Finite Group Databasebeta

← #27288590

Subgroups — #27288590

Order 13568.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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#20282987C211yesmaximal, characteristic
2#20274884C211yesmaximal, characteristic
3#20274883C212yesmaximal
4#20272597C212yesmaximal
5C53×#1295C211yesmaximal, characteristic
6C2×#13133151—22no—
7C2×#13133150—21no—
8C2×#13133149—22no—
9C2×#13133148—21no—
10C2×#13133128C2211yescharacteristic
11#16810565—22no—
12#16810562—21no—
13#16810544—22no—
14#16810543—21no—
15#16810493C2211yescharacteristic
16#16810489C2212yes—
17C2×#97×C53C2212yes—
18C53×#382C2211yescharacteristic
19C22×Dic106—21no—
20C22×#88322—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27#13133151—44no—
28#13133150—42no—
29#13133149—44no—
30#13133148—42no—
31#13133128—21no—
32#13133128Q812yescomplemented
33#13133124—21no—
34#13133124—22no—
35#13133124—21no—
36#13133124—22no—
37#13133124—22no—
38C22×C8×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C42×C53C2311yescharacteristic
42#97×C53—22no—
43#97×C53Q814yescomplemented
44#95×C53—22no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—41no—
52C2×Dic106—82no—
53C2×Dic106—81no—
54C2×C8×C53—24no—
55C2×C8×C53—24no—
56C2×C8×C53—24no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—41no—
59C42×C53C2×Q812yes—
60C42×C53C2×Q811yescharacteristic
61C42×C53#4111yescharacteristic
62C4⋊Dic53—42no—
63C4⋊Dic53—42no—
64C4⋊Dic53—41no—
65C4⋊Dic53—42no—
66C4⋊Dic53—41no—
67C4⋊Dic53—42no—
68C4⋊Dic53—81no—
69#11719919—82no—
70C2×#88322—22no—
71C2×#88322—22no—
72C2×#88322—22no—
73C23×C53#12011yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—42no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—42no—
82C2×C4×C53C22⋊Q812yes—
83C2×C4×C53C22⋊Q811yescharacteristic
84C2×C4×C53C4⋊Q812yes—
85C2×C4×C53C4⋊Q811yescharacteristic
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—82no—
93C2×Dic53—81no—
94Dic106—84no—
95C8×C53—28no—
96#88322—24no—
97#10583—531nomaximal, Sylow
98C22×C53#39511yescharacteristic
99C22×C53#45312yes—
100C22×C53#45311yescharacteristic
101C22×C53#45612yes—
102C22×C53#45611yescharacteristic
103C4×C53—21no—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—81no—
108C4×C53—81no—
109Dic53—82no—
110Dic53—82no—
111Dic53—161no—
112#2844—531no—
113#1847—532no—
114#1847—531no—
115#1295—532no—
116#1295D5311yescharacteristic
117C2×C53#165511yescharacteristic
118C2×C53#166011yescharacteristic
119C2×C53#166012yes—
120C2×C53#206012yes—
121C2×C53#206011yescharacteristic
122C2×#107—1062no—
123C2×#107—1061no—
124C2×#106—1062no—
125C2×#106—1061no—
126C2×#97—531no—
127C2×#97C2×D5312yes—
128#400—1062no—
129#400—1061no—
130#395—1062no—
131#395—1061no—
132#384—531no—
133#382—532no—
134#382C2×D5311yescharacteristic
135C53#1058311yescharacteristic, complemented, Sylow
136C22×Q8—1061no—
137C22×C8—22no—
138C22×C8—1061no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—1062no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—1062no—
145C2×C4⋊C4D21211yescharacteristic
146C2×C4⋊C4D21211yescharacteristic
147C2×C42C22×D5311yescharacteristic
148#107—2122no—
149#107—2124no—
150#106—2122no—
151#106—2124no—
152#97—22no—
153#97—532no—
154#97—1061no—
155#97Dic10614yescomplemented
156#95—22no—
157#95—1062no—
158#95—1061no—
159#95—1061no—
160#95—1062no—
161#95—1062no—
162C2×Q8—4242no—
163C2×Q8—4241no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—1062no—
167C22×C4—1062no—
168C22×C4—2121no—
169C22×C4C2×D21211yescharacteristic
170C22×C4D4×D5312yes—
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—1062no—
175C2×C8—1062no—
176C2×C8—1062no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—4242no—
180C4⋊C4—4241no—
181C4⋊C4—2122no—
182C4⋊C4—2121no—
183C4⋊C4—2122no—
184C4⋊C4—2121no—
185C4⋊C4—2122no—
186C4⋊C4—2122no—
187C42C2×Dic10612yes—
188C42#1171994711yescharacteristic
189C42Q8×D5311yescharacteristic
190C23#1313318711yescharacteristic
191Q8—4244no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—4242no—
201C2×C4—4241no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4#1313319412yes—
209C2×C4#1313319711yescharacteristic
210C2×C4#1313321612yes—
211C2×C4#1313321711yescharacteristic
212C8—28no—
213C8—1064no—
214C22#1681054411yescharacteristic
215C22#1681079212yes—
216C22#1681079311yescharacteristic
217C22#1681080712yes—
218C22#1681080811yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—8481no—
226C4—4242no—
227C4—4242no—
228C2#2027400511yescharacteristic
229C2#2027402711yescharacteristic
230C2#2027402812yes—
231C2#2027605912yes—
232C2#2027606111yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10583
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
32. #13133128Q8semidirect
43. #97×C53Q8semidirect
135. C53#10583semidirect
155. #97Dic106semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202748842
#202748832
#202725972
C53×#12952
#1058353