Finite Group Databasebeta

← #27278500

Subgroups — #27278500

Order 13568.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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#20275697C211yesmaximal, characteristic
2#20273483C212yesmaximal
3#20273310C211yesmaximal, characteristic
4#20272607C212yesmaximal
5C53×#2987C211yesmaximal, characteristic
6#16810647C2211yescharacteristic
7C4×#11719917C2212yes—
8#16810469C2211yescharacteristic
9#16810382—22no—
10#16810375—22no—
11#16810373—22no—
12#16810365C411yescharacteristic, complemented
13#16810365C412yescomplemented
14#16810365C411yescharacteristic, complemented
15C53×#533C2211yescharacteristic
16C53×#446—22no—
17C53×#446—22no—
18C53×#433—22no—
19C53×#432C2212yes—
20C2×#11719917—21no—
21C2×#11719917—22no—
22C2×C4×Dic53—21no—
23C4⋊#88322—22no—
24#13133128C2×C412yes—
25#13133128C2×C412yes—
26C4×#88322—22no—
27#13133124—22no—
28C4⋊Q8×C53Q812yes—
29#125×C53C2×C411yescharacteristic
30#125×C53C2×C411yescharacteristic
31C22⋊Q8×C53—22no—
32C4×Q8×C53—22no—
33#117×C53D412yes—
34C2×C42×C53C2311yescharacteristic
35C4⋊C8×C53—22no—
36C4⋊C8×C53—22no—
37C4⋊C8×C53D412yes—
38C4⋊C8×C53D412yes—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41#97×C53—22no—
42C4×C8×C53—22no—
43C2×Q8×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53#4112yes—
46C22×Dic53—41no—
47C2×C8×C53—22no—
48C2×C8×C53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53C4⋊C412yes—
53C4⋊C4×C53C4⋊C412yes—
54C22⋊C4×C53—22no—
55C42×C53—22no—
56C42×C53C22×C411yescharacteristic
57C42×C53C2×D412yes—
58C42×C53C2×Q811yescharacteristic
59C4×Dic53—22no—
60C4×Dic53—41no—
61#11719917—44no—
62#11719917—42no—
63C2×#88322—22no—
64C2×#88322—22no—
65C2×#88322—41no—
66C23×C53#12311yescharacteristic
67Q8×C53—82no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53C2×C4⋊C411yescharacteristic
74C2×C4×C53C4×D412yes—
75C2×C4×C53#12011yescharacteristic
76C2×C4×C53C22⋊Q812yes—
77C2×Dic53—42no—
78C2×Dic53—82no—
79C8×C53—42no—
80C8×C53—42no—
81#88322—24no—
82#88322—42no—
83#88322—81no—
84#8919—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—22no—
87C22×C53#38511yescharacteristic
88C4×C53—21no—
89C4×C53—22no—
90C4×C53—22no—
91C4×C53—42no—
92C4×C53—42no—
93C4×C53#45411yescharacteristic
94C4×C53#45611yescharacteristic
95Dic53—82no—
96#2987D5311yescharacteristic
97#1995—531no—
98#1569—532no—
99#1526—531no—
100#1300—532no—
101C2×C53—41no—
102C2×C53#185711yescharacteristic
103C2×C53#186112yes—
104#533C2×D5311yescharacteristic
105#446—22no—
106#446—22no—
107#433—22no—
108#432C2×D5312yes—
109#423—531no—
110C4×M16—532no—
111#376—531no—
112#339—1062no—
113#335—1062no—
114#334—1062no—
115#330—531no—
116#330—532no—
117#330—531no—
118C53#891911yescharacteristic, complemented, Sylow
119C2×M16—1062no—
120C2×M16—1061no—
121C4⋊Q8Dic10612yes—
122#125C4×D5311yescharacteristic
123#125C4×D5311yescharacteristic
124C22⋊Q8—22no—
125C4×Q8—22no—
126#117C22⋊D5312yes—
127C2×C42—1061no—
128C2×C42C22×D5311yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—1062no—
132C4⋊C8C22⋊D5312yes—
133C4⋊C8C22⋊D5312yes—
134C22⋊C8—22no—
135#97—22no—
136#97—22no—
137#97—532no—
138#97—532no—
139C4×C8—22no—
140C4×C8—1062no—
141#95—1062no—
142C2×Q8—22no—
143C22×C4—2121no—
144C22×C4#1171994512yes—
145C22×C4C2×C22⋊D5311yescharacteristic
146M16—2124no—
147M16—2122no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—1062no—
151C2×C8—1062no—
152C2×C8—2121no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4#1171991912yes—
157C4⋊C4#1171991912yes—
158C22⋊C4—22no—
159C42—22no—
160C42—1062no—
161C42—2121no—
162C42C2×Dic10611yescharacteristic
163C42C2×C4×D5311yescharacteristic
164C42C2×C22⋊D5312yes—
165C23#1313320011yescharacteristic
166Q8—82no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—4242no—
173C2×C4—2122no—
174C2×C4C2×#1171991911yescharacteristic
175C2×C4C4×C22⋊D5312yes—
176C2×C4#1313318611yescharacteristic
177C2×C4#1313319612yes—
178C8—42no—
179C8—42no—
180C8—1064no—
181C8—4241no—
182C8—2122no—
183C22—22no—
184C22—21no—
185C22#1681049611yescharacteristic
186C4—22no—
187C4—22no—
188C4—21no—
189C4—42no—
190C4—42no—
191C4—4242no—
192C4#1681079611yescharacteristic
193C4#1681080611yescharacteristic
194C2—41no—
195C2#2027493011yescharacteristic
196C2#2027495512yes—
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2987C2
socleC22×C53#385
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#2987D53
hypocenterC53#8919
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
12. #16810365C4semidirect
13. #16810365C4semidirect
14. #16810365C4semidirect
118. C53#8919semidirect

Maximal subgroups

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

subgroupindex
#202756972
#202734832
#202733102
#202726072
C53×#29872
#891953