Finite Group Databasebeta

← #27288188

Subgroups — #27288188

Order 13568.

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

normal subgroups only

This group has 344 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#20274834C211yesmaximal, characteristic
2#20273415C212yesmaximal
3#20273227C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8#16810469C2211yescharacteristic
9#16810377—22no—
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—
22C22⋊#88322—22no—
23#13133128—22no—
24#13133124—21no—
25#13133124D412yes—
26C2×Q16×C53—22no—
27C22×C8×C53D412yes—
28C4⋊Q8×C53D412yes—
29C4⋊Q8×C53D412yes—
30#125×C53—22no—
31C22⋊Q8×C53—22no—
32C22⋊Q8×C53—22no—
33C4×Q8×C53—22no—
34C4×Q8×C53—22no—
35#117×C53—22no—
36C2×C42×C53C2311yescharacteristic
37#107×C53—22no—
38C4⋊C8×C53—22no—
39#103×C53—24no—
40C2×Q8×C53—22no—
41C2×Q8×C53—22no—
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53C2×D411yescharacteristic
44C22×C4×C53#4111yescharacteristic
45C22×Dic53—22no—
46Q16×C53—82no—
47C2×C8×C53—22no—
48C2×C8×C53—42no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C22⋊C4×C53—42no—
56C42×C53—42no—
57C42×C53C2×D412yes—
58C42×C53#4112yes—
59C4×Dic53—41no—
60C4×Dic53—41no—
61C2×#88322—42no—
62C2×#88322—41no—
63C2×#88322—42no—
64C23×C53#12111yescharacteristic
65Q8×C53—42no—
66Q8×C53—42no—
67Q8×C53—82no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53#12011yescharacteristic
75C2×C4×C53#12112yes—
76C2×C4×C53#12312yes—
77C2×C4×C53#12411yescharacteristic
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81C8×C53—42no—
82#88322—82no—
83#88322—81no—
84#10517—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#39411yescharacteristic
88C22×C53#47312yes—
89C4×C53—22no—
90C4×C53—21no—
91C4×C53—22no—
92C4×C53—82no—
93C4×C53—82no—
94C4×C53#45411yescharacteristic
95C4×C53#45511yescharacteristic
96Dic53—82no—
97#3249D5311yescharacteristic
98#1832—531no—
99#1552—532no—
100#1506—532no—
101#1290—531no—
102C2×C53—22no—
103C2×C53#204011yescharacteristic
104C2×C53#204711yescharacteristic
105C2×C53#205311yescharacteristic
106#533C2×D5312yes—
107#477—22no—
108#467—22no—
109#462—22no—
110#432—532no—
111C2×C4⋊C8—531no—
112C2×C4⋊C8C2×D5311yescharacteristic
113#376—531no—
114#336—1062no—
115#332—1062no—
116#332—1062no—
117#328—1064no—
118C53#1051711yescharacteristic, complemented, Sylow
119C2×Q16—22no—
120C22×C8—1061no—
121C22×C8C22⋊D5312yes—
122C4⋊Q8C22⋊D5312yes—
123C4⋊Q8C22⋊D5312yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42—1061no—
131C2×C42C22×D5311yescharacteristic
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—1062no—
135C4⋊C8—1062no—
136#103—24no—
137C22⋊C8—1062no—
138#97—1062no—
139#95—532no—
140#95—1061no—
141C2×Q8—22no—
142C2×Q8—22no—
143C22×C4—1062no—
144C22×C4D4×D5311yescharacteristic
145C22×C4#1171994711yescharacteristic
146C22×C4C2×C22⋊D5311yescharacteristic
147Q16—82no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—2122no—
151C2×C8—2121no—
152C2×C8—2122no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C42—42no—
161C42—2121no—
162C42—2121no—
163C42#1171994712yes—
164C42C2×C22⋊D5312yes—
165C23#1313318811yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—82no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—2122no—
176C2×C4—2122no—
177C2×C4—2122no—
178C2×C4#1313318611yescharacteristic
179C2×C4#1313318912yes—
180C2×C4#1313319812yes—
181C2×C4#1313320411yescharacteristic
182C8—42no—
183C8—4242no—
184C8—4241no—
185C22—21no—
186C22—21no—
187C22#1681053711yescharacteristic
188C22#1681089512yes—
189C4—22no—
190C4—22no—
191C4—21no—
192C4—82no—
193C4—82no—
194C4—4242no—
195C4#1681079611yescharacteristic
196C4#1681080111yescharacteristic
197C2—22no—
198C2#2027594111yescharacteristic
199C2#2027598011yescharacteristic
200C2#2027601811yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3249C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#3249D53
hypocenterC53#10517
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
118. C53#10517semidirect

Maximal subgroups

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

subgroupindex
#202748342
#202734152
#202732272
#202725852
C53×#32492
#1051753