Finite Group Databasebeta

← #27298339

Subgroups — #27298339

Order 13568.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276793C211yesmaximal, characteristic, complemented
2#20276788C211yesmaximal, characteristic
3#20276553C211yesmaximal, characteristic, complemented
4#20272791C211yesmaximal, characteristic, complemented
5#20272754C211yesmaximal, characteristic, complemented
6#20272752C211yesmaximal, characteristic, complemented
7C53×#2248C211yesmaximal, characteristic, complemented
8#16811064—21no—
9#16811059—21no—
10#16811039—21no—
11#16810741C2211yescharacteristic
12#16810444C2211yescharacteristic
13#16810444C2211yescharacteristic
14#16810444C2211yescharacteristic
15#16810439—21no—
16#16810436—21no—
17#16810406—21no—
18C2×SD32×C53—21no—
19C2×SD32×C53—21no—
20C53×#494C2211yescharacteristic
21C53×#493C2211yescharacteristic
22C4×C16×C53C2211yescharacteristic
23#13133239D411yescharacteristic
24#13133223—21no—
25#13133163—41no—
26#13133161—41no—
27#13133160—41no—
28#13133158—41no—
29#13133157—21no—
30#13133157—21no—
31#13133157—21no—
32#13133156—41no—
33C16×D53—41no—
34#13133153D411yescharacteristic
35#13133145—21no—
36C2×Q16×C53—21no—
37C2×Q16×C53—21no—
38C2×D8×C53—21no—
39C2×D8×C53—21no—
40C4⋊Q8×C53—21no—
41#127×C53—21no—
42SD32×C53—42no—
43SD32×C53—41no—
44SD32×C53—41no—
45C2×C16×C53D411yescharacteristic
46C2×C16×C53D411yescharacteristic
47C4×C8×C53C2311yescharacteristic
48C2×Q8×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—81no—
51#11719945—21no—
52Q16×C53—41no—
53Q16×C53—41no—
54Q16×C53—41no—
55Q16×C53—41no—
56D8×C53—41no—
57D8×C53—41no—
58D8×C53—41no—
59D8×C53—41no—
60C2×C8×C53C2×D411yescharacteristic
61C2×C8×C53C2×D411yescharacteristic
62C4⋊C4×C53—81no—
63C42×C53#4111yescharacteristic
64#11719917—21no—
65Dic212D811yescharacteristic
66D424D811yescharacteristic
67#11719913—21no—
68#11719912—41no—
69C8×D53—41no—
70C16×C53—21no—
71C16×C53D811yescharacteristic
72C16×C53D811yescharacteristic
73#11719908—41no—
74#11719908—41no—
75#11719908—41no—
76C23×C53—81no—
77Q8×C53—81no—
78Q8×C53—81no—
79D4×C53—81no—
80D4×C53—81no—
81D4×C53—162no—
82C2×C4×C53—21no—
83C2×C4×C53—81no—
84C2×C4×C53#12111yescharacteristic
85C22⋊D53—41no—
86D212—21no—
87C4×D53—41no—
88Dic106—21no—
89C8×C53C2×D811yescharacteristic
90C8×C53C2×D811yescharacteristic
91C8×C53#13511yescharacteristic
92C8×C53AGL(1,ℤ/8)11yescharacteristic
93#88322—41no—
94#12438—531nomaximal, Sylow
95C22×C53—162no—
96C22×C53—161no—
97C22×C53#47211yescharacteristic
98C2×D53—41no—
99C4×C53—22no—
100C4×C53—161no—
101C4×C53#45911yescharacteristic
102C4×C53#46611yescharacteristic
103Dic53—41no—
104#2248D5311yescharacteristic, complemented
105#2214—531no—
106#2213—531no—
107#2170—531no—
108#1373—531no—
109#1358—531no—
110#1357—531no—
111C2×C53—21no—
112C2×C53—321no—
113C2×C53#166711yescharacteristic
114D53—81no—
115#510—1061no—
116#509—1061no—
117C2×SD32—21no—
118C2×SD32—21no—
119#504—1061no—
120#494C2×D5311yescharacteristic
121#493C2×D5311yescharacteristic
122#443—531no—
123#364—531no—
124#364—531no—
125#364—531no—
126#362—1061no—
127#361—1061no—
128#350—1061no—
129C4×C16C2×D5311yescharacteristic
130C53#1243811yescharacteristic, complemented, Sylow
131#135—531no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—21no—
135C2×D8—21no—
136#131—1061no—
137C4⋊Q8—21no—
138#127—21no—
139Q32—2121no—
140SD32—42no—
141SD32—41no—
142SD32—41no—
143SD32—2121no—
144SD32—2121no—
145D16—2121no—
146M32—1061no—
147M32—1061no—
148M32—1061no—
149M32—2121no—
150C2×C16—2121no—
151C2×C16C22⋊D5311yescharacteristic
152C2×C16C22⋊D5311yescharacteristic
153#108—531no—
154#104—1061no—
155C4×C8C22×D5311yescharacteristic
156#41—1061no—
157C2×Q8—41no—
158C2×D4—41no—
159C2×D4—81no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16—531no—
165SD16—1061no—
166D8—41no—
167D8—41no—
168D8—41no—
169D8—41no—
170D8—531no—
171M16—1061no—
172M16—2121no—
173C2×C8—2121no—
174C2×C8D4×D5311yescharacteristic
175C2×C8C2×C22⋊D5311yescharacteristic
176C4⋊C4—81no—
177C42#1171994711yescharacteristic
178C16—21no—
179C16—2121no—
180C16—2121no—
181C16—2121no—
182C16#1171992211yescharacteristic
183C16#1171992211yescharacteristic
184C23—81no—
185Q8—81no—
186Q8—81no—
187Q8—1061no—
188D4—81no—
189D4—81no—
190D4—162no—
191D4—1061no—
192D4—2121no—
193C2×C4—21no—
194C2×C4—81no—
195C2×C4—2121no—
196C2×C4#1313318811yescharacteristic
197C8—2121no—
198C8D8×D5311yescharacteristic
199C8C2×#1171992211yescharacteristic
200C8#1313323611yescharacteristic
201C8#1313324311yescharacteristic
202C22—162no—
203C22—161no—
204C22—2121no—
205C22#1681089011yescharacteristic
206C4—22no—
207C4—161no—
208C4—2121no—
209C4#1681081511yescharacteristic
210C4#1681085611yescharacteristic
211C2—21no—
212C2—321no—
213C2—4241no—
214C2#2027406711yescharacteristic
subgroupG/H
centerC2#20274067
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2248C2
socleC2×C53#1667
prime radical#20276788C2
hypercenter#2248D53
hypocenterC53#12438
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #20276793C2semidirect
3. #20276553C2semidirect
4. #20272791C2semidirect
5. #20272754C2semidirect
6. #20272752C2semidirect
7. C53×#2248C2semidirect
104. #2248D53semidirect
130. C53#12438semidirect

Maximal subgroups

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

subgroupindex
#202767932
#202767882
#202765532
#202727912
#202727542
#202727522
C53×#22482
#1243853