Finite Group Databasebeta

← #27278478

Subgroups — #27278478

Order 13568.

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

normal subgroups only

This group has 396 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#20274849C212yesmaximal
4#20272597C212yesmaximal
5C2×C53×#330C211yesmaximal, characteristic
6C2×#13133128C2211yescharacteristic
7#16810522—22no—
8#16810509—22no—
9#16810509—22no—
10#16810493C2211yescharacteristic
11#16810489C2212yes—
12#16810386—22no—
13#16810386—22no—
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#13133128—22no—
26#13133124—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—22no—
30C22×C8×C53—22no—
31#125×C53—22no—
32#125×C53C2×C412yes—
33#125×C53Q812yes—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37#97×C53—22no—
38#97×C53C2×C414yes—
39C22×C4×C53—22no—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53C2×D412yes—
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53—41no—
45C2×C8×C53—24no—
46C2×C8×C53—42no—
47C2×C8×C53—42no—
48C4⋊C4×C53—24no—
49C4⋊C4×C53—24no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53C4⋊C414yes—
52C42×C53C22×C411yescharacteristic
53C42×C53C2×Q811yescharacteristic
54C42×C53#4112yes—
55C4⋊Dic53—42no—
56C4⋊Dic53—42no—
57C4⋊Dic53—42no—
58C4⋊Dic53—44no—
59C4⋊Dic53—82no—
60C4×Dic53—82no—
61C2×#88322—22no—
62C2×#88322—41no—
63C2×#88322—41no—
64C23×C53#12011yescharacteristic
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—24no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53C2×C4⋊C411yescharacteristic
73C2×C4×C53C4×D412yes—
74C2×C4×C53C22⋊Q812yes—
75C2×C4×C53#12311yescharacteristic
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—82no—
83C2×Dic53—81no—
84C8×C53—44no—
85#88322—42no—
86#8915—531nomaximal, Sylow
87C22×C53#38511yescharacteristic
88C22×C53#45412yes—
89C22×C53#45411yescharacteristic
90C22×C53#45512yes—
91C22×C53#45611yescharacteristic
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—44no—
97Dic53—82no—
98Dic53—82no—
99Dic53—161no—
100#2382—531no—
101#1992—531no—
102#1836—532no—
103C2×#330D5311yescharacteristic
104#1295—532no—
105C2×C53#165811yescharacteristic
106C2×C53#166211yescharacteristic
107C2×C53#166412yes—
108C2×C53#185711yescharacteristic
109C2×C53#186012yes—
110C2×#125C2×D5311yescharacteristic
111C2×#97—531no—
112C2×#97C2×D5312yes—
113#389—1062no—
114#387—1062no—
115#387—1062no—
116#384—531no—
117#382—532no—
118#340—1062no—
119#340—1062no—
120#340—1062no—
121#330Dic5314yescomplemented
122C53#891511yescharacteristic, complemented, Sylow
123C22×C8—22no—
124C22×C8—1061no—
125#125—22no—
126#125Dic10612yes—
127#125C2×Dic5312yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—1062no—
132C2×C4⋊C4—1061no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4D21212yes—
135C2×C42—1062no—
136C2×C42C22×D5311yescharacteristic
137#97—22no—
138#97—1062no—
139#97C2×Dic5314yes—
140#95—1062no—
141#95—1062no—
142#95—1062no—
143#95—1062no—
144C22×C4—22no—
145C22×C4—1062no—
146C22×C4—1062no—
147C22×C4—2121no—
148C22×C4C2×D21211yescharacteristic
149C22×C4D4×D5312yes—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—1062no—
154C2×C8—2121no—
155C2×C8—2121no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—4242no—
160C4⋊C4—2124no—
161C4⋊C4—2122no—
162C4⋊C4—2122no—
163C4⋊C4—2122no—
164C4⋊C4C4⋊Dic5314yes—
165C42—4242no—
166C42C2×Dic10611yescharacteristic
167C42#1171994712yes—
168C42C22×Dic5311yescharacteristic
169C23#1313318711yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—4242no—
178C2×C4—4241no—
179C2×C4—2122no—
180C2×C4—2122no—
181C2×C4—2122no—
182C2×C4—2122no—
183C2×C4—2122no—
184C2×C4—2122no—
185C2×C4C2×C4⋊Dic5311yescharacteristic
186C2×C4D4×Dic5312yes—
187C2×C4#1313319412yes—
188C2×C4#1313320111yescharacteristic
189C8—44no—
190C8—2122no—
191C22#1681049811yescharacteristic
192C22#1681079712yes—
193C22#1681079811yescharacteristic
194C22#1681080212yes—
195C22#1681080811yescharacteristic
196C4—22no—
197C4—22no—
198C4—21no—
199C4—21no—
200C4—44no—
201C4—8481no—
202C4—4242no—
203C4—4242no—
204C2#2027401811yescharacteristic
205C2#2027403911yescharacteristic
206C2#2027405112yes—
207C2#2027493211yescharacteristic
208C2#2027495012yes—
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#330C2
socle, prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenterC2×#330D53
hypocenterC53#8915
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
121. #330Dic53semidirect
122. C53#8915semidirect

Maximal subgroups

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

subgroupindex
#202779012
#202756852
#202748492
#202725972
C2×C53×#3302
#891553