Finite Group Databasebeta

← #27286384

Subgroups — #27286384

Order 13568.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#20285856C212yesmaximal
2#20275535C211yesmaximal, characteristic
3#20273279C212yesmaximal
4#20272560C211yesmaximal, characteristic
5C53×#3162C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16811019—22no—
8#16810973—22no—
9#16810951—24no—
10#16810908—24no—
11#16810676C2212yes—
12C2×#13133142—22no—
13#16810489C2211yescharacteristic
14#16810360—24no—
15C53×#533C2212yes—
16C53×#501—22no—
17C53×#501—22no—
18C53×#427—22no—
19C2×#97×C53C2211yescharacteristic
20C22×Dic106—21no—
21#13133217—21no—
22#13133217—22no—
23#13133217D412yes—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—21no—
26#13133148—28no—
27C4⋊#88322—22no—
28#13133142—44no—
29#13133141—28no—
30C22⋊#88322D414yes—
31C4×#88322—22no—
32#13133124—22no—
33C22×C8×C53D412yes—
34C4⋊Q8×C53D412yes—
35C4⋊Q8×C53D412yes—
36#125×C53—22no—
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41#117×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#107×C53—24no—
44#106×C53—22no—
45#106×C53—22no—
46#97×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53—22no—
49C22×C4×C53C2×D412yes—
50C22×C4×C53C2×D411yescharacteristic
51C22×Dic53—41no—
52C22×Dic53—41no—
53C2×Dic106—22no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×Dic106—41no—
57C2×C8×C53—22no—
58C2×C8×C53—22no—
59C2×C8×C53SD1614yes—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53—42no—
68C42×C53C2×D411yescharacteristic
69C42×C53C2×D412yes—
70C42×C53#4111yescharacteristic
71C4⋊Dic53—24no—
72C4⋊Dic53—42no—
73C4⋊Dic53—42no—
74C2×#88322—24no—
75C23×C53#12711yescharacteristic
76Q8×C53—44no—
77Q8×C53—82no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53#12012yes—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53#12111yescharacteristic
84C2×C4×C53#12112yes—
85C2×C4×C53C2×SD1612yes—
86C2×C4×C53#13712yes—
87C2×Dic53—42no—
88C2×Dic53—81no—
89C2×Dic53—81no—
90C2×Dic53—82no—
91C2×Dic53—81no—
92Dic106—44no—
93Dic106—82no—
94C8×C53—24no—
95#88322—44no—
96#10227—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53#39211yescharacteristic
99C22×C53#49211yescharacteristic
100C22×C53#49711yescharacteristic
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—82no—
104C4×C53—82no—
105C4×C53#44812yes—
106C4×C53#46112yes—
107Dic53—82no—
108Dic53—161no—
109#3229—532no—
110#3162D5311yescharacteristic
111#1960—531no—
112#1518—532no—
113#1281—531no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#200912yes—
117C2×C53#201311yescharacteristic
118#533C2×D5312yes—
119C2×C4⋊Q8—531no—
120#501—22no—
121#501—22no—
122#498—1062no—
123#487—1062no—
124#483—1064no—
125#475—1064no—
126#431—532no—
127#427—22no—
128C2×#103—1062no—
129C2×#97C2×D5311yescharacteristic
130#382—531no—
131#328—1064no—
132C53#1022711yescharacteristic, complemented, Sylow
133C22×Q8—1061no—
134C22×C8C22⋊D5312yes—
135C4⋊Q8—532no—
136C4⋊Q8—1061no—
137C4⋊Q8—1062no—
138C4⋊Q8C22⋊D5312yes—
139C4⋊Q8C22⋊D5312yes—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—1062no—
147C2×C4⋊C4—1061no—
148C2×C42C22×D5311yescharacteristic
149#107—24no—
150#106—22no—
151#106—22no—
152#106—1068no—
153C4⋊C8—1062no—
154#103—1068no—
155#103—2124no—
156C22⋊C8—534no—
157#97—22no—
158C4×C8—1062no—
159#95—1062no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1062no—
163C2×Q8—2122no—
164C2×Q8—2122no—
165C2×Q8—2121no—
166C22×C4—2121no—
167C22×C4—2121no—
168C22×C4D4×D5312yes—
169C22×C4C2×C22⋊D5311yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1064no—
173C2×C8#1171992414yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—1064no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C22⋊C4—42no—
184C42—42no—
185C42D4×D5311yescharacteristic
186C42#1171994711yescharacteristic
187C42C2×C22⋊D5312yes—
188C23#1313321311yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—4242no—
192Q8—2124no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—4242no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—2122no—
201C2×C4#1313318512yes—
202C2×C4#1313318611yescharacteristic
203C2×C4#1313318811yescharacteristic
204C2×C4#1313318912yes—
205C2×C4C2×#1171992412yes—
206C2×C4#1313324812yes—
207C8—24no—
208C8—2124no—
209C22—22no—
210C22#1681053111yescharacteristic
211C22#1681099711yescharacteristic
212C22#1681101511yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—8481no—
218C4—4242no—
219C4#1681076212yes—
220C4#1681082512yes—
221C2—21no—
222C2—21no—
223C2#2027577212yes—
224C2#2027579811yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3162C2
socleC22×C53#392
prime radicalC23×C53#127
hypercenter#3162D53
hypocenterC53#10227
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
132. C53#10227semidirect

Maximal subgroups

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

subgroupindex
#202858562
#202755352
#202732792
#202725602
C53×#31622
#1022753