Finite Group Databasebeta

← #27287583

Subgroups — #27287583

Order 13568.

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

normal subgroups only

This group has 386 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#20275685C212yesmaximal
2#20274848C211yesmaximal, characteristic
3C2×#16810365C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#2822C211yesmaximal, characteristic
6C2×#13133128C2212yes—
7C2×#13133128C2211yescharacteristic
8#16810493C2211yescharacteristic
9#16810385—22no—
10#16810385—22no—
11#16810385—22no—
12#16810365—24no—
13C2×#125×C53C2212yes—
14C53×#400—22no—
15C53×#398—22no—
16C53×#398—22no—
17C53×#382C2211yescharacteristic
18C22×#88322—22no—
19C22×#88322—21no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53—21no—
22#13133128—21no—
23#13133128—22no—
24#13133128—22no—
25#13133128D412yes—
26#125×C53—22no—
27#125×C53—22no—
28#125×C53—22no—
29#125×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D412yes—
35C2×C4⋊C4×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37#95×C53—24no—
38#95×C53—22no—
39#95×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×D411yescharacteristic
45C22×Dic53—41no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—82no—
56C42×C53C2×D411yescharacteristic
57C42×C53#4111yescharacteristic
58C42×C53#4112yes—
59C4⋊Dic53—42no—
60C4⋊Dic53—44no—
61C2×#88322—24no—
62C2×#88322—22no—
63C2×#88322—42no—
64C2×#88322—41no—
65C2×#88322—41no—
66C2×#88322—42no—
67C23×C53#12011yescharacteristic
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
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×C53—42no—
81C2×C4×C53#12111yescharacteristic
82C2×C4×C53#12112yes—
83C2×C4×C53#12312yes—
84C2×C4×C53#12411yescharacteristic
85C2×Dic53—81no—
86C2×Dic53—82no—
87#88322—44no—
88#88322—42no—
89#10428—531nomaximal, Sylow
90C22×C53#39411yescharacteristic
91C22×C53#45411yescharacteristic
92C22×C53#45412yes—
93C22×C53#45511yescharacteristic
94C22×C53#45612yes—
95C4×C53—22no—
96C4×C53—22no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—82no—
100C4×C53—82no—
101C4×C53—82no—
102Dic53—161no—
103#2822D5311yescharacteristic
104#1992—532no—
105#1836—531no—
106C2×#330—532no—
107#1295—531no—
108C2×C53#165811yescharacteristic
109C2×C53#166311yescharacteristic
110C2×C53#166412yes—
111C2×C53#203812yes—
112C2×C53#204011yescharacteristic
113C2×#125C2×D5312yes—
114C2×#97—531no—
115C2×#97—532no—
116#400—22no—
117#398—22no—
118#398—22no—
119#384—531no—
120#382C2×D5311yescharacteristic
121#340—1062no—
122#340—1062no—
123#340—1062no—
124#330—1064no—
125C53#1042811yescharacteristic, complemented, Sylow
126C22×C8—1062no—
127C22×C8—1061no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—1061no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4C22⋊D5312yes—
138C2×C4⋊C4C22⋊D5312yes—
139C2×C4⋊C4C22⋊D5312yes—
140C2×C42C22×D5311yescharacteristic
141#97—532no—
142#97—1062no—
143#97—1061no—
144#97—1062no—
145#95—24no—
146#95—22no—
147#95—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—2121no—
152C22×C4C2×C22⋊D5311yescharacteristic
153C22×C4C2×C22⋊D5312yes—
154C2×C8—1064no—
155C2×C8—1062no—
156C2×C8—2122no—
157C2×C8—2121no—
158C2×C8—2121no—
159C2×C8—2122no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—2124no—
171C4⋊C4—2122no—
172C42D4×D5311yescharacteristic
173C42#1171994711yescharacteristic
174C42#1171994712yes—
175C23#1313318611yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—4241no—
190C2×C4—4242no—
191C2×C4#1313318811yescharacteristic
192C2×C4#1313318912yes—
193C2×C4#1313319812yes—
194C2×C4#1313320411yescharacteristic
195C8—2124no—
196C8—2122no—
197C22#1681053711yescharacteristic
198C22#1681079611yescharacteristic
199C22#1681079612yes—
200C22#1681080111yescharacteristic
201C22#1681080612yes—
202C4—22no—
203C4—22no—
204C4—21no—
205C4—21no—
206C4—82no—
207C4—82no—
208C4—82no—
209C4—8481no—
210C2#2027401711yescharacteristic
211C2#2027404311yescharacteristic
212C2#2027404912yes—
213C2#2027592912yes—
214C2#2027594111yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2822C2
socle, prime radicalC23×C53#120
hypercenter#2822D53
hypocenterC53#10428
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
125. C53#10428semidirect

Maximal subgroups

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

subgroupindex
#202756852
#202748482
C2×#168103652
#202725962
C53×#28222
#1042853