Finite Group Databasebeta

← #27302636

Subgroups — #27302636

Order 13568.

308 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 115 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#20283653C211yesmaximal, characteristic
2#20276481C211yesmaximal, characteristic
3#20272800C212yesmaximal
4#20272794C212yesmaximal
5C53×#2110C211yesmaximal, characteristic
6#16811176C411yescharacteristic
7C2×#13133157C2212yes—
8#16810753—22no—
9#16810751—22no—
10#16810651C411yescharacteristic
11#16810650C411yescharacteristic
12#16810646C411yescharacteristic
13#16810578C2211yescharacteristic
14C2×#13133127C2211yescharacteristic
15#16810444—22no—
16C2×M32×C53C2212yes—
17C53×#405C2211yescharacteristic
18C53×#347—22no—
19C2×#11719917C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21#13133216—21no—
22#13133208—21no—
23#13133194—22no—
24C4×Dic106—22no—
25#13133176C2×C411yescharacteristic
26C2×C4×Dic53C2×C411yescharacteristic
27#13133157—22no—
28#13133157—22no—
29C2×#11719908—22no—
30C4⋊#88322—22no—
31#13133146—22no—
32#13133146—22no—
33#13133127—21no—
34#13133127—21no—
35#13133127—21no—
36C8×Dic53—21no—
37C2×M16×C53C2×C411yescharacteristic
38C22×C8×C53C2311yescharacteristic
39#117×C53C2×C411yescharacteristic
40M32×C53—22no—
41M32×C53—22no—
42C2×C16×C53Q814yescomplemented
43#97×C53D412yes—
44C4×C8×C53D412yes—
45C22×C4×C53C22×C411yescharacteristic
46C22×Dic53—21no—
47C2×Dic106—22no—
48M16×C53—22no—
49C2×C8×C53C22⋊C412yes—
50C2×C8×C53C2×D411yescharacteristic
51C2×C8×C53C2×D411yescharacteristic
52C2×C8×C53C2×Q812yes—
53C2×C8×C53#4112yes—
54C4⋊C4×C53—21no—
55C22⋊C4×C53—21no—
56C42×C53C22⋊C412yes—
57C22⋊Dic53M1612yes—
58C4⋊Dic53—22no—
59C4⋊Dic53—21no—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53M1611yescharacteristic
65C4×Dic53M1611yescharacteristic
66#11719917—22no—
67C2×#88322—21no—
68C2×#88322—21no—
69C2×#88322—22no—
70C16×C53—24no—
71#11719908—42no—
72C23×C53C2×M1611yescharacteristic
73C2×C4×C53—22no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53C4×Q812yes—
76C2×C4×C53C2×M1611yescharacteristic
77C2×C4×C53#13112yes—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53—21no—
81C2×Dic53—81no—
82Dic106—82no—
83C8×C53—22no—
84C8×C53C22⋊Q812yes—
85C8×C53#12411yescharacteristic
86C8×C53C4⋊Q811yescharacteristic
87#88322—41no—
88#88322—41no—
89#13269—531nomaximal, Sylow
90C22×C53—41no—
91C22×C53#40711yescharacteristic
92C22×C53#44612yes—
93C4×C53—42no—
94C4×C53#39111yescharacteristic
95C4×C53#40811yescharacteristic
96C4×C53#44612yes—
97Dic53—41no—
98Dic53—42no—
99Dic53—41no—
100#2961—531no—
101#2153—531no—
102#2110D5311yescharacteristic
103#1378—532no—
104#1375—532no—
105C2×C53—21no—
106C2×C53—22no—
107C2×C53#198511yescharacteristic
108#533—531no—
109C2×M32—532no—
110C2×M32C2×D5312yes—
111#446—1062no—
112#446—1062no—
113#424—531no—
114#424—531no—
115#423—531no—
116#405—531no—
117#405C2×D5311yescharacteristic
118C2×#97—531no—
119#364—1062no—
120#347—22no—
121C53#1326911yescharacteristic, complemented, Sylow
122C2×M16—531no—
123C2×M16C4×D5311yescharacteristic
124C22×C8—531no—
125C22×C8C22×D5311yescharacteristic
126C4⋊Q8—1061no—
127#125—1061no—
128C22⋊Q8—1062no—
129C4×Q8—1062no—
130#117—531no—
131#117C4×D5311yescharacteristic
132C2×C42—531no—
133M32—22no—
134M32—22no—
135M32—1062no—
136M32—1062no—
137C2×C16—1062no—
138C2×C16Dic10614yescomplemented
139C4⋊C8—1062no—
140C4⋊C8—1062no—
141C4⋊C8—1062no—
142#97—1061no—
143#97—1061no—
144#97—1061no—
145#97C22⋊D5312yes—
146C4×C8—1061no—
147C4×C8D21212yes—
148C2×Q8—1062no—
149C22×C4—1061no—
150C22×C4C2×C4×D5311yescharacteristic
151M16—22no—
152M16—1062no—
153C2×C8—1061no—
154C2×C8—1061no—
155C2×C8—1062no—
156C2×C8#1171992112yes—
157C2×C8C2×Dic10612yes—
158C2×C8C2×D21211yescharacteristic
159C2×C8#1171994512yes—
160C2×C8C2×C22⋊D5311yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1061no—
166C22⋊C4—21no—
167C22⋊C4—532no—
168C42—531no—
169C42—531no—
170C42—1061no—
171C42—1061no—
172C42#1171992112yes—
173C16—24no—
174C16—2122no—
175C23M16×D5311yescharacteristic
176Q8—4242no—
177C2×C4—22no—
178C2×C4—1061no—
179C2×C4—1061no—
180C2×C4—1062no—
181C2×C4—4241no—
182C2×C4C2×#1171992111yescharacteristic
183C2×C4C4×Dic10612yes—
184C2×C4M16×D5311yescharacteristic
185C2×C4#1313322512yes—
186C8—22no—
187C8—2121no—
188C8—2121no—
189C8#1313319612yes—
190C8#1313320511yescharacteristic
191C8#1313321711yescharacteristic
192C22—41no—
193C22#1681058611yescharacteristic
194C22#1681075112yes—
195C4—42no—
196C4—2121no—
197C4—2121no—
198C4—2122no—
199C4#1681052911yescharacteristic
200C4#1681059111yescharacteristic
201C4#1681075112yes—
202C2—21no—
203C2—22no—
204C2#2027565511yescharacteristic
subgroupG/H
centerC4#16810529
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2110C2
socleC2×C53#1985
prime radicalC23×C53C2×M16
cosocleC22×C8×C53C23
hypercenter#2110D53
hypocenterC53#13269
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
42. C2×C16×C53Q8semidirect
121. C53#13269semidirect
138. C2×C16Dic106semidirect

Maximal subgroups

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

subgroupindex
#202836532
#202764812
#202728002
#202727942
C53×#21102
#1326953