Finite Group Databasebeta

← #27281622

Subgroups — #27281622

Order 13568.

436 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 95 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#20285091C211yesmaximal, characteristic
2#20283821C211yesmaximal, characteristic, complemented
3#20273428C212yesmaximal
4#20272588C212yesmaximal, complemented
5C53×#1537C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7#16810989—22no—
8#16810971C414yes—
9#16810754—24no—
10#16810684—22no—
11#16810680C2212yes—
12#16810627—22no—
13C4×#11719917C2211yescharacteristic
14#16810366—24no—
15C53×#533C2211yescharacteristic
16C53×#432C2212yes—
17C53×#330—22no—
18C22×D212—21no—
19C22×D212—22no—
20C2×#11719917—21no—
21#13133214—22no—
22#13133214—21no—
23#13133214C2×C412yes—
24C4⋊#88322—24no—
25C4⋊#88322D414yes—
26#13133137—28no—
27C22⋊#88322—22no—
28#13133128—22no—
29#13133128—22no—
30#13133128—21no—
31C4×#88322—22no—
32C4×#88322C2×C412yes—
33C4⋊Q8×C53C2×C412yes—
34#125×C53D412yes—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53—22no—
40C22⋊C8×C53—22no—
41#97×C53—22no—
42C23×D53—41no—
43C2×Q8×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53#4112yes—
46C2×D212—22no—
47C2×D212—24no—
48C2×D212—44no—
49C2×D212—41no—
50C2×D212—42no—
51C2×D212—84no—
52C2×C8×C53—42no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53C22⋊C414yes—
56C22⋊C4×C53—22no—
57C42×C53—22no—
58C42×C53C22×C411yescharacteristic
59C42×C53C2×D411yescharacteristic
60C42×C53C2×D412yes—
61#11719917—42no—
62C2×#88322—22no—
63C2×#88322—24no—
64C23×C53#12311yescharacteristic
65C22×D53—42no—
66C22×D53—82no—
67C22×D53—84no—
68Q8×C53—82no—
69C2×C4×C53—22no—
70C2×C4×C53—24no—
71C2×C4×C53—42no—
72C2×C4×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53C4×D412yes—
74C2×C4×C53#12011yescharacteristic
75C2×C4×C53#12112yes—
76D212—44no—
77D212—84no—
78D212—84no—
79D212—82no—
80C8×C53—82no—
81#88322—42no—
82#88322—44no—
83#9424—531nomaximal, Sylow
84C22×C53—21no—
85C22×C53—22no—
86C22×C53#38611yescharacteristic
87C2×D53—82no—
88C2×D53—84no—
89C2×D53—164no—
90C4×C53—21no—
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—44no—
94C4×C53#45412yes—
95#3138—531no—
96#2987—531no—
97#1555—532no—
98#1537D5311yescharacteristic, complemented
99#1291—532no—
100C2×C53—41no—
101C2×C53#188811yescharacteristic
102C2×C53#190312yes—
103D53—162no—
104#533C2×D5311yescharacteristic
105C2×#127—531no—
106#490—1062no—
107#486—534no—
108#446—1064no—
109#433—1062no—
110#432—532no—
111#432C2×D5312yes—
112#418—1062no—
113C4×M16—531no—
114#331—1064no—
115#330—22no—
116C53#942411yescharacteristic, complemented, Sylow
117C22×D4—1061no—
118C22×D4—1062no—
119C2×M16—1061no—
120C4⋊Q8C4×D5312yes—
121#127—532no—
122#127—1062no—
123#127—1061no—
124#125C22⋊D5312yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117D21212yes—
128C2×C42C22×D5311yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—534no—
131C4⋊C8—1064no—
132#102—1068no—
133C22⋊C8—22no—
134C22⋊C8—1062no—
135#97—22no—
136#97—1062no—
137#97—1062no—
138#97—1061no—
139C4×C8—532no—
140C4×C8—1062no—
141C24—2121no—
142C2×Q8—22no—
143C2×D4—1062no—
144C2×D4—1064no—
145C2×D4—4244no—
146C2×D4—2124no—
147C2×D4—2121no—
148C2×D4—2122no—
149C22×C4C2×D21211yescharacteristic
150C22×C4#1171994712yes—
151M16—2122no—
152C2×C8—42no—
153C2×C8—1062no—
154C2×C8—1064no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4#1171992114yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D5311yescharacteristic
161C42D4×D5312yes—
162C42C2×C22⋊D5311yescharacteristic
163C23—4244no—
164C23—4242no—
165C23—2122no—
166C23#1313320111yescharacteristic
167Q8—82no—
168D4—4244no—
169D4—4244no—
170D4—4242no—
171D4—2124no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×#1171992111yescharacteristic
176C2×C4#1313317712yes—
177C2×C4#1313318711yescharacteristic
178C2×C4#1313318912yes—
179C8—82no—
180C8—2122no—
181C8—2124no—
182C22—21no—
183C22—22no—
184C22—8484no—
185C22—4242no—
186C22—4244no—
187C22#1681050011yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1681079812yes—
193C2—41no—
194C2—8482no—
195C2#2027511611yescharacteristic
196C2#2027519312yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1537C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1537D53
hypocenterC53#9424
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #20283821C2semidirect
4. #20272588C2semidirect
5. C53×#1537C2semidirect
98. #1537D53semidirect
116. C53#9424semidirect

Maximal subgroups

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

subgroupindex
#202850912
#202838212
#202734282
#202725882
C53×#15372
#942453