121no—
169#125—2122no—
170C4×Q8—21no—
171C4×Q8—531no—
172C4×Q8—1062no—
173C4×Q8—2121no—
174SD32—42no—
175SD32—2124no—
176C2×C16—21no—
177C2×C16—1061no—
178#107C22⋊D5311yescharacteristic
179C4⋊C8—531no—
180C4⋊C8—1061no—
181C4⋊C8—1061no—
182#103—21no—
183#103—1062no—
184#102—2121no—
185C4×C8—1061no—
186C4×C8C22×D5311yescharacteristic
187C2×Q8—21no—
188C2×Q8—531no—
189C2×Q8—1062no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193Q16—21no—
194Q16—21no—
195Q16—4242no—
196Q16—2122no—
197Q16—2121no—
198Q16—2121no—
199SD16—4242no—
200D8—42no—
201D8—41no—
202D8—41no—
203C2×C8—1061no—
204C2×C8—2121no—
205C2×C8D4×D5311yescharacteristic
206C2×C8C2×C22⋊D5311yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—531no—
210C4⋊C4—1061no—
211C4⋊C4—2122no—
212C4⋊C4—2122no—
213C4⋊C4—2122no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C42—41no—
217C42—1061no—
218C42—2122no—
219C42D4×D5311yescharacteristic
220C16—41no—
221C16—1062no—
222C23—81no—
223Q8—41no—
224Q8—41no—
225Q8—532no—
226Q8—1061no—
227Q8—2122no—
228Q8—2122no—
229D4—81no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—161no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—2122no—
239C2×C4#1313318511yescharacteristic
240C2×C4D8×D5311yescharacteristic
241C2×C4#1313324111yescharacteristic
242C8—21no—
243C8—4241no—
244C8—2121no—
245C8#1313323711yescharacteristic
246C8#1313324311yescharacteristic
247C22—161no—
248C22—161no—
249C22—161no—
250C22#1681075511yescharacteristic
251C4—21no—
252C4—81no—
253C4—81no—
254C4—1062no—
255C4—4242no—
256C4—2121no—
257C4#1681076611yescharacteristic
258C4#1681078911yescharacteristic
259C4SD32×D5311yescharacteristic
260C4#1681106211yescharacteristic
261C2—321no—
262C2#2027377911yescharacteristic
263C2#2027663011yescharacteristic
264C2#2027667611yescharacteristic
subgroupG/H
centerC22#16810755
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2207C2
socleC22×C53#447
prime radicalC53×#493C22
hypercenter#2207D53
hypocenterC53#12330
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #20287720C2semidirect
4. #20272765C2semidirect
5. #20272693C2semidirect
6. #20272679C2semidirect
108. Dic106SD32semidirect
157. C53#12330semidirect

Maximal subgroups

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

subgroupindex
#202877202
#202769702
#202740722
#202727652
#202726932
#202726792
C53×#22072
#1233053