Finite Group Databasebeta

← #27278595

Subgroups — #27278595

Order 13568.

412 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 107 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#20275544C211yesmaximal, characteristic
2#20273503C212yesmaximal, complemented
3#20273377C211yesmaximal, characteristic
4#20272547C212yesmaximal, complemented
5C53×#2979C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2211yescharacteristic
8C2×#13133144—22no—
9C42×Dic53C2211yescharacteristic
10#16810375—22no—
11#16810373—22no—
12#16810372C412yescomplemented
13#16810367C411yescharacteristic, complemented
14#16810367C411yescharacteristic, complemented
15C53×#532C2211yescharacteristic
16C53×#436—22no—
17C8×D4×C53—22no—
18C53×#431C2212yes—
19C53×#408—22no—
20C2×#11719917—21no—
21C2×C4×Dic53—21no—
22C2×C4×Dic53—22no—
23C4⋊#88322—22no—
24C4⋊#88322C2×C412yes—
25C4⋊#88322C2×C412yes—
26#13133144—44no—
27C22⋊#88322—22no—
28C4×#88322—22no—
29C2×#41×C53D412yes—
30C2×M16×C53—22no—
31C22×C8×C53—22no—
32C4⋊Q8×C53C2×C411yescharacteristic
33#127×C53C2×C411yescharacteristic
34#124×C53Q812yes—
35#121×C53—22no—
36C4×D4×C53—22no—
37C2×C42×C53C2311yescharacteristic
38C4⋊C8×C53D412yes—
39C4⋊C8×C53D412yes—
40C22⋊C8×C53—24no—
41C22⋊C8×C53—22no—
42C4×C8×C53—22no—
43#41×C53—42no—
44#41×C53—42no—
45C2×Q8×C53C4⋊C412yes—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53C4⋊C412yes—
49C22×C4×C53—22no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53#4112yes—
52C22×Dic53—41no—
53M16×C53—82no—
54C2×C8×C53—22no—
55C2×C8×C53—22no—
56C2×C8×C53—82no—
57C4⋊C4×C53—22no—
58C22⋊C4×C53—24no—
59C42×C53C22×C411yescharacteristic
60C42×C53C2×D412yes—
61C42×C53C2×Q811yescharacteristic
62C4×Dic53—22no—
63C4×Dic53—24no—
64C4×Dic53—42no—
65C4×Dic53—41no—
66#11719917—42no—
67C2×#88322—22no—
68C2×#88322—22no—
69C2×#88322—41no—
70C23×C53—22no—
71C23×C53#12311yescharacteristic
72Q8×C53—42no—
73D4×C53—42no—
74D4×C53—42no—
75D4×C53—42no—
76D4×C53—82no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53C2×C4⋊C411yescharacteristic
85C2×C4×C53C4×D412yes—
86C2×C4×C53#12011yescharacteristic
87C2×C4×C53C22⋊Q812yes—
88C2×Dic53—42no—
89C2×Dic53—44no—
90C8×C53—42no—
91C8×C53—42no—
92#88322—42no—
93#88322—42no—
94#88322—81no—
95#8934—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—22no—
98C22×C53—42no—
99C22×C53—44no—
100C22×C53#38511yescharacteristic
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53#44312yes—
105C4×C53#44412yes—
106C4×C53#45311yescharacteristic
107C4×C53#45511yescharacteristic
108Dic53—44no—
109#2979D5311yescharacteristic
110#1962—531no—
111#1574—532no—
112#1543—531no—
113#1276—532no—
114C2×C53—41no—
115C2×C53—82no—
116C2×C53#185611yescharacteristic
117C2×C53#186612yes—
118#532C2×D5311yescharacteristic
119#436—22no—
120C8×D4—22no—
121#431—532no—
122#431C2×D5312yes—
123#423—531no—
124C2×#104—1062no—
125#408—22no—
126C43—531no—
127#335—1062no—
128#334—1062no—
129#333—532no—
130#331—531no—
131#331—531no—
132C53#893411yescharacteristic, complemented, Sylow
133C2×#41C22⋊D5312yes—
134C2×M16—22no—
135C2×M16—1061no—
136C22×C8—22no—
137C4⋊Q8C4×D5311yescharacteristic
138#127C4×D5311yescharacteristic
139#124Dic10612yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—1062no—
143C2×C42—1061no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—532no—
146C4⋊C8—532no—
147C4⋊C8—1062no—
148C4⋊C8C22⋊D5312yes—
149C4⋊C8C22⋊D5312yes—
150#104—2124no—
151C22⋊C8—24no—
152C22⋊C8—22no—
153C22⋊C8—1062no—
154C4×C8—22no—
155C4×C8—1062no—
156#41—42no—
157#41—42no—
158C2×Q8#1171991912yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4#1171991912yes—
162C22×C4—22no—
163C22×C4—2121no—
164C22×C4#1171994512yes—
165C22×C4C2×C22⋊D5311yescharacteristic
166M16—82no—
167M16—2122no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—82no—
171C2×C8—1062no—
172C2×C8—1062no—
173C2×C8—2121no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—1062no—
177C42—1064no—
178C42—2121no—
179C42—2122no—
180C42C2×Dic10611yescharacteristic
181C42C2×C4×D5311yescharacteristic
182C42C2×C22⋊D5312yes—
183C23—22no—
184C23#1313320011yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—2124no—
198C2×C4—2122no—
199C2×C4C2×#1171991911yescharacteristic
200C2×C4C4×C22⋊D5312yes—
201C2×C4#1313318611yescharacteristic
202C2×C4#1313319612yes—
203C8—42no—
204C8—42no—
205C8—4241no—
206C8—2122no—
207C8—2122no—
208C22—22no—
209C22—21no—
210C22—44no—
211C22—42no—
212C22#1681049611yescharacteristic
213C4—22no—
214C4—21no—
215C4—42no—
216C4—2124no—
217C4#1681073812yes—
218C4#1681074312yes—
219C4#1681079111yescharacteristic
220C4#1681080111yescharacteristic
221C2—41no—
222C2—82no—
223C2#2027492511yescharacteristic
224C2#2027499012yes—
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2979C2
socleC22×C53#385
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#2979D53
hypocenterC53#8934
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #20273503C2semidirect
4. #20272547C2semidirect
12. #16810372C4semidirect
13. #16810367C4semidirect
14. #16810367C4semidirect
132. C53#8934semidirect

Maximal subgroups

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

subgroupindex
#202755442
#202735032
#202733772
#202725472
C53×#29792
#893453