Finite Group Databasebeta

← #27288164

Subgroups — #27288164

Order 13568.

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

normal subgroups only

This group has 388 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#20274834C211yesmaximal, characteristic, complemented
2#20273410C212yesmaximal, complemented
3#20273223C212yesmaximal, complemented
4#20272585C211yesmaximal, characteristic, complemented
5C53×#3245C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8#16810469C2211yescharacteristic
9#16810377—22no—
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—
22C22⋊#88322—22no—
23#13133128—22no—
24#13133124—21no—
25#13133124D412yes—
26C2×#41×C53—22no—
27C2×D8×C53—22no—
28C22×C8×C53D412yes—
29C4⋊Q8×C53D412yes—
30#127×C53D412yes—
31#124×C53—22no—
32#121×C53—22no—
33#121×C53—22no—
34C4×D4×C53—22no—
35C4×D4×C53—22no—
36C2×C42×C53C2311yescharacteristic
37#107×C53—22no—
38C4⋊C8×C53—22no—
39#102×C53—24no—
40#41×C53—84no—
41C2×Q8×C53—22no—
42C2×D4×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53—42no—
46C22×C4×C53—42no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53#4111yescharacteristic
50C22×Dic53—22no—
51D8×C53—82no—
52C2×C8×C53—22no—
53C2×C8×C53—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—42no—
57C22⋊C4×C53—42no—
58C42×C53C2×D412yes—
59C42×C53#4112yes—
60C4×Dic53—41no—
61C4×Dic53—41no—
62C2×#88322—42no—
63C2×#88322—41no—
64C2×#88322—42no—
65C23×C53—42no—
66C23×C53#12111yescharacteristic
67Q8×C53—82no—
68D4×C53—42no—
69D4×C53—42no—
70D4×C53—82no—
71D4×C53—82no—
72D4×C53—84no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—42no—
78C2×C4×C53—84no—
79C2×C4×C53#12011yescharacteristic
80C2×C4×C53#12112yes—
81C2×C4×C53#12312yes—
82C2×C4×C53#12411yescharacteristic
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C8×C53—42no—
87#88322—82no—
88#88322—81no—
89#10513—531nomaximal, Sylow
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—82no—
93C22×C53—82no—
94C22×C53—82no—
95C22×C53#39411yescharacteristic
96C22×C53#47212yes—
97C4×C53—22no—
98C4×C53—21no—
99C4×C53—22no—
100C4×C53—82no—
101C4×C53#45411yescharacteristic
102C4×C53#45511yescharacteristic
103Dic53—82no—
104#3245D5311yescharacteristic
105#1832—531no—
106#1551—532no—
107#1505—532no—
108#1290—531no—
109C2×C53—22no—
110C2×C53—162no—
111C2×C53#204011yescharacteristic
112C2×C53#204511yescharacteristic
113C2×C53#205311yescharacteristic
114#532C2×D5312yes—
115#474—22no—
116#466—22no—
117#459—22no—
118#432—532no—
119C2×C4⋊C8—531no—
120C2×C4⋊C8C2×D5311yescharacteristic
121#376—531no—
122#336—1062no—
123#333—1062no—
124#331—1062no—
125#327—1064no—
126C53#1051311yescharacteristic, complemented, Sylow
127C2×#41—22no—
128C2×D8—22no—
129C22×C8—1061no—
130C22×C8C22⋊D5312yes—
131C4⋊Q8C22⋊D5312yes—
132#127C22⋊D5312yes—
133#124—22no—
134#121—22no—
135#121—22no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C42—1061no—
139C2×C42C22×D5311yescharacteristic
140#107—22no—
141C4⋊C8—22no—
142C4⋊C8—1062no—
143C4⋊C8—1062no—
144#102—24no—
145C22⋊C8—1062no—
146#97—1062no—
147#95—532no—
148#95—1061no—
149#41—84no—
150C2×Q8—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—42no—
155C22×C4—42no—
156C22×C4—1062no—
157C22×C4D4×D5311yescharacteristic
158C22×C4#1171994711yescharacteristic
159C22×C4C2×C22⋊D5311yescharacteristic
160D8—82no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—2122no—
164C2×C8—2121no—
165C2×C8—2122no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—2121no—
171C42—2121no—
172C42#1171994712yes—
173C42C2×C22⋊D5312yes—
174C23—42no—
175C23#1313318811yescharacteristic
176Q8—82no—
177D4—42no—
178D4—42no—
179D4—82no—
180D4—82no—
181D4—84no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4#1313318611yescharacteristic
192C2×C4#1313318912yes—
193C2×C4#1313319812yes—
194C2×C4#1313320411yescharacteristic
195C8—42no—
196C8—4242no—
197C8—4241no—
198C22—21no—
199C22—21no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22#1681053711yescharacteristic
204C22#1681089012yes—
205C4—22no—
206C4—22no—
207C4—21no—
208C4—82no—
209C4—4242no—
210C4#1681079611yescharacteristic
211C4#1681080111yescharacteristic
212C2—22no—
213C2—162no—
214C2#2027594111yescharacteristic
215C2#2027596811yescharacteristic
216C2#2027601811yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3245C2
socleC22×C53#394
hypercenter#3245D53
hypocenterC53#10513
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. #20274834C2semidirect
2. #20273410C2semidirect
3. #20273223C2semidirect
4. #20272585C2semidirect
126. C53#10513semidirect

Maximal subgroups

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

subgroupindex
#202748342
#202734102
#202732232
#202725852
C53×#32452
#1051353