Finite Group Databasebeta

← #28309709

Subgroups — #28309709

Order 13568.

306 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 78 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#20291078C211yesmaximal, characteristic
2D53⋊#335C211yesmaximal, characteristic
3#20289733C212yesmaximal, complemented
4#20289730C212yesmaximal, complemented
5#20285298C211yesmaximal, characteristic
6D53⋊(C4⋊C8)—21no—
7#16811544C2212yes—
8#16811518—21no—
9#16811511—21no—
10#16811487C2212yes—
11#16811486—21no—
12#16811177—21no—
13#16811027C2211yescharacteristic
14#16811001C2211yescharacteristic
15#16810972—21no—
16#16810865C411yescharacteristic
17#16810863—21no—
18#16810861C411yescharacteristic
19C4×Dic212C411yescharacteristic
20C4×#11719925—21no—
21C4×C8×D53C2211yescharacteristic
22C4×Q16×C53C411yescharacteristic
23C2×D53⋊C8—41no—
24C2×#11719910—22no—
25#13133284—41no—
26C4⋊#88324—22no—
27C2×Dic212C2×C411yescharacteristic
28C2×#11719925—21no—
29C2×C8×D53C2×C411yescharacteristic
30C2×C8×D53D411yescharacteristic
31#13133216—21no—
32#13133215—21no—
33#13133207—41no—
34#13133197—21no—
35#13133195—41no—
36#13133193—21no—
37C4×Dic106—21no—
38C4×Dic106—41no—
39#13133174—41no—
40C42×D53D411yescharacteristic
41#13133151C2×C411yescharacteristic
42#13133150—21no—
43#13133142—21no—
44#13133141—21no—
45#13133140—21no—
46C4×#88322D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53C2311yescharacteristic
49C2×Q16×C53C2×C411yescharacteristic
50C4×Q8×C53—21no—
51#107×C53C2×C411yescharacteristic
52#103×C53—21no—
53C4×C8×C53C2×C411yescharacteristic
54C2×Q8×C53—21no—
55C2×C4×D53C22⋊C411yescharacteristic
56C2×C4×D53D811yescharacteristic
57C2×C4×D53SD1611yescharacteristic
58C2×Dic106—21no—
59C2×Dic106—41no—
60C2×#88324—42no—
61D53⋊C8—82no—
62Q16×C53—41no—
63C2×C8×C53C22⋊C411yescharacteristic
64C2×C8×C53C22×C411yescharacteristic
65C4⋊C4×C53—21no—
66C4⋊C4×C53—41no—
67C42×C53—41no—
68C42×C53C22⋊C411yescharacteristic
69#11719925—42no—
70#11719921—41no—
71#11719921—41no—
72C4⋊Dic53—21no—
73C4⋊Dic53—41no—
74#11719919—41no—
75#11719919—41no—
76#11719919—41no—
77#11719919—41no—
78#11719919—41no—
79C4×Dic53—41no—
80C4×Dic53D811yescharacteristic
81C4×Dic53SD1611yescharacteristic
82C4×Dic53C2×D411yescharacteristic
83C2×#88322C22⋊C411yescharacteristic
84C2×#88322C2×D411yescharacteristic
85Dic212—41no—
86C8×D53—22no—
87C8×D53—41no—
88#11719910—42no—
89C22×D53#10211yescharacteristic
90Q8×C53—41no—
91Q8×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53#10211yescharacteristic
95C2×C4×C53#10211yescharacteristic
96C2×C4×C53C2×C22⋊C411yescharacteristic
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53#10211yescharacteristic
100C2×Dic53C2×D811yescharacteristic
101C2×Dic53C2×SD1611yescharacteristic
102C4×D53—22no—
103C4×D53—21no—
104C4×D53—21no—
105C4×D53—21no—
106C4×D53—21no—
107Dic106—41no—
108Dic106—41no—
109Dic106—81no—
110Dic106—81no—
111#88324—82no—
112C8×C53—21no—
113C8×C53—21no—
114#88322—21no—
115#88322#13512yes—
116#11954—531nomaximal, Sylow
117C22×C53C2×#10211yescharacteristic
118C2×D53—21no—
119C2×D53—21no—
120C2×D53#36212yes—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—81no—
