Finite Group Databasebeta

← #27288183

Subgroups — #27288183

Order 13568.

364 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 81 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#20274868C211yesmaximal, characteristic
2#20273415C212yesmaximal
3#20273227C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×#13133146—22no—
9#16810469C2211yescharacteristic
10#16810369—22no—
11#16810369—22no—
12#16810360—24no—
13C53×#533C2212yes—
14C53×#477—22no—
15C53×#467—22no—
16C53×#462—22no—
17C2×C4⋊C8×C53C2211yescharacteristic
18C22×#88322—21no—
19C2×C4×Dic53—21no—
20C4⋊#88322—22no—
21C4⋊#88322—22no—
22#13133146—44no—
23C22⋊#88322—22no—
24#13133128—22no—
25#13133124—21no—
26#13133124D412yes—
27C2×Q16×C53—22no—
28C22×C8×C53D412yes—
29C4⋊Q8×C53D412yes—
30C4⋊Q8×C53D412yes—
31#125×C53—22no—
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—22no—
34C4×Q8×C53—22no—
35C4×Q8×C53—22no—
36#117×C53—22no—
37C2×C42×C53C2311yescharacteristic
38#107×C53—22no—
39C4⋊C8×C53—22no—
40#103×C53—24no—
41C2×Q8×C53—22no—
42C2×Q8×C53—22no—
43C22×C4×C53C2×D411yescharacteristic
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53#4111yescharacteristic
46C22×Dic53—22no—
47Q16×C53—82no—
48C2×C8×C53—22no—
49C2×C8×C53—42no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—42no—
56C22⋊C4×C53—42no—
57C42×C53—42no—
58C42×C53C2×D412yes—
59C42×C53#4112yes—
60C4×Dic53—22no—
61C4×Dic53—22no—
62C2×#88322—42no—
63C2×#88322—41no—
64C2×#88322—42no—
65C23×C53#12111yescharacteristic
66Q8×C53—42no—
67Q8×C53—42no—
68Q8×C53—82no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—41no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53#12011yescharacteristic
76C2×C4×C53#12112yes—
77C2×C4×C53#12312yes—
78C2×C4×C53#12411yescharacteristic
79C2×Dic53—24no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C8×C53—42no—
83#88322—82no—
84#88322—81no—
85#10516—531nomaximal, Sylow
86C22×C53—21no—
87C22×C53—21no—
88C22×C53#39411yescharacteristic
89C22×C53#47312yes—
90C4×C53—22no—
91C4×C53—21no—
92C4×C53—22no—
93C4×C53—82no—
94C4×C53—82no—
95C4×C53#45311yescharacteristic
96C4×C53#45611yescharacteristic
97Dic53—44no—
98#3249D5311yescharacteristic
99#1843—531no—
100#1552—532no—
101#1506—532no—
102#1290—531no—
103C2×C53—22no—
104C2×C53#203911yescharacteristic
105C2×C53#204711yescharacteristic
106C2×C53#205311yescharacteristic
107#533C2×D5312yes—
108#477—22no—
109#467—22no—
110#462—22no—
111#432—532no—
112C2×C4⋊C8—531no—
113C2×C4⋊C8—1062no—
114C2×C4⋊C8C2×D5311yescharacteristic
115#376—531no—
116#332—1062no—
117#332—1062no—
118#328—1064no—
119C53#1051611yescharacteristic, complemented, Sylow
120C2×Q16—22no—
121C22×C8—1061no—
122C22×C8C22⋊D5312yes—
123C4⋊Q8C22⋊D5312yes—
124C4⋊Q8C22⋊D5312yes—
125#125—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130#117—22no—
131C2×C42—1061no—
132C2×C42C22×D5311yescharacteristic
133#107—22no—
134C4⋊C8—22no—
135C4⋊C8—1062no—
136C4⋊C8—1062no—
137C4⋊C8—2124no—
138#103—24no—
139C22⋊C8—1062no—
140#97—1062no—
141#95—532no—
142#95—1061no—
143C2×Q8—22no—
144C2×Q8—22no—
145C22×C4—1062no—
146C22×C4D4×D5311yescharacteristic
147C22×C4#1171994711yescharacteristic
148C22×C4C2×C22⋊D5311yescharacteristic
149Q16—82no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—2122no—
153C2×C8—2121no—
154C2×C8—2122no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C22⋊C4—42no—
162C42—42no—
163C42—1062no—
164C42—1062no—
165C42#1171994712yes—
166C42C2×C22⋊D5312yes—
167C23#1313318811yescharacteristic
168Q8—42no—
169Q8—42no—
170Q8—82no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—1064no—
178C2×C4—2122no—
179C2×C4—2122no—
180C2×C4#1313318611yescharacteristic
181C2×C4#1313318912yes—
182C2×C4#1313319812yes—
183C2×C4#1313320411yescharacteristic
184C8—42no—
185C8—4242no—
186C8—4241no—
187C22—21no—
188C22—21no—
189C22#1681053711yescharacteristic
190C22#1681089512yes—
191C4—22no—
192C4—22no—
193C4—21no—
194C4—82no—
195C4—82no—
196C4—2124no—
197C4#1681079111yescharacteristic
198C4#1681080611yescharacteristic
199C2—22no—
200C2#2027593611yescharacteristic
201C2#2027598011yescharacteristic
202C2#2027601811yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3249C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#3249D53
hypocenterC53#10516
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
119. C53#10516semidirect

Maximal subgroups

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

subgroupindex
#202748682
#202734152
#202732272
#202725852
C53×#32492
#1051653