Finite Group Databasebeta

← #27288862

Subgroups — #27288862

Order 13568.

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

normal subgroups only

This group has 330 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#20274869C211yesmaximal, characteristic, complemented
3#20273362C212yesmaximal
4#20272611C212yesmaximal, complemented
5C53×#1274C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810647C2211yescharacteristic
8C2×C4⋊#88322—21no—
9C2×C4⋊#88322—21no—
10C2×C4⋊#88322C2212yes—
11C2×#13133145—21no—
12C2×#13133145—21no—
13#16810368—22no—
14#16810366—22no—
15C53×#431C2212yes—
16C53×#376C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719917—21no—
19C22×#88322—22no—
20#13133217D411yescharacteristic
21#13133214D411yescharacteristic
22#13133205—21no—
23#13133191—22no—
24C4×D212—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322—42no—
29C4⋊#88322—42no—
30#13133145—42no—
31#13133145—42no—
32C2×C42×C53D411yescharacteristic
33C2×C42×C53D411yescharacteristic
34C2×C42×C53C2311yescharacteristic
35C4⋊C8×C53—22no—
36C22⋊C8×C53—22no—
37C4×C8×C53—22no—
38#95×C53—22no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4112yes—
43C2×C22⋊D53—41no—
44#11719945—41no—
45#11719945—41no—
46#11719945—41no—
47#11719945—41no—
48C2×D212—21no—
49C2×D212—22no—
50C2×C4×D53—41no—
51C2×Dic106—21no—
52C2×C8×C53—42no—
53C42×C53—21no—
54C42×C53—21no—
55C42×C53D812yes—
56C42×C53SD1612yes—
57C42×C53C2×D411yescharacteristic
58C42×C53#4111yescharacteristic
59C42×C53#4112yes—
60#11719921—42no—
61C4⋊Dic53—22no—
62#11719917—41no—
63#11719917—41no—
64C2×#88322—42no—
65C2×#88322—41no—
66C2×#88322—42no—
67C2×#88322—42no—
68C23×C53#12311yescharacteristic
69C22×D53—41no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53#12111yescharacteristic
76C2×C4×C53#12312yes—
77C2×C4×C53#12411yescharacteristic
78C2×C4×C53#12612yes—
79C2×C4×C53C2×D811yescharacteristic
80C2×C4×C53C2×SD1611yescharacteristic
81C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C53#13711yescharacteristic
83C22⋊D53—81no—
84C22⋊D53—81no—
85C2×Dic53—41no—
86D212—41no—
87D212—41no—
88D212—82no—
89C4×D53—81no—
90C4×D53—81no—
91Dic106—41no—
92Dic106—41no—
93C8×C53—82no—
94#88322—82no—
95#88322—81no—
96#10627—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53#39711yescharacteristic
99C22×C53#48011yescharacteristic
100C22×C53#48111yescharacteristic
101C2×D53—81no—
102C2×D53—82no—
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—42no—
107C4×C53#45911yescharacteristic
108C4×C53#46111yescharacteristic
109C4×C53#48611yescharacteristic
110C4×C53#48911yescharacteristic
111Dic53—81no—
112#1963—531no—
113#1843—531no—
114#1540—532no—
115#1301—532no—
116#1274D5311yescharacteristic, complemented
117C2×C53—21no—
118C2×C53—21no—
119C2×C53#207111yescharacteristic
120C2×C53#208012yes—
121D53—161no—
122#532—531no—
123#431C2×D5312yes—
124#423—531no—
125C2×C4⋊C8—532no—
126C2×C4⋊C8—1061no—
127C2×C4⋊C8—1061no—
128C2×#104—1061no—
129C2×#104—1061no—
130#376C2×D5311yescharacteristic
131#332—1062no—
132#331—1062no—
133C53#1062711yescharacteristic, complemented, Sylow
134C2×#41—1061no—
135C2×M16—1061no—
136C22×C8—1062no—
137C4⋊Q8—531no—
138#127—531no—
139#124—1061no—
140#121—1062no—
141C4×D4—1062no—
142C2×C42D21211yescharacteristic
143C2×C42D21211yescharacteristic
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—1062no—
147C4⋊C8—1062no—
148C4⋊C8—1062no—
149C4⋊C8—2122no—
150C4⋊C8—2122no—
151#104—2122no—
152#104—2122no—
153C22⋊C8—22no—
154C4×C8—22no—
155#95—22no—
156#41—2121no—
157#41—2121no—
158#41—2121no—
159#41—2121no—
160C2×Q8—1061no—
161C2×D4—1061no—
162C2×D4—1062no—
163C2×D4—2121no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2121no—
167C22×C4C2×D21211yescharacteristic
168C22×C4#1171994712yes—
169M16—2121no—
170M16—2121no—
171C2×C8—42no—
172C2×C8—2122no—
173C2×C8—2121no—
174C2×C8—2122no—
175C2×C8—2122no—
176C4⋊C4—1062no—
177C22⋊C4—2122no—
178C42—21no—
179C42—21no—
180C42#1171991312yes—
181C42D42412yes—
182C42#1171994512yes—
183C42D4×D5311yescharacteristic
184C42#1171994911yescharacteristic
185C23—2121no—
186C23#1313320111yescharacteristic
187Q8—2121no—
188Q8—2121no—
189D4—4242no—
190D4—4241no—
191D4—4241no—
192D4—2121no—
193D4—2121no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—4241no—
200C2×C4—4241no—
201C2×C4—2121no—
202C2×C4#1313319211yescharacteristic
203C2×C4#1313319912yes—
204C2×C4#1313320511yescharacteristic
205C2×C4#1313321212yes—
206C2×C4C2×D42411yescharacteristic
207C2×C4C2×#1171991311yescharacteristic
208C2×C4#1313324511yescharacteristic
209C2×C4#1313325011yescharacteristic
210C8—82no—
211C8—4242no—
212C8—4241no—
213C22—22no—
214C22—4241no—
215C22—4242no—
216C22#1681055011yescharacteristic
217C22#1681093711yescharacteristic
218C22#1681094211yescharacteristic
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—4241no—
224C4#1681081911yescharacteristic
225C4#1681082911yescharacteristic
226C4#1681097011yescharacteristic
227C4#1681098311yescharacteristic
228C2—21no—
229C2—21no—
230C2—8481no—
231C2#2027612011yescharacteristic
232C2#2027616712yes—
subgroupG/H
centerC2×C4#13133205
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1274C2
socleC22×C53#397
prime radical#16811173C22
hypercenter#1274D53
hypocenterC53#10627
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
2. #20274869C2semidirect
4. #20272611C2semidirect
5. C53×#1274C2semidirect
116. #1274D53semidirect
133. C53#10627semidirect

Maximal subgroups

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

subgroupindex
#202755472
#202748692
#202733622
#202726112
C53×#12742
#1062753