Finite Group Databasebeta

← #27287172

Subgroups — #27287172

Order 13568.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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#20275687C212yesmaximal
2#20275534C211yesmaximal, characteristic
3#20272596C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#2845C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133140—22no—
8C2×#13133128C2211yescharacteristic
9C2×#13133128C2212yes—
10#16810385—22no—
11#16810385—22no—
12C53×#398—22no—
13C53×#398—22no—
14C53×#398—22no—
15C53×#382C2211yescharacteristic
16C53×#382C2212yes—
17C22×Dic106—21no—
18C22×#88322—21no—
19C22×#88322—22no—
20#13133217—21no—
21#13133217—22no—
22#13133217Q812yes—
23C2×C4⋊Dic53—22no—
24#13133140—44no—
25#13133128—22no—
26#13133128—22no—
27#13133128—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D412yes—
34C2×C42×C53C2311yescharacteristic
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38#95×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53C2×D412yes—
43C22×C4×C53C2×D411yescharacteristic
44C22×Dic53—41no—
45C2×Dic106—22no—
46C2×Dic106—42no—
47C2×Dic106—42no—
48C2×Dic106—41no—
49C4⋊C4×C53—42no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—82no—
53C4⋊C4×C53—82no—
54C4⋊C4×C53—82no—
55C42×C53C2×Q811yescharacteristic
56C42×C53#4111yescharacteristic
57C42×C53#4112yes—
58C4⋊Dic53—24no—
59C4⋊Dic53—42no—
60C2×#88322—22no—
61C2×#88322—24no—
62C2×#88322—41no—
63C2×#88322—42no—
64C2×#88322—41no—
65C2×#88322—42no—
66C23×C53#12011yescharacteristic
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
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×Dic53—42no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87Dic106—44no—
88Dic106—82no—
89#88322—42no—
90#88322—44no—
91#10361—531nomaximal, Sylow
92C22×C53#39311yescharacteristic
93C22×C53#45311yescharacteristic
94C22×C53#45411yescharacteristic
95C22×C53#45512yes—
96C22×C53#45612yes—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—82no—
102C4×C53—82no—
103C4×C53—82no—
104Dic53—82no—
105#2845D5311yescharacteristic
106#1993—532no—
107#1960—531no—
108#1295—531no—
109#1295—532no—
110C2×C53#166112yes—
111C2×C53#166211yescharacteristic
112C2×C53#166311yescharacteristic
113C2×C53#202711yescharacteristic
114C2×C53#202812yes—
115C2×C4⋊Q8—531no—
116C2×#103—1062no—
117C2×#97—531no—
118C2×#97—532no—
119#398—22no—
120#398—22no—
121#398—22no—
122#382C2×D5311yescharacteristic
123#382C2×D5312yes—
124#340—1062no—
125#340—1062no—
126C53#1036111yescharacteristic, complemented, Sylow
127C22×Q8—1061no—
128C22×C8—1062no—
129C22×C8—1061no—
130C4⋊Q8—532no—
131C4⋊Q8—1062no—
132C4⋊Q8—1061no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4C22⋊D5312yes—
138C2×C4⋊C4C22⋊D5312yes—
139C2×C4⋊C4C22⋊D5312yes—
140C2×C42C22×D5311yescharacteristic
141#103—2124no—
142#97—1062no—
143#97—1062no—
144#97—1062no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149C2×Q8—1062no—
150C2×Q8—2122no—
151C2×Q8—2122no—
152C2×Q8—2121no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—2121no—
157C22×C4C2×C22⋊D5312yes—
158C22×C4C2×C22⋊D5311yescharacteristic
159C2×C8—1062no—
160C2×C8—1064no—
161C2×C8—2121no—
162C2×C8—2122no—
163C2×C8—2122no—
164C2×C8—2121no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—1064no—
172C4⋊C4—2122no—
173C42Q8×D5311yescharacteristic
174C42#1171994912yes—
175C42#1171994911yescharacteristic
176C23#1313318611yescharacteristic
177Q8—4242no—
178Q8—2124no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—4241no—
193C2×C4—4241no—
194C2×C4—2122no—
195C2×C4#1313319312yes—
196C2×C4#1313319311yescharacteristic
197C2×C4#1313320612yes—
198C2×C4#1313320611yescharacteristic
199C8—2122no—
200C8—2124no—
201C22#1681053411yescharacteristic
202C22#1681079111yescharacteristic
203C22#1681079611yescharacteristic
204C22#1681080112yes—
205C22#1681080612yes—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—82no—
212C4—82no—
213C4—4242no—
214C2#2027403212yes—
215C2#2027403811yescharacteristic
216C2#2027404311yescharacteristic
217C2#2027586811yescharacteristic
218C2#2027587312yes—
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2845C2
socle, prime radicalC23×C53#120
hypercenter#2845D53
hypocenterC53#10361
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2845

Complements

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

NH ≅ G/Ntype
126. C53#10361semidirect

Maximal subgroups

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

subgroupindex
#202756872
#202755342
#202725962
C53×#28452
#1036153