124C4×C53—81no—
125C4×C53#41311yescharacteristic
126C4×C53#41611yescharacteristic
127Dic53—21no—
128Dic53—21no—
129Dic53—81no—
130Dic53—81no—
131Dic53#50911yescharacteristic
132Dic53#51011yescharacteristic
133#3170—531no—
134#2163—531no—
135C2×#335—531no—
136#1364—532no—
137#1362—532no—
138C2×C53#154011yescharacteristic
139C2×C53#213911yescharacteristic
140C2×C53C2×#36211yescharacteristic
141D53—22no—
142#533—1061no—
143#500—531no—
144#494—531no—
145#487—1061no—
146#467—531no—
147#467—531no—
148#467—1061no—
149C4×Q16—531no—
150C4×Q16—1061no—
151C4×Q16#615711yescharacteristic
152C2×C4⋊C8—1061no—
153C2×C4×C8—531no—
154C4⋊C16—532no—
155C22⋊C16—1061no—
156#346—1061no—
157#335—532no—
158#335—1061no—
159C53#1195411yescharacteristic, complemented, Sylow
160C2×Q16—531no—
161C2×Q16—1061no—
162C2×Q16C2×#615711yescharacteristic
163C22×C8—531no—
164C22×C8—531no—
165C22×C8—2121no—
166C4⋊Q8—1061no—
167C4⋊Q8—1061no—
168#125—2121no—
169C22⋊Q8—1061no—
170C22⋊Q8—1061no—
171C22⋊Q8—2121no—
172C4×Q8—21no—
173C4×Q8—1061no—
174C4×Q8—2121no—
175#117—2121no—
176C2×C42—531no—
177C2×C16—1062no—
178#107—531no—
179#107—1061no—
180#107C2×#615711yescharacteristic
181C4⋊C8—1062no—
182C4⋊C8—2121no—
183#103—21no—
184#103—1061no—
185#103—1061no—
186#103—1061no—
187C4×C8—531no—
188C4×C8—531no—
189C4×C8—531no—
190C4×C8C2×#615711yescharacteristic
191C2×Q8—21no—
192C2×Q8—1061no—
193C2×Q8—2121no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197Q16—41no—
198Q16—2121no—
199Q16—2122no—
200C2×C8—531no—
201C2×C8—531no—
202C2×C8—1062no—
203C2×C8—4242no—
204C2×C8—2121no—
205C2×C8—2122no—
206C2×C8C22⋊#615711yescharacteristic
207C2×C8C22×#615711yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—1061no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C22⋊C4—2121no—
218C22⋊C4—2121no—
219C42—41no—
220C42—531no—
221C42—531no—
222C42—531no—
223C42—2121no—
224C42C22⋊#615711yescharacteristic
225C16—2122no—
226C23—531no—
227Q8—41no—
228Q8—41no—
229Q8—4241no—
230Q8—4241no—
231Q8—2121no—
232Q8—2121no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—531no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4#1313327911yescharacteristic
246C2×C4#1313327911yescharacteristic
247C2×C4C2×C22⋊#615711yescharacteristic
248C8—21no—
249C8—21no—
250C8—532no—
251C8—1061no—
252C8—4242no—
253C22—532no—
254C22—1061no—
255C22—1061no—
256C22C2×#1313327911yescharacteristic
257C4—21no—
258C4—21no—
259C4—81no—
260C4—81no—
261C4—531no—
262C4—531no—
263C4—1061no—
264C4—1061no—
265C4—4241no—
266C4—4241no—
267C4#1681160911yescharacteristic
268C4#1681161311yescharacteristic
269C2—1062no—
270C2#2029021211yescharacteristic
271C2#2029102311yescharacteristic
272C2D53⋊#36211yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC4×Q16×C53C4
socleC22×C53C2×#102
prime radicalC22×D53#102
cosocleC8×Dic53C23
hypercenterC4×Q16#6157
hypocenterC53#11954
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #20289733C2semidirect
4. #20289730C2semidirect
159. C53#11954semidirect

Maximal subgroups

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

subgroupindex
#202910782
D53⋊#3352
#202897332
#202897302
#202852982
#1195453