Finite Group Databasebeta

← #27285427

Subgroups — #27285427

Order 13568.

500 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16811029C212yesmaximal
2#20282783C211yesmaximal, characteristic
3C2×#16810364C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#1960C211yesmaximal, characteristic
6C2×#13133209C2212yes—
7#16811029—24no—
8#16811029—24no—
9C2×#13133150—22no—
10C2×#13133148—22no—
11C2×#13133128C2212yes—
12#16810563—22no—
13#16810555—22no—
14#16810553—22no—
15#16810364—24no—
16C2×C4⋊Q8×C53C2211yescharacteristic
17C2×#103×C53—22no—
18C2×#97×C53C2211yescharacteristic
19C53×#382C2211yescharacteristic
20C22×#88322—22no—
21#13133209—22no—
22#13133209—22no—
23#13133209—22no—
24#13133209—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30#13133150—28no—
31#13133148—28no—
32#13133128—22no—
33#13133128D414yes—
34#13133124—24no—
35#13133124—22no—
36C22×Q8×C53—21no—
37C22×C8×C53—21no—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53D412yes—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53C2311yescharacteristic
44#103×C53—44no—
45#97×C53—22no—
46#95×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53—42no—
49C2×Q8×C53—42no—
50C2×Q8×C53—41no—
51C22×C4×C53—22no—
52C22×C4×C53—41no—
53C22×C4×C53C2×D412yes—
54C22×C4×C53C2×D411yescharacteristic
55C22×Dic53—22no—
56C22×Dic53—22no—
57C2×C8×C53—22no—
58C2×C8×C53—41no—
59C2×C8×C53—41no—
60C4⋊C4×C53—24no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C42×C53C2×D411yescharacteristic
64C42×C53C2×D412yes—
65C42×C53#4111yescharacteristic
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—82no—
71#11719919—42no—
72#11719919—42no—
73#11719919—42no—
74#11719919—42no—
75C2×#88322—28no—
76C2×#88322—24no—
77C23×C53#12011yescharacteristic
78Q8×C53—44no—
79Q8×C53—82no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—24no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—44no—
86C2×C4×C53—81no—
87C2×C4×C53—81no—
88C2×C4×C53#12012yes—
89C2×C4×C53#12112yes—
90C2×C4×C53#12111yescharacteristic
91C2×C4×C53#12711yescharacteristic
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—44no—
97C8×C53—42no—
98#88322—28no—
99#10087—531nomaximal, Sylow
100C22×C53#39211yescharacteristic
101C22×C53#45311yescharacteristic
102C22×C53#45411yescharacteristic
103C22×C53#45614yes—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—82no—
109C4×C53—82no—
110Dic53—82no—
111Dic53—82no—
112C2×#501—532no—
113#2822—531no—
114#1960D5311yescharacteristic
115C2×#330—532no—
116#1295—531no—
117C2×C53#166012yes—
118C2×C53#166312yes—
119C2×C53#200512yes—
120C2×C53#200611yescharacteristic
121C2×C4⋊Q8C2×D5311yescharacteristic
122C2×#125—532no—
123#501—1064no—
124#501—1064no—
125C2×#107—1062no—
126C2×#106—1062no—
127C2×#103—22no—
128C2×#97—532no—
129C2×#97C2×D5311yescharacteristic
130#400—1062no—
131#398—1062no—
132#398—1062no—
133#382C2×D5311yescharacteristic
134#330—1064no—
135C53#1008711yescharacteristic, complemented, Sylow
136C22×Q8—21no—
137C22×C8—21no—
138C22×C8—1062no—
139C4⋊Q8—21no—
140C4⋊Q8—22no—
141C4⋊Q8D21212yes—
142#125—1062no—
143#125—1062no—
144#125—1062no—
145#125—1062no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4C22⋊D5312yes—
153C2×C42C22×D5311yescharacteristic
154#107—1068no—
155#106—1068no—
156#103—44no—
157#97—22no—
158#97—534no—
159#97—1062no—
160#95—22no—
161#95—1064no—
162#95—1062no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—1062no—
170C22×C4—1062no—
171C22×C4D4×D5312yes—
172C22×C4C2×C22⋊D5311yescharacteristic
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—1068no—
177C2×C8—1064no—
178C4⋊C4—24no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—4242no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C42C2×D21211yescharacteristic
191C42#1171994511yescharacteristic
192C42D4×D5312yes—
193C23#1313318511yescharacteristic
194Q8—44no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2124no—
209C2×C4#1313318712yes—
210C2×C4#1313319012yes—
211C2×C4#1313319111yescharacteristic
212C2×C4#1313321311yescharacteristic
213C8—42no—
214C8—1068no—
215C22#1681053211yescharacteristic
216C22#1681079011yescharacteristic
217C22#1681079511yescharacteristic
218C22#1681080414yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—4242no—
226C4—4242no—
227C2#2027402412yes—
228C2#2027404112yes—
229C2#2027575212yes—
230C2#2027575611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1960C2
socle, prime radicalC23×C53#120
hypercenter#1960D53
hypocenterC53#10087
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
135. C53#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
#202827832
C2×#168103642
#202725962
C53×#19602
#1008753