Finite Group Databasebeta

← #27288159

Subgroups — #27288159

Order 13568.

408 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 81 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#20274868C211yesmaximal, characteristic, complemented
2#20273410C212yesmaximal, complemented
3#20273223C212yesmaximal, complemented
4#20272585C211yesmaximal, characteristic, complemented
5C53×#3245C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×#13133146—22no—
9#16810469C2211yescharacteristic
10#16810372—22no—
11#16810367—22no—
12#16810357—24no—
13C53×#532C2212yes—
14C53×#474—22no—
15C53×#466—22no—
16C53×#459—22no—
17C2×C4⋊C8×C53C2211yescharacteristic
18C22×#88322—21no—
19C2×C4×Dic53—21no—
20C4⋊#88322—22no—
21C4⋊#88322—22no—
22#13133146—44no—
23C22⋊#88322—22no—
24#13133128—22no—
25#13133124—21no—
26#13133124D412yes—
27C2×#41×C53—22no—
28C2×D8×C53—22no—
29C22×C8×C53D412yes—
30C4⋊Q8×C53D412yes—
31#127×C53D412yes—
32#124×C53—22no—
33#121×C53—22no—
34#121×C53—22no—
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C2×C42×C53C2311yescharacteristic
38#107×C53—22no—
39C4⋊C8×C53—22no—
40#102×C53—24no—
41#41×C53—84no—
42C2×Q8×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—42no—
47C22×C4×C53—42no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53#4111yescharacteristic
51C22×Dic53—22no—
52D8×C53—82no—
53C2×C8×C53—22no—
54C2×C8×C53—42no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C42×C53C2×D412yes—
60C42×C53#4112yes—
61C4×Dic53—22no—
62C4×Dic53—22no—
63C2×#88322—42no—
64C2×#88322—41no—
65C2×#88322—42no—
66C23×C53—42no—
67C23×C53#12111yescharacteristic
68Q8×C53—82no—
69D4×C53—42no—
70D4×C53—42no—
71D4×C53—82no—
72D4×C53—82no—
73D4×C53—84no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—42no—
79C2×C4×C53—84no—
80C2×C4×C53#12011yescharacteristic
81C2×C4×C53#12112yes—
82C2×C4×C53#12312yes—
83C2×C4×C53#12411yescharacteristic
84C2×Dic53—24no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C8×C53—42no—
88#88322—82no—
89#88322—81no—
90#10512—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—82no—
94C22×C53—82no—
95C22×C53—82no—
96C22×C53#39411yescharacteristic
97C22×C53#47212yes—
98C4×C53—22no—
99C4×C53—21no—
100C4×C53—22no—
101C4×C53—82no—
102C4×C53#45311yescharacteristic
103C4×C53#45611yescharacteristic
104Dic53—44no—
105#3245D5311yescharacteristic
106#1843—531no—
107#1551—532no—
108#1505—532no—
109#1290—531no—
110C2×C53—22no—
111C2×C53—162no—
112C2×C53#203911yescharacteristic
113C2×C53#204511yescharacteristic
114C2×C53#205311yescharacteristic
115#532C2×D5312yes—
116#474—22no—
117#466—22no—
118#459—22no—
119#432—532no—
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—22no—
129C2×D8—22no—
130C22×C8—1061no—
131C22×C8C22⋊D5312yes—
132C4⋊Q8C22⋊D5312yes—
133#127C22⋊D5312yes—
134#124—22no—
135#121—22no—
136#121—22no—
137C4×D4—22no—
138C4×D4—22no—
139C2×C42—1061no—
140C2×C42C22×D5311yescharacteristic
141#107—22no—
142C4⋊C8—22no—
143C4⋊C8—1062no—
144C4⋊C8—1062no—
145C4⋊C8—2124no—
146#102—24no—
147C22⋊C8—1062no—
148#97—1062no—
149#95—532no—
150#95—1061no—
151#41—84no—
152C2×Q8—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—42no—
157C22×C4—42no—
158C22×C4—1062no—
159C22×C4D4×D5311yescharacteristic
160C22×C4#1171994711yescharacteristic
161C22×C4C2×C22⋊D5311yescharacteristic
162D8—82no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—2122no—
166C2×C8—2121no—
167C2×C8—2122no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—1062no—
173C42—1062no—
174C42#1171994712yes—
175C42C2×C22⋊D5312yes—
176C23—42no—
177C23#1313318811yescharacteristic
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—82no—
182D4—82no—
183D4—84no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4—1064no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4#1313318611yescharacteristic
194C2×C4#1313318912yes—
195C2×C4#1313319812yes—
196C2×C4#1313320411yescharacteristic
197C8—42no—
198C8—4242no—
199C8—4241no—
200C22—21no—
201C22—21no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1681053711yescharacteristic
206C22#1681089012yes—
207C4—22no—
208C4—22no—
209C4—21no—
210C4—82no—
211C4—2124no—
212C4#1681079111yescharacteristic
213C4#1681080611yescharacteristic
214C2—22no—
215C2—162no—
216C2#2027593611yescharacteristic
217C2#2027596811yescharacteristic
218C2#2027601811yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3245C2
socleC22×C53#394
hypercenter#3245D53
hypocenterC53#10512
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20274868C2semidirect
2. #20273410C2semidirect
3. #20273223C2semidirect
4. #20272585C2semidirect
127. C53#10512semidirect

Maximal subgroups

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

subgroupindex
#202748682
#202734102
#202732232
#202725852
C53×#32452
#1051253