Finite Group Databasebeta

← #28306171

Subgroups — #28306171

Order 13568.

422 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290900C212yesmaximal
2#20290428C211yesmaximal, characteristic
3#20290271C211yesmaximal, characteristic
4#20290065C212yesmaximal
5#20285299C211yesmaximal, characteristic
6#16811627C2212yes—
7C4×D53⋊C8C2211yescharacteristic
8C42×#6157C2211yescharacteristic
9#16811493—24no—
10(C2×#6157)⋊C8—22no—
11#16811485—22no—
12#16811473—24no—
13#16811178C2212yes—
14#16811028C411yescharacteristic, complemented
15#16810973C412yescomplemented
16#16810861—22no—
17Q16×Dic53—22no—
18C4×C8×D53C2211yescharacteristic
19C53×#494C411yescharacteristic, complemented
20C2×#11719936—22no—
21C2×D53⋊C8—21no—
22C2×C4×#6157—21no—
23C2×C4×#6157—21no—
24C2×C4×#6157D412yes—
25C4⋊#88324—22no—
26C4⋊#88324—22no—
27C4×#88324—21no—
28C4×#88324D412yes—
29C2×C8×D53D412yes—
30#13133217C2×C412yes—
31#13133208—22no—
32#13133197—22no—
33#13133193—22no—
34Q8×Dic53—22no—
35#13133182—22no—
36#13133175—22no—
37C42×D53C2311yescharacteristic
38#13133151—22no—
39#13133141—24no—
40C4×#88322C2×C411yescharacteristic
41C8×Dic53D412yes—
42C2×Q16×C53—22no—
43C4⋊Q8×C53C2×C412yes—
44C4×C8×C53C2×C411yescharacteristic
45C22×#6157—22no—
46C2×Q8×C53—22no—
47C2×C4×D53C2×D411yescharacteristic
48C2×C4×D53C2×D411yescharacteristic
49C2×C4×D53#4111yescharacteristic
50C2×Dic106—22no—
51C2×#88324—22no—
52C2×#88324—42no—
53C4×#6157—24no—
54C4×#6157—24no—
55C4×#6157—42no—
56#11719936—44no—
57D53⋊C8—42no—
58Q16×C53—44no—
59C2×C8×C53C22⋊C412yes—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C42×C53C22×C411yescharacteristic
63#11719921—42no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53—42no—
67#11719919—42no—
68C4×Dic53—42no—
69C4×Dic53C2×D411yescharacteristic
70C4×Dic53C2×D411yescharacteristic
71C4×Dic53#4111yescharacteristic
72C2×#88322C22⋊C412yes—
73C8×D53—42no—
74C22×D53#12111yescharacteristic
75C2×#6157—24no—
76C2×#6157—42no—
77C2×#6157—42no—
78Q8×C53—44no—
79C2×C4×C53—42no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C4×D411yescharacteristic
82C2×C4×C53C4×D411yescharacteristic
83C2×Dic53—42no—
84C2×Dic53#12111yescharacteristic
85C2×Dic53#12411yescharacteristic
86C2×Dic53#12711yescharacteristic
87C4×D53—22no—
88C4×D53—22no—
89C4×D53#13714yes—
90Dic106—44no—
91#88324—42no—
92#88324—82no—
93C8×C53—22no—
94#88322—22no—
95#9853—531nomaximal, Sylow
96C22×C53#39011yescharacteristic
97C2×D53—21no—
98C2×D53—21no—
99C2×D53#47012yes—
100#6157—44no—
101C4×C53—82no—
102C4×C53#42112yes—
103C4×C53#44112yes—
104C4×C53#44312yes—
105Dic53—21no—
106Dic53—21no—
107Dic53—82no—
108Dic53#49712yes—
109#3170—531no—
110#1994—532no—
111#1744—531no—
112#1566—531no—
113#1493—532no—
114C2×C53#196311yescharacteristic
115C2×C53#197111yescharacteristic
116C2×C53#197411yescharacteristic
117D53—22no—
118#533—532no—
119#500—531no—
120#494#615711yescharacteristic, complemented
121#487—532no—
122#467—1062no—
123C4×Q16—1062no—
124#423—532no—
125C2×C4×C8—531no—
126C2×C4×C8—531no—
127C43—531no—
128#339—1064no—
129#336—1062no—
130#334—1062no—
131Q8⋊C8—1064no—
132C53#985311yescharacteristic, complemented, Sylow
133C2×Q16—22no—
134C2×M16—1062no—
135C22×C8—532no—
136C22×C8—1061no—
137C4⋊Q8—532no—
138C4⋊Q8C2×#615712yes—
139#125—1062no—
140C22⋊Q8—1062no—
141C22⋊Q8—1062no—
142C4×Q8—1062no—
143C4×Q8—1062no—
144#117—1062no—
145C2×C42—531no—
146C2×C42—532no—
147C2×C42—1061no—
148C2×C42—1061no—
149#107—1062no—
150C4⋊C8—1062no—
151C4⋊C8—1062no—
152#103—1064no—
153C4×C8—532no—
154C4×C8—532no—
155C4×C8—531no—
156C4×C8—1061no—
157C4×C8C2×#615711yescharacteristic
158C2×Q8—22no—
159C2×Q8—1062no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—1062no—
164Q16—44no—
165M16—2124no—
166C2×C8—532no—
167C2×C8—1062no—
168C2×C8—2122no—
169C2×C8—2122no—
170C2×C8—2122no—
171C2×C8C22⋊#615712yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C22⋊C4—2122no—
179C42—531no—
180C42—531no—
181C42—531no—
182C42—1064no—
183C42—1064no—
184C42—2122no—
185C42—2122no—
186C42C22×#615711yescharacteristic
187C23—531no—
188Q8—44no—
189Q8—2124no—
190C2×C4—42no—
191C2×C4—531no—
192C2×C4—534no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1064no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4C2×C22⋊#615711yescharacteristic
202C2×C4D4×#615711yescharacteristic
203C2×C4D4×#615711yescharacteristic
204C8—22no—
205C8—1062no—
206C8—4242no—
207C8—2122no—
208C22—532no—
209C22—1061no—
210C22—1061no—
211C22#1681157511yescharacteristic
212C4—82no—
213C4—532no—
214C4—1061no—
215C4—1061no—
216C4—4242no—
217C4—2124no—
218C4#1681162112yes—
219C4#1681166312yes—
220C4#1681166712yes—
221C2—1062no—
222C2#2029085311yescharacteristic
223C2#2029086311yescharacteristic
224C2#2029086611yescharacteristic
subgroupG/H
centerC22#16811575
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#494C4
socleC22×C53#390
prime radicalC22×D53#121
cosocleC42×D53C23
hypercenter#494#6157
hypocenterC53#9853
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
14. #16811028C4semidirect
15. #16810973C4semidirect
19. C53×#494C4semidirect
120. #494#6157semidirect
132. C53#9853semidirect

Maximal subgroups

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

subgroupindex
#202909002
#202904282
#202902712
#202900652
#202852992
#985353