Finite Group Databasebeta

← #27278468

Subgroups — #27278468

Order 13568.

428 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 119 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#20277901C211yesmaximal, characteristic
2#20275685C211yesmaximal, characteristic
3#20274883C212yesmaximal
4#20272597C212yesmaximal
5C2×C53×#330C211yesmaximal, characteristic
6C2×#13133151—22no—
7C2×#13133149—22no—
8C2×#13133128C2211yescharacteristic
9#16810522—22no—
10#16810509—22no—
11#16810509—22no—
12#16810493C2211yescharacteristic
13#16810489C2212yes—
14#16810385—22no—
15C2×#125×C53C2211yescharacteristic
16C2×#97×C53C2212yes—
17C53×#330C414yescomplemented
18C22×#88322—21no—
19C2×C4⋊Dic53—22no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53D412yes—
24C2×C4×Dic53—22no—
25#13133151—44no—
26#13133149—44no—
27#13133128—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124—22no—
32C22×C8×C53—22no—
33#125×C53—22no—
34#125×C53C2×C412yes—
35#125×C53Q812yes—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39#97×C53—22no—
40#97×C53C2×C414yes—
41C22×C4×C53—22no—
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53C2×D412yes—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53—41no—
47C2×C8×C53—24no—
48C2×C8×C53—24no—
49C2×C8×C53—24no—
50C4⋊C4×C53—24no—
51C4⋊C4×C53—24no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53C4⋊C414yes—
54C42×C53C22×C411yescharacteristic
55C42×C53C2×Q811yescharacteristic
56C42×C53#4112yes—
57C4⋊Dic53—42no—
58C4⋊Dic53—42no—
59C4⋊Dic53—42no—
60C4⋊Dic53—42no—
61C4⋊Dic53—42no—
62C4⋊Dic53—82no—
63C4×Dic53—82no—
64C2×#88322—22no—
65C2×#88322—41no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—24no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53C2×C4⋊C411yescharacteristic
76C2×C4×C53C4×D412yes—
77C2×C4×C53C22⋊Q812yes—
78C2×C4×C53#12311yescharacteristic
79C2×Dic53—42no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—82no—
86C2×Dic53—81no—
87C8×C53—28no—
88#88322—42no—
89#8913—531nomaximal, Sylow
90C22×C53#38511yescharacteristic
91C22×C53#45312yes—
92C22×C53#45411yescharacteristic
93C22×C53#45612yes—
94C22×C53#45611yescharacteristic
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—22no—
98C4×C53—22no—
99C4×C53—44no—
100Dic53—82no—
101Dic53—82no—
102Dic53—161no—
103#2382—531no—
104#1992—531no—
105#1847—532no—
106C2×#330D5311yescharacteristic
107#1295—532no—
108C2×C53#165611yescharacteristic
109C2×C53#166012yes—
110C2×C53#166311yescharacteristic
111C2×C53#185711yescharacteristic
112C2×C53#185912yes—
113C2×#125C2×D5311yescharacteristic
114C2×#107—1062no—
115C2×#106—1062no—
116C2×#97—531no—
117C2×#97C2×D5312yes—
118#389—1062no—
119#387—1062no—
120#387—1062no—
121#384—531no—
122#382—532no—
123#340—1062no—
124#330Dic5314yescomplemented
125C53#891311yescharacteristic, complemented, Sylow
126C22×C8—22no—
127C22×C8—1061no—
128#125—22no—
129#125Dic10612yes—
130#125C2×Dic5312yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1061no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4D21212yes—
138C2×C42—1062no—
139C2×C42C22×D5311yescharacteristic
140#107—2124no—
141#106—2124no—
142#97—22no—
143#97—1062no—
144#97C2×Dic5314yes—
145#95—1062no—
146#95—1062no—
147#95—1062no—
148#95—1062no—
149C22×C4—22no—
150C22×C4—1062no—
151C22×C4—1062no—
152C22×C4—2121no—
153C22×C4C2×D21211yescharacteristic
154C22×C4D4×D5312yes—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—1062no—
159C2×C8—2121no—
160C2×C8—2121no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—4242no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C4⋊C4—2122no—
168C4⋊C4—2122no—
169C4⋊C4—2122no—
170C4⋊C4C4⋊Dic5314yes—
171C42—4242no—
172C42C2×Dic10611yescharacteristic
173C42#1171994712yes—
174C42C22×Dic5311yescharacteristic
175C23#1313318711yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—4242no—
184C2×C4—4241no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4C2×C4⋊Dic5311yescharacteristic
192C2×C4D4×Dic5312yes—
193C2×C4#1313319412yes—
194C2×C4#1313320111yescharacteristic
195C8—28no—
196C8—2122no—
197C22#1681049811yescharacteristic
198C22#1681079212yes—
199C22#1681079811yescharacteristic
200C22#1681080712yes—
201C22#1681080811yescharacteristic
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—8481no—
208C4—4242no—
209C4—4242no—
210C2#2027401011yescharacteristic
211C2#2027402812yes—
212C2#2027404411yescharacteristic
213C2#2027493211yescharacteristic
214C2#2027494312yes—
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#330C2
socle, prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenterC2×#330D53
hypocenterC53#8913
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
17. C53×#330C4semidirect
124. #330Dic53semidirect
125. C53#8913semidirect

Maximal subgroups

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

subgroupindex
#202779012
#202756852
#202748832
#202725972
C2×C53×#3302
#891353