Finite Group Databasebeta

← #27288163

Subgroups — #27288163

Order 13568.

408 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 77 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#20286025C211yesmaximal, characteristic
2#20274868C211yesmaximal, characteristic, complemented
3#20273411C212yesmaximal, complemented
4#20273224C212yesmaximal, complemented
5C53×#1290C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810904—22no—
8#16810859—22no—
9#16810819—22no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133146—22no—
12#16810469C2211yescharacteristic
13#16810371—22no—
14#16810366—22no—
15#16810358—24no—
16C53×#432C2212yes—
17C2×C4⋊C8×C53C2211yescharacteristic
18C2×#11719945—22no—
19C2×D424—22no—
20C22×#88322—21no—
21#13133217D412yes—
22#13133214D412yes—
23#13133205—22no—
24#13133191—22no—
25#13133191—22no—
26C4×D212—22no—
27C4×D212—22no—
28C2×C4×Dic53—21no—
29#13133151—22no—
30C4⋊#88322—22no—
31#13133146—44no—
32#13133139—24no—
33#13133124—21no—
34#13133124D412yes—
35C22×C8×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38C4⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53#4111yescharacteristic
44C2×C22⋊D53—42no—
45C22×Dic53—22no—
46#11719945—84no—
47C2×D212—22no—
48C2×D212—22no—
49C2×D212—22no—
50C2×C4×D53—42no—
51C2×Dic106—22no—
52C2×C8×C53—22no—
53C2×C8×C53—42no—
54C2×C8×C53—42no—
55C42×C53C2×D412yes—
56C42×C53#4112yes—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61C4×Dic53—22no—
62C4×Dic53—22no—
63C2×#88322—42no—
64C2×#88322—41no—
65D424—82no—
66C23×C53#12111yescharacteristic
67C22×D53—42no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53#12011yescharacteristic
73C2×C4×C53#12112yes—
74C2×C4×C53#12312yes—
75C2×C4×C53#12411yescharacteristic
76C22⋊D53—84no—
77C2×Dic53—24no—
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81D212—42no—
82D212—42no—
83D212—82no—
84D212—82no—
85C4×D53—84no—
86Dic106—82no—
87C8×C53—42no—
88C8×C53—82no—
89#88322—81no—
90#10512—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—21no—
93C22×C53#39411yescharacteristic
94C22×C53#47212yes—
95C2×D53—82no—
96C2×D53—82no—
97C2×D53—82no—
98C4×C53—22no—
99C4×C53—21no—
100C4×C53—22no—
101C4×C53#45311yescharacteristic
102C4×C53#45611yescharacteristic
103Dic53—44no—
104Dic53—82no—
105#3245—531no—
106#1843—531no—
107#1551—532no—
108#1505—532no—
109#1290D5311yescharacteristic, complemented
110C2×C53—22no—
111C2×C53#203911yescharacteristic
112C2×C53#204511yescharacteristic
113C2×C53#205311yescharacteristic
114D53—162no—
115#532—532no—
116#474—1062no—
117#466—1062no—
118#459—1062no—
119#432C2×D5312yes—
120C2×C4⋊C8—531no—
121C2×C4⋊C8—1062no—
122C2×C4⋊C8C2×D5311yescharacteristic
123#376—531no—
124#333—1062no—
125#331—1062no—
126#327—1064no—
127C53#1051211yescharacteristic, complemented, Sylow
128C2×#41—1062no—
129C2×D8—1062no—
130C22×C8—1061no—
131C22×C8D21212yes—
132C4⋊Q8—532no—
133#127—532no—
134#124—1062no—
135#121—1062no—
136#121—1062no—
137C4×D4—1062no—
138C4×D4—1062no—
139C2×C42—1061no—
140C2×C42C22×D5311yescharacteristic
141#107—1062no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—1062no—
145C4⋊C8—2124no—
146#102—1064no—
147C22⋊C8—22no—
148#97—22no—
149#95—532no—
150#95—1061no—
151#41—4244no—
152C2×Q8—1062no—
153C2×D4—1062no—
154C2×D4—1062no—
155C2×D4—1062no—
156C2×D4—2122no—
157C22×C4—1062no—
158C22×C4—2122no—
159C22×C4C2×D21211yescharacteristic
160C22×C4D4×D5311yescharacteristic
161C22×C4#1171994911yescharacteristic
162D8—4242no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—2122no—
167C2×C8—2121no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C22⋊C4—2122no—
171C22⋊C4—2122no—
172C42—1062no—
173C42—1062no—
174C42#1171994512yes—
175C42D4×D5312yes—
176C23—2122no—
177C23#1313319211yescharacteristic
178Q8—4242no—
179D4—4242no—
180D4—4242no—
181D4—4244no—
182D4—2122no—
183D4—2122no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—1064no—
189C2×C4—4244no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4#1313318711yescharacteristic
194C2×C4#1313319012yes—
195C2×C4#1313320212yes—
196C2×C4#1313320511yescharacteristic
197C8—42no—
198C8—82no—
199C8—4241no—
200C22—21no—
201C22—21no—
202C22—4242no—
203C22—4242no—
204C22—4242no—
205C22#1681054111yescharacteristic
206C22#1681089412yes—
207C4—22no—
208C4—22no—
209C4—21no—
210C4—4242no—
211C4—2124no—
212C4#1681079311yescharacteristic
213C4#1681080811yescharacteristic
214C2—22no—
215C2—8482no—
216C2#2027594011yescharacteristic
217C2#2027597211yescharacteristic
218C2#2027602211yescharacteristic
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1290C2
socleC22×C53#394
prime radical#20286025C2
hypercenter#1290D53
hypocenterC53#10512
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
2. #20274868C2semidirect
3. #20273411C2semidirect
4. #20273224C2semidirect
5. C53×#1290C2semidirect
109. #1290D53semidirect
127. C53#10512semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202748682
#202734112
#202732242
C53×#12902
#1051253