Finite Group Databasebeta

← #27287227

Subgroups — #27287227

Order 13568.

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

normal subgroups only

This group has 402 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#20275547C211yesmaximal, characteristic
2#20273209C212yesmaximal
3#20272618C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#2402C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9C2×#13133145—22no—
10#16810356—22no—
11#16810356—22no—
12C53×#400—22no—
13C53×#398—22no—
14C4×C4⋊C4×C53—22no—
15C4×C4⋊C4×C53C2212yes—
16C53×#376C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719917—21no—
19#13133217Q811yescharacteristic
20#13133214Q811yescharacteristic
21#13133205—21no—
22#13133191—22no—
23C4×D212—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133145—44no—
27C22⋊#88322—22no—
28C4×#88322—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C42×C53—22no—
33C2×C42×C53D412yes—
34C2×C42×C53C2311yescharacteristic
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38#95×C53D412yes—
39#95×C53D412yes—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×D411yescharacteristic
45C2×C22⋊D53—41no—
46#11719945—42no—
47#11719945—42no—
48C2×D212—21no—
49C2×D212—22no—
50C2×C4×D53—41no—
51C2×Dic106—21no—
52C4⋊C4×C53—44no—
53C4⋊C4×C53—44no—
54C4⋊C4×C53—44no—
55C42×C53—22no—
56C42×C53—22no—
57C42×C53—42no—
58C42×C53—42no—
59C42×C53C2×Q811yescharacteristic
60C42×C53#4112yes—
61C42×C53#4111yescharacteristic
62#11719921—42no—
63C4⋊Dic53—22no—
64#11719917—42no—
65C2×#88322—42no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68C22×D53—41no—
69C2×C4×C53—24no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
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×C53C22⋊Q812yes—
80C2×C4×C53C22⋊Q811yescharacteristic
81C2×C4×C53#12412yes—
82C2×C4×C53#12411yescharacteristic
83C2×C4×C53#13512yes—
84C2×C4×C53AGL(1,ℤ/8)12yes—
85C22⋊D53—82no—
86C2×Dic53—41no—
87D212—42no—
88D212—82no—
89C4×D53—81no—
90C4×D53—81no—
91Dic106—42no—
92#88322—82no—
93#88322—81no—
94#10372—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53#39311yescharacteristic
97C22×C53#44912yes—
98C2×D53—81no—
99C2×D53—82no—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—44no—
103C4×C53—82no—
104C4×C53—82no—
105C4×C53#47812yes—
106C4×C53#48811yescharacteristic
107C4×C53#49011yescharacteristic
108Dic53—81no—
109#2402D5311yescharacteristic, complemented
110#1963—531no—
111#1502—532no—
112#1303—532no—
113#1274—531no—
114C2×C53—22no—
115C2×C53#202511yescharacteristic
116C2×C53#203112yes—
117D53—161no—
118#532—531no—
119#431—532no—
120#423—531no—
121C2×#104—1062no—
122#400—22no—
123#398—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×D5312yes—
126#376C2×D5311yescharacteristic
127#327—1062no—
128#327—1062no—
129C53#1037211yescharacteristic, complemented, Sylow
130C2×#41—1061no—
131C2×M16—1061no—
132C4⋊Q8—531no—
133#127—531no—
134#124—1061no—
135#121—1062no—
136C4×D4—1062no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C42—22no—
141C2×C42C22⋊D5312yes—
142C2×C42C22×D5311yescharacteristic
143C4⋊C8—1062no—
144C4⋊C8—1062no—
145#104—2124no—
146C22⋊C8—1062no—
147C4×C8—1062no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95C22⋊D5312yes—
152#95C22⋊D5312yes—
153#41—2122no—
154#41—2122no—
155C2×Q8—1061no—
156C2×D4—1062no—
157C2×D4—1061no—
158C2×D4—2121no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2121no—
163C22×C4C2×C22⋊D5312yes—
164C22×C4C2×C22⋊D5311yescharacteristic
165M16—2122no—
166C2×C8—2122no—
167C2×C8—2121no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—1062no—
172C22⋊C4—2122no—
173C42—22no—
174C42—22no—
175C42—42no—
176C42—42no—
177C42Q8×D5311yescharacteristic
178C42#1171994911yescharacteristic
179C42#1171994912yes—
180C23—2121no—
181C23#1313318611yescharacteristic
182Q8—2122no—
183D4—4242no—
184D4—4242no—
185D4—2122no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—4241no—
197C2×C4—4241no—
198C2×C4—2121no—
199C2×C4#1313319312yes—
200C2×C4#1313319311yescharacteristic
201C2×C4#1313320612yes—
202C2×C4#1313320611yescharacteristic
203C2×C4#1313323712yes—
204C2×C4#1313324212yes—
205C8—4242no—
206C8—4241no—
207C22—22no—
208C22—4241no—
209C22—4242no—
210C22#1681053411yescharacteristic
211C22#1681076812yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—82no—
216C4—82no—
217C4—4241no—
218C4#1681092012yes—
219C4#1681098011yescharacteristic
220C4#1681098911yescharacteristic
221C2—22no—
222C2—8481no—
223C2#2027585811yescharacteristic
224C2#2027588412yes—
subgroupG/H
centerC2×C4#13133206
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2402C2
socleC22×C53#393
prime radical#16811173C22
hypercenter#2402D53
hypocenterC53#10372
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
3. #20272618C2semidirect
4. #20272544C2semidirect
5. C53×#2402C2semidirect
109. #2402D53semidirect
129. C53#10372semidirect

Maximal subgroups

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

subgroupindex
#202755472
#202732092
#202726182
#202725442
C53×#24022
#1037253