Finite Group Databasebeta

← #27287123

Subgroups — #27287123

Order 13568.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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#20279957C211yesmaximal, characteristic
2#20275662C212yesmaximal
3#20275516C211yesmaximal, characteristic
4#20272554C212yesmaximal
5C53×#1296C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133140—22no—
8C2×C4×#88322C2211yescharacteristic
9#16810562—22no—
10#16810522—22no—
11#16810488—22no—
12C4×C4⋊Dic53C2212yes—
13#16810386—22no—
14#16810386—22no—
15C2×C4⋊C8×C53C2212yes—
16C53×#384C2211yescharacteristic
17C22×Dic106—21no—
18C22×#88322—21no—
19#13133217—21no—
20#13133217—22no—
21#13133217Q812yes—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×C4×Dic53—22no—
27#13133140—44no—
28C4×#88322—22no—
29#13133124—24no—
30#13133124—22no—
31#13133124—22no—
32C22×C8×C53—22no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53D412yes—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53C2×D411yescharacteristic
40C22×C4×C53C2×D412yes—
41C22×Dic53—22no—
42C22×Dic53—22no—
43C22×Dic53—41no—
44C2×Dic106—22no—
45C2×Dic106—42no—
46C2×Dic106—42no—
47C2×Dic106—41no—
48C2×C8×C53—42no—
49C2×C8×C53—42no—
50C2×C8×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C42×C53C2×Q811yescharacteristic
55C42×C53#4112yes—
56C42×C53#4111yescharacteristic
57C4⋊Dic53—24no—
58C4⋊Dic53—22no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—42no—
63C4⋊Dic53—82no—
64C4×Dic53—42no—
65C4×Dic53—42no—
66C2×#88322—22no—
67C2×#88322—41no—
68C2×#88322—41no—
69C23×C53#12011yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—44no—
73C2×C4×C53—42no—
74C2×C4×C53C22⋊Q812yes—
75C2×C4×C53C22⋊Q811yescharacteristic
76C2×C4×C53#12412yes—
77C2×C4×C53#12411yescharacteristic
78C2×C4×C53#13512yes—
79C2×C4×C53#13512yes—
80C2×C4×C53#13712yes—
81C2×C4×C53#13712yes—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—81no—
90C2×Dic53—81no—
91Dic106—44no—
92Dic106—82no—
93C8×C53—82no—
94#88322—42no—
95#10352—531nomaximal, Sylow
96C22×C53#39311yescharacteristic
97C22×C53#45212yes—
98C22×C53#45212yes—
99C22×C53#45311yescharacteristic
100C22×C53#45411yescharacteristic
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—82no—
104C4×C53#47912yes—
105C4×C53#47912yes—
106C4×C53#48812yes—
107C4×C53#49112yes—
108Dic53—82no—
109Dic53—82no—
110Dic53—82no—
111#2571—531no—
112#1987—532no—
113#1956—531no—
114#1296D5311yescharacteristic
115#1279—532no—
116C2×C53#163011yescharacteristic
117C2×C53#163411yescharacteristic
118C2×C53#163812yes—
119C2×C53#202612yes—
120C2×C53#202711yescharacteristic
121C2×C4⋊Q8—531no—
122C2×C4⋊C8C2×D5312yes—
123C2×#103—1062no—
124C2×C4×C8—531no—
125#400—1062no—
126#389—1062no—
127#384C2×D5311yescharacteristic
128#382—1062no—
129C4×C4⋊C4—532no—
130#340—1062no—
131#340—1062no—
132C53#1035211yescharacteristic, complemented, Sylow
133C22×Q8—1061no—
134C22×C8—22no—
135C22×C8—1061no—
136C4⋊Q8—532no—
137C4⋊Q8—1062no—
138C4⋊Q8—1061no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—1062no—
143C2×C4⋊C4—1062no—
144C2×C4⋊C4D21212yes—
145C2×C42—1062no—
146C2×C42C22×D5311yescharacteristic
147C4⋊C8—22no—
148C4⋊C8—22no—
149#103—2124no—
150C4×C8—1062no—
151#95—1062no—
152#95—1064no—
153#95—1062no—
154C2×Q8—1062no—
155C2×Q8—2122no—
156C2×Q8—2122no—
157C2×Q8—2121no—
158C22×C4—22no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—2121no—
162C22×C4C2×D21211yescharacteristic
163C22×C4D4×D5312yes—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—1062no—
168C2×C8—2121no—
169C2×C8—2121no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—1064no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—4242no—
179C4⋊C4—2122no—
180C42—2122no—
181C42—2122no—
182C42#1171994512yes—
183C42#1171994711yescharacteristic
184C42Q8×D5311yescharacteristic
185C23#1313318711yescharacteristic
186Q8—4242no—
187Q8—2124no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—4241no—
193C2×C4—4241no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4#1313319512yes—
202C2×C4#1313319711yescharacteristic
203C2×C4#1313320312yes—
204C2×C4#1313320511yescharacteristic
205C2×C4#1313323612yes—
206C2×C4#1313323812yes—
207C2×C4#1313324612yes—
208C2×C4#1313324912yes—
209C8—82no—
210C8—2122no—
211C22#1681053611yescharacteristic
212C22#1681078612yes—
213C22#1681078812yes—
214C22#1681079311yescharacteristic
215C22#1681079811yescharacteristic
216C4—21no—
217C4—21no—
218C4—82no—
219C4—4242no—
220C4—4242no—
221C4—4242no—
222C4#1681092912yes—
223C4#1681093312yes—
224C4#1681097612yes—
225C4#1681099012yes—
226C2#2027383711yescharacteristic
227C2#2027386111yescharacteristic
228C2#2027388812yes—
229C2#2027586512yes—
230C2#2027587211yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socle, prime radicalC23×C53#120
hypercenter#1296D53
hypocenterC53#10352
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
132. C53#10352semidirect

Maximal subgroups

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

subgroupindex
#202799572
#202756622
#202755162
#202725542
C53×#12962
#1035253