Finite Group Databasebeta

← #27285088

Subgroups — #27285088

Order 13568.

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

normal subgroups only

This group has 476 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#20285239C211yesmaximal, characteristic
2#20284755C212yesmaximal
3#20275655C211yesmaximal, characteristic
4#20273119C212yesmaximal
5C53×#1275C211yesmaximal, characteristic
6#16811180C2212yes—
7C2×C4×Dic106C2211yescharacteristic
8#16811032—22no—
9#16811032—22no—
10#16810959—22no—
11#16810919—22no—
12#16810834—22no—
13#16810781—22no—
14C4×Dic212—24no—
15#16810661—22no—
16C2×C4⋊#88322C2211yescharacteristic
17#16810377—22no—
18#16810355—24no—
19C53×#431C2212yes—
20C2×#97×C53C2211yescharacteristic
21C22×Dic106D411yescharacteristic
22C2×Dic212—24no—
23C22×#88322—21no—
24#13133217D412yes—
25#13133217D412yes—
26#13133209—22no—
27#13133196—22no—
28#13133196—22no—
29C4×Dic106—22no—
30C4×Dic106—22no—
31C4×Dic106—22no—
32C4×Dic106—22no—
33#13133176—22no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×C4×Dic53—21no—
37#13133151—22no—
38#13133151—22no—
39#13133151—24no—
40#13133149—24no—
41C4⋊#88322—22no—
42#13133142—24no—
43#13133142—24no—
44C22×C8×C53D412yes—
45C2×C42×C53C2311yescharacteristic
46C4⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48#97×C53—22no—
49C4×C8×C53D414yes—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D411yescharacteristic
52C22×C4×C53#4111yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—21no—
55C2×Dic106—22no—
56C2×Dic106—22no—
57C2×Dic106—22no—
58C2×Dic106—42no—
59C2×C8×C53—22no—
60C2×C8×C53—22no—
61C2×C8×C53—24no—
62C2×C8×C53Q1614yescomplemented
63C42×C53C2×D412yes—
64C42×C53C2×D412yes—
65C22⋊Dic53—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—22no—
71#11719919—42no—
72#11719919—42no—
73#11719919—42no—
74C4×Dic53—41no—
75C4×Dic53—41no—
76C4×Dic53—42no—
77C2×#88322—41no—
78C2×#88322—42no—
79Dic212—48no—
80C23×C53#12111yescharacteristic
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53#12012yes—
84C2×C4×C53#12011yescharacteristic
85C2×C4×C53#12112yes—
86C2×C4×C53#12711yescharacteristic
87C2×C4×C53C2×Q1612yes—
88C2×C4×C53#13712yes—
89C2×Dic53—22no—
90C2×Dic53—41no—
91C2×Dic53—41no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—41no—
96Dic106—24no—
97Dic106—42no—
98Dic106—44no—
99Dic106—44no—
100C8×C53—24no—
101C8×C53—44no—
102#88322—81no—
103#10037—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—21no—
106C22×C53#39211yescharacteristic
107C22×C53#46212yes—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53#45112yes—
111C4×C53#49411yescharacteristic
112C4×C53#49711yescharacteristic
113Dic53—42no—
114Dic53—82no—
115Dic53—82no—
116Dic53—81no—
117#3162—531no—
118#3104—532no—
119#1985—531no—
120#1481—532no—
121#1275D5311yescharacteristic
122C2×C53—22no—
123C2×C53#201011yescharacteristic
124C2×C53#201611yescharacteristic
125C2×C53#201711yescharacteristic
126#533—532no—
127C2×C4×Q8—531no—
128#501—1062no—
129#501—1062no—
130#484—1062no—
131#477—1062no—
132#462—1062no—
133#451—1062no—
134C4×Q16—1064no—
135#431C2×D5312yes—
136#427—1062no—
137C2×C4⋊C8—531no—
138C2×#97C2×D5311yescharacteristic
139#336—1062no—
140Q8⋊C8—1064no—
141C53#1003711yescharacteristic, complemented, Sylow
142C22×Q8—531no—
143C2×Q16—1064no—
144C22×C8—1061no—
145C22×C8D21212yes—
146C4⋊Q8—532no—
147C4⋊Q8—532no—
148#125—1062no—
149C22⋊Q8—1062no—
150C22⋊Q8—1062no—
151C4×Q8—1062no—
152C4×Q8—1062no—
153C4×Q8—1062no—
154C4×Q8—1062no—
155#117—1062no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—1061no—
158C2×C42—1061no—
159C2×C42C22×D5311yescharacteristic
160#107—1062no—
161#107—1062no—
162#107—1064no—
163#106—1064no—
164C4⋊C8—22no—
165C4⋊C8—1062no—
166#103—1064no—
167#103—1064no—
168C22⋊C8—22no—
169#97—22no—
170C4×C8D21214yes—
171C2×Q8—1062no—
172C2×Q8—1062no—
173C2×Q8—1062no—
174C2×Q8—2122no—
175C22×C4—1061no—
176C22×C4—1061no—
177C22×C4C2×D21211yescharacteristic
178C22×C4D4×D5311yescharacteristic
179C22×C4#1171994911yescharacteristic
180Q16—2128no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—2121no—
185C2×C8—2122no—
186C2×C8Dic21214yescomplemented
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C22⋊C4—2122no—
196C42—2121no—
197C42—2121no—
198C42—2122no—
199C42C2×D21212yes—
200C42D4×D5312yes—
201C23#1313319211yescharacteristic
202Q8—1064no—
203Q8—2124no—
204Q8—2124no—
205Q8—2122no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1062no—
209C2×C4—2122no—
210C2×C4—2121no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4#1313318712yes—
216C2×C4#1313318711yescharacteristic
217C2×C4#1313319212yes—
218C2×C4#1313321411yescharacteristic
219C2×C4C2×Dic21212yes—
220C2×C4#1313325012yes—
221C8—24no—
222C8—44no—
223C8—4241no—
224C22—21no—
225C22—21no—
226C22#1681053311yescharacteristic
227C22#1681083412yes—
228C4—22no—
229C4—22no—
230C4—4242no—
231C4—4241no—
232C4—4242no—
233C4—2122no—
234C4#1681078112yes—
235C4#1681100311yescharacteristic
236C4#1681101711yescharacteristic
237C2—22no—
238C2#2027578311yescharacteristic
239C2#2027581711yescharacteristic
240C2#2027582011yescharacteristic
subgroupG/H
centerC2×C4#13133214
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1275C2
socleC22×C53#392
prime radicalC23×C53#121
hypercenter#1275D53
hypocenterC53#10037
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
62. C2×C8×C53Q16semidirect
141. C53#10037semidirect
186. C2×C8Dic212semidirect

Maximal subgroups

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

subgroupindex
#202852392
#202847552
#202756552
#202731192
C53×#12752
#1003753