Finite Group Databasebeta

← #27287121

Subgroups — #27287121

Order 13568.

416 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 125 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#20275663C212yesmaximal
2#20275516C211yesmaximal, characteristic
3#20272598C211yesmaximal, characteristic
4#20272556C212yesmaximal
5C53×#2571C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×C4⋊#88322C2212yes—
8C2×#13133140—22no—
9C2×C4×#88322C2211yescharacteristic
10#16810385—22no—
11#16810385—22no—
12C53×#400—22no—
13C53×#389—22no—
14C53×#384C2211yescharacteristic
15C53×#382—22no—
16C4×C4⋊C4×C53C2212yes—
17C22×Dic106—21no—
18C22×#88322—21no—
19C22×#88322—22no—
20#13133217—21no—
21#13133217—22no—
22#13133217Q812yes—
23C2×C4⋊Dic53—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133140—44no—
27C4×#88322—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53D412yes—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D412yes—
33C2×C42×C53—22no—
34C2×C42×C53C2311yescharacteristic
35#95×C53—22no—
36#95×C53—24no—
37#95×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53C2×D412yes—
43C22×Dic53—41no—
44C2×Dic106—22no—
45C2×Dic106—42no—
46C2×Dic106—42no—
47C2×Dic106—41no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—82no—
56C42×C53—42no—
57C42×C53—42no—
58C42×C53C2×Q811yescharacteristic
59C42×C53#4112yes—
60C42×C53#4111yescharacteristic
61C4⋊Dic53—24no—
62C4⋊Dic53—42no—
63C2×#88322—22no—
64C2×#88322—41no—
65C2×#88322—41no—
66C2×#88322—42no—
67C2×#88322—42no—
68C2×#88322—42no—
69C23×C53#12011yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—44no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53C22⋊Q811yescharacteristic
81C2×C4×C53C22⋊Q812yes—
82C2×C4×C53#12411yescharacteristic
83C2×C4×C53#12412yes—
84C2×C4×C53#13512yes—
85C2×C4×C53#13512yes—
86C2×C4×C53#13712yes—
87C2×C4×C53#13712yes—
88C2×Dic53—42no—
89C2×Dic53—81no—
90C2×Dic53—81no—
91Dic106—44no—
92Dic106—82no—
93#88322—42no—
94#88322—82no—
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—82no—
105C4×C53—82no—
106C4×C53#47912yes—
107C4×C53#47912yes—
108C4×C53#48812yes—
109C4×C53#49112yes—
110Dic53—82no—
111#2571D5311yescharacteristic
112#1987—532no—
113#1956—531no—
114#1296—531no—
115#1279—532no—
116C2×C53#163011yescharacteristic
117C2×C53#163411yescharacteristic
118C2×C53#163812yes—
119C2×C53#202612yes—
120C2×C53#202711yescharacteristic
121C2×C4⋊Q8—531no—
122C2×C4⋊C8—532no—
123C2×#103—1062no—
124C2×C4×C8—531no—
125#400—22no—
126#389—22no—
127#384C2×D5311yescharacteristic
128#382—22no—
129C4×C4⋊C4C2×D5312yes—
130#340—1062no—
131#340—1062no—
132C53#1035211yescharacteristic, complemented, Sylow
133C22×Q8—1061no—
134C22×C8—1061no—
135C22×C8—1062no—
136C4⋊Q8—532no—
137C4⋊Q8—1062no—
138C4⋊Q8—1061no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—1062no—
142C2×C4⋊C4C22⋊D5312yes—
143C2×C4⋊C4C22⋊D5312yes—
144C2×C4⋊C4C22⋊D5312yes—
145C2×C42—22no—
146C2×C42C22×D5311yescharacteristic
147C4⋊C8—1062no—
148C4⋊C8—1062no—
149#103—2124no—
150C4×C8—1062no—
151#95—22no—
152#95—24no—
153#95—22no—
154C2×Q8—1062no—
155C2×Q8—2122no—
156C2×Q8—2122no—
157C2×Q8—2121no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—2121no—
162C22×C4C2×C22⋊D5311yescharacteristic
163C22×C4C2×C22⋊D5312yes—
164C2×C8—1062no—
165C2×C8—2122no—
166C2×C8—2122no—
167C2×C8—2122no—
168C2×C8—2121no—
169C2×C8—2121no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—1064no—
179C4⋊C4—2122no—
180C42—42no—
181C42—42no—
182C42Q8×D5311yescharacteristic
183C42#1171994912yes—
184C42#1171994911yescharacteristic
185C23#1313318611yescharacteristic
186Q8—4242no—
187Q8—2124no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—2122no—
201C2×C4#1313319311yescharacteristic
202C2×C4#1313319312yes—
203C2×C4#1313320611yescharacteristic
204C2×C4#1313320612yes—
205C2×C4#1313323712yes—
206C2×C4#1313323712yes—
207C2×C4#1313324712yes—
208C2×C4#1313324712yes—
209C8—4242no—
210C8—2122no—
211C22#1681053411yescharacteristic
212C22#1681078512yes—
213C22#1681078512yes—
214C22#1681079111yescharacteristic
215C22#1681079611yescharacteristic
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—4242no—
222C4#1681092712yes—
223C4#1681092712yes—
224C4#1681098012yes—
225C4#1681099312yes—
226C2#2027383611yescharacteristic
227C2#2027386011yescharacteristic
228C2#2027388712yes—
229C2#2027586312yes—
230C2#2027586811yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2571C2
socle, prime radicalC23×C53#120
hypercenter#2571D53
hypocenterC53#10352
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

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
#202756632
#202755162
#202725982
#202725562
C53×#25712
#1035253