Finite Group Databasebeta

← #27298330

Subgroups — #27298330

Order 13568.

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

normal subgroups only

This group has 234 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#20276964C211yesmaximal, characteristic, complemented
2#20276791C211yesmaximal, characteristic, complemented
3#20276552C211yesmaximal, characteristic, complemented
4#20272790C211yesmaximal, characteristic, complemented
5#20272752C211yesmaximal, characteristic, complemented
6#20272751C211yesmaximal, characteristic, complemented
7C53×#2213C211yesmaximal, characteristic, complemented
8#16811058—21no—
9C2×#13133158—21no—
10C2×#13133158—21no—
11#16811040—21no—
12#16811000C2211yescharacteristic
13#16810444C2211yescharacteristic
14#16810444C2211yescharacteristic
15#16810435—21no—
16#16810407—21no—
17C4×#11719908C2211yescharacteristic
18C53×#509—21no—
19C53×#493C2211yescharacteristic
20C53×#443C2211yescharacteristic
21C53×#364C2211yescharacteristic
22C53×#361—21no—
23C2×D424—21no—
24C2×D424—21no—
25#13133214—21no—
26#13133160—41no—
27#13133158—42no—
28#13133158—41no—
29#13133158—41no—
30#13133158—41no—
31#13133157—21no—
32#13133157—21no—
33#13133157—41no—
34C2×#11719908—41no—
35C2×#11719908D411yescharacteristic
36C2×#11719908D411yescharacteristic
37#135×C53D411yescharacteristic
38C2×D8×C53—21no—
39C2×D8×C53—21no—
40#131×C53—21no—
41#127×C53—21no—
42SD32×C53—41no—
43D16×C53—41no—
44M32×C53—21no—
45#108×C53D411yescharacteristic
46#104×C53—21no—
47C4×C8×C53C2311yescharacteristic
48#41×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—81no—
51C2×D212—41no—
52C2×D212—81no—
53Q16×C53D811yescharacteristic
54SD16×C53—21no—
55D8×C53—41no—
56D8×C53—41no—
57D8×C53—41no—
58D8×C53—41no—
59D8×C53D811yescharacteristic
60M16×C53—21no—
61M16×C53—41no—
62C2×C8×C53—41no—
63C2×C8×C53C2×D411yescharacteristic
64C2×C8×C53C2×D411yescharacteristic
65C42×C53#4111yescharacteristic
66D424—41no—
67D424—41no—
68D424—41no—
69D424—41no—
70C16×C53—41no—
71#11719908—21no—
72#11719908—41no—
73#11719908—41no—
74#11719908D811yescharacteristic
75#11719908D811yescharacteristic
76C23×C53—81no—
77C22×D53—81no—
78Q8×C53—21no—
79D4×C53—21no—
80D4×C53—41no—
81D4×C53—81no—
82D4×C53—81no—
83D4×C53—162no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53#12111yescharacteristic
87D212—81no—
88D212—81no—
89D212—162no—
90C8×C53—41no—
91C8×C53C2×D811yescharacteristic
92C8×C53C2×D811yescharacteristic
93C8×C53#13511yescharacteristic
94C8×C53AGL(1,ℤ/8)11yescharacteristic
95#12436—531nomaximal, Sylow
96C22×C53—41no—
97C22×C53—162no—
98C22×C53—161no—
99C22×C53#47211yescharacteristic
100C2×D53—162no—
101C2×D53—161no—
102C4×C53—22no—
103C4×C53—41no—
104C4×C53#45911yescharacteristic
105C4×C53#46611yescharacteristic
106#2246—531no—
107#2213—531no—
108#2213D5311yescharacteristic, complemented
109#2170—531no—
110#1373—531no—
111#1357—531no—
112#1357—531no—
113C2×C53—21no—
114C2×C53—81no—
115C2×C53—321no—
116C2×C53#166711yescharacteristic
117D53—321no—
118#509—21no—
119#509—1061no—
120C2×D16—1061no—
121C2×D16—1061no—
122#504—1061no—
123#493—531no—
124#493C2×D5311yescharacteristic
125#443C2×D5311yescharacteristic
126#364—531no—
127#364—531no—
128#364C2×D5311yescharacteristic
129#361—21no—
130#361—1061no—
131#350—1061no—
132C4×C16—531no—
133C53#1243611yescharacteristic, complemented, Sylow
134#135C22⋊D5311yescharacteristic
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—1061no—
138C2×D8—1061no—
139#131—21no—
140#127—21no—
141#127—1061no—
142SD32—41no—
143SD32—2121no—
144D16—41no—
145D16—2121no—
146D16—2121no—
147D16—2122no—
148D16—2121no—
149M32—21no—
150M32—1061no—
151M32—1061no—
152M32—2121no—
153C2×C16—531no—
154C2×C16—531no—
155C2×C16—2121no—
156#108C22⋊D5311yescharacteristic
157#104—21no—
158C4×C8C22×D5311yescharacteristic
159#41—21no—
160C2×D4—41no—
161C2×D4—81no—
162C2×D4—4241no—
163C2×D4—2121no—
164Q16#1171992211yescharacteristic
165SD16—21no—
166D8—41no—
167D8—41no—
168D8—41no—
169D8—41no—
170D8—2121no—
171D8—2121no—
172D8—2121no—
173D8—2121no—
174D8#1171992211yescharacteristic
175M16—21no—
176M16—41no—
177C2×C8—41no—
178C2×C8D4×D5311yescharacteristic
179C2×C8C2×C22⋊D5311yescharacteristic
180C42#1171994711yescharacteristic
181C16—41no—
182C16—531no—
183C16—531no—
184C16—1061no—
185C16—2121no—
186C16—2121no—
187C23—81no—
188C23—4241no—
189Q8—21no—
190D4—21no—
191D4—41no—
192D4—81no—
193D4—81no—
194D4—162no—
195D4—8482no—
196D4—4241no—
197D4—4241no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4#1313318911yescharacteristic
201C8—41no—
202C8D8×D5311yescharacteristic
203C8C2×#1171992211yescharacteristic
204C8#1313323711yescharacteristic
205C8#1313324111yescharacteristic
206C22—41no—
207C22—162no—
208C22—161no—
209C22—8482no—
210C22—8481no—
211C22#1681089111yescharacteristic
212C4—22no—
213C4—41no—
214C4#1681081611yescharacteristic
215C4#1681085711yescharacteristic
216C2—21no—
217C2—81no—
218C2—321no—
219C2—16961no—
220C2#2027406811yescharacteristic
subgroupG/H
centerC2#20274068
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2213C2
socleC2×C53#1667
hypercenter#2213D53
hypocenterC53#12436
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #20276964C2semidirect
2. #20276791C2semidirect
3. #20276552C2semidirect
4. #20272790C2semidirect
5. #20272752C2semidirect
6. #20272751C2semidirect
7. C53×#2213C2semidirect
108. #2213D53semidirect
133. C53#12436semidirect

Maximal subgroups

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

subgroupindex
#202769642
#202767912
#202765522
#202727902
#202727522
#202727512
C53×#22132
#1243653