Finite Group Databasebeta

← #27287232

Subgroups — #27287232

Order 13568.

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

normal subgroups only

This group has 364 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#20275696C211yesmaximal, characteristic
2#20273213C212yesmaximal
3#20272618C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#2402C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9#16810383—22no—
10#16810359—22no—
11#16810359—22no—
12C53×#400—22no—
13C53×#398—22no—
14C4×C4⋊C4×C53—22no—
15C4×C4⋊C4×C53C2212yes—
16C53×#376C2211yescharacteristic
17C2×#11719917—21no—
18#13133217Q812yes—
19#13133209—21no—
20#13133196—22no—
21C4×Dic106—22no—
22#13133176—21no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C22⋊#88322—22no—
26C4×#88322—22no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C42×C53—22no—
31C2×C42×C53D412yes—
32C2×C42×C53C2311yescharacteristic
33#95×C53—22no—
34#95×C53—22no—
35#95×C53—22no—
36#95×C53D412yes—
37#95×C53D412yes—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53C2×D411yescharacteristic
43C2×Dic106—22no—
44C4⋊C4×C53—44no—
45C4⋊C4×C53—44no—
46C4⋊C4×C53—44no—
47C42×C53—22no—
48C42×C53—22no—
49C42×C53—42no—
50C42×C53—42no—
51C42×C53C2×Q811yescharacteristic
52C42×C53#4112yes—
53C42×C53#4111yescharacteristic
54C22⋊Dic53—41no—
55C4⋊Dic53—22no—
56C4⋊Dic53—22no—
57#11719919—42no—
58C4×Dic53—41no—
59#11719917—42no—
60C2×#88322—42no—
61C2×#88322—41no—
62C23×C53#12011yescharacteristic
63C2×C4×C53—24no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53C22⋊Q812yes—
74C2×C4×C53C22⋊Q811yescharacteristic
75C2×C4×C53#12412yes—
76C2×C4×C53#12411yescharacteristic
77C2×C4×C53#13512yes—
78C2×C4×C53#13712yes—
79C2×Dic53—42no—
80Dic106—82no—
81#88322—82no—
82#88322—81no—
83#10373—531nomaximal, Sylow
84C22×C53—22no—
85C22×C53#39311yescharacteristic
86C22×C53#45212yes—
87C4×C53—22no—
88C4×C53—22no—
89C4×C53—44no—
90C4×C53—82no—
91C4×C53—82no—
92C4×C53#47912yes—
93C4×C53#48811yescharacteristic
94C4×C53#49111yescharacteristic
95Dic53—82no—
96#2402D5311yescharacteristic
97#1995—531no—
98#1503—532no—
99#1303—532no—
100#1274—531no—
101C2×C53—22no—
102C2×C53#202611yescharacteristic
103C2×C53#203112yes—
104#533—531no—
105#431—532no—
106#423—531no—
107#400—22no—
108#398—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×D5312yes—
111#376C2×D5311yescharacteristic
112#339—1062no—
113#328—1062no—
114#328—1062no—
115C53#1037311yescharacteristic, complemented, Sylow
116C2×M16—1061no—
117C4⋊Q8—532no—
118#125—1061no—
119C22⋊Q8—1062no—
120C4×Q8—1062no—
121#117—1061no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C22⋊D5312yes—
127C2×C42C22×D5311yescharacteristic
128C4⋊C8—1062no—
129C4⋊C8—1062no—
130C22⋊C8—1062no—
131C4×C8—1062no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95C22⋊D5312yes—
136#95C22⋊D5312yes—
137C2×Q8—1062no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4C2×C22⋊D5312yes—
142C22×C4C2×C22⋊D5311yescharacteristic
143M16—2122no—
144C2×C8—2122no—
145C2×C8—2121no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—1062no—
150C4⋊C4—1062no—
151C4⋊C4—2122no—
152C22⋊C4—2121no—
153C42—22no—
154C42—22no—
155C42—42no—
156C42—42no—
157C42—2121no—
158C42Q8×D5311yescharacteristic
159C42#1171994911yescharacteristic
160C42#1171994912yes—
161C23#1313318611yescharacteristic
162Q8—4242no—
163C2×C4—24no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—2122no—
174C2×C4#1313319312yes—
175C2×C4#1313319311yescharacteristic
176C2×C4#1313320612yes—
177C2×C4#1313320611yescharacteristic
178C2×C4#1313323712yes—
179C2×C4#1313324712yes—
180C8—4242no—
181C8—4241no—
182C22—22no—
183C22#1681053411yescharacteristic
184C22#1681078512yes—
185C4—22no—
186C4—22no—
187C4—44no—
188C4—82no—
189C4—82no—
190C4—4242no—
191C4#1681092712yes—
192C4#1681098011yescharacteristic
193C4#1681099311yescharacteristic
194C2—22no—
195C2#2027586311yescharacteristic
196C2#2027588412yes—
subgroupG/H
centerC2×C4#13133206
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2402C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#2402D53
hypocenterC53#10373
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2402

Complements

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

NH ≅ G/Ntype
115. C53#10373semidirect

Maximal subgroups

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

subgroupindex
#202756962
#202732132
#202726182
#202725442
C53×#24022
#1037353