Finite Group Databasebeta

← #27298344

Subgroups — #27298344

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#20276972C211yesmaximal, characteristic, complemented
2#20276791C211yesmaximal, characteristic
3#20276552C211yesmaximal, characteristic, complemented
4#20272790C211yesmaximal, characteristic, complemented
5#20272754C211yesmaximal, characteristic, complemented
6#20272751C211yesmaximal, characteristic, complemented
7C53×#2214C211yesmaximal, characteristic, complemented
8#16811058—21no—
9C2×#13133160—21no—
10C2×#13133160—21no—
11#16811040—21no—
12#16811000C2211yescharacteristic
13#16810444C2211yescharacteristic
14#16810444C2211yescharacteristic
15#16810435—21no—
16#16810407—21no—
17C4×#11719908C2211yescharacteristic
18C53×#510—21no—
19C53×#494C2211yescharacteristic
20C53×#443C2211yescharacteristic
21C53×#364C2211yescharacteristic
22C53×#362—21no—
23C2×D424—21no—
24C2×D424—21no—
25#13133214—21no—
26#13133160—42no—
27#13133160—41no—
28#13133160—41no—
29#13133160—41no—
30#13133158—41no—
31#13133157—21no—
32#13133157—21no—
33#13133157—41no—
34C2×#11719908—41no—
35C2×#11719908D411yescharacteristic
36C2×#11719908D411yescharacteristic
37#135×C53D411yescharacteristic
38C2×Q16×C53—21no—
39C2×Q16×C53—21no—
40#131×C53—21no—
41C4⋊Q8×C53—21no—
42Q32×C53—41no—
43SD32×C53—41no—
44M32×C53—21no—
45#108×C53D411yescharacteristic
46#104×C53—21no—
47C4×C8×C53C2311yescharacteristic
48#41×C53—21no—
49C2×Q8×C53—41no—
50C2×D212—41no—
51C2×D212—81no—
52Q16×C53—41no—
53Q16×C53—41no—
54Q16×C53—41no—
55Q16×C53—41no—
56Q16×C53D811yescharacteristic
57SD16×C53—21no—
58D8×C53D811yescharacteristic
59M16×C53—21no—
60M16×C53—41no—
61C2×C8×C53—41no—
62C2×C8×C53C2×D411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—81no—
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
76C22×D53—81no—
77Q8×C53—21no—
78Q8×C53—81no—
79Q8×C53—81no—
80D4×C53—21no—
81D4×C53—41no—
82C2×C4×C53—21no—
83C2×C4×C53—41no—
84C2×C4×C53—81no—
85C2×C4×C53#12111yescharacteristic
86D212—81no—
87D212—81no—
88D212—162no—
89C8×C53—41no—
90C8×C53C2×D811yescharacteristic
91C8×C53C2×D811yescharacteristic
92C8×C53#13511yescharacteristic
93C8×C53AGL(1,ℤ/8)11yescharacteristic
94#12438—531nomaximal, Sylow
95C22×C53—41no—
96C22×C53#47211yescharacteristic
97C2×D53—162no—
98C2×D53—161no—
99C4×C53—22no—
100C4×C53—41no—
101C4×C53—161no—
102C4×C53#45911yescharacteristic
103C4×C53#46611yescharacteristic
104#2248—531no—
105#2214D5311yescharacteristic, complemented
106#2213—531no—
107#2170—531no—
108#1373—531no—
109#1358—531no—
110#1357—531no—
111C2×C53—21no—
112C2×C53—81no—
113C2×C53#166711yescharacteristic
114D53—321no—
115#510—21no—
116#509—1061no—
117C2×SD32—1061no—
118C2×SD32—1061no—
119#504—1061no—
120#494C2×D5311yescharacteristic
121#493—531no—
122#443C2×D5311yescharacteristic
123#364—531no—
124#364—531no—
125#364C2×D5311yescharacteristic
126#362—21no—
127#361—1061no—
128#350—1061no—
129C4×C16—531no—
130C53#1243811yescharacteristic, complemented, Sylow
131#135C22⋊D5311yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1061no—
135C2×D8—1061no—
136#131—21no—
137C4⋊Q8—21no—
138#127—1061no—
139Q32—41no—
140SD32—41no—
141SD32—2121no—
142SD32—2121no—
143SD32—2122no—
144SD32—2121no—
145D16—2121no—
146M32—21no—
147M32—1061no—
148M32—1061no—
149M32—2121no—
150C2×C16—531no—
151C2×C16—531no—
152C2×C16—2121no—
153#108C22⋊D5311yescharacteristic
154#104—21no—
155C4×C8C22×D5311yescharacteristic
156#41—21no—
157C2×Q8—41no—
158C2×D4—4241no—
159C2×D4—2121no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16#1171992211yescharacteristic
165SD16—21no—
166D8—2121no—
167D8—2121no—
168D8—2121no—
169D8—2121no—
170D8#1171992211yescharacteristic
171M16—21no—
172M16—41no—
173C2×C8—41no—
174C2×C8D4×D5311yescharacteristic
175C2×C8C2×C22⋊D5311yescharacteristic
176C4⋊C4—81no—
177C42#1171994711yescharacteristic
178C16—41no—
179C16—531no—
180C16—531no—
181C16—1061no—
182C16—2121no—
183C16—2121no—
184C23—4241no—
185Q8—21no—
186Q8—81no—
187Q8—81no—
188D4—21no—
189D4—41no—
190D4—8482no—
191D4—4241no—
192D4—4241no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4#1313318911yescharacteristic
197C8—41no—
198C8D8×D5311yescharacteristic
199C8C2×#1171992211yescharacteristic
200C8#1313323711yescharacteristic
201C8#1313324111yescharacteristic
202C22—41no—
203C22—8482no—
204C22—8481no—
205C22#1681089111yescharacteristic
206C4—22no—
207C4—41no—
208C4—161no—
209C4#1681081611yescharacteristic
210C4#1681085711yescharacteristic
211C2—21no—
212C2—81no—
213C2—16961no—
214C2#2027406811yescharacteristic
subgroupG/H
centerC2#20274068
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2214C2
socleC2×C53#1667
prime radical#20276791C2
hypercenter#2214D53
hypocenterC53#12438
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #20276972C2semidirect
3. #20276552C2semidirect
4. #20272790C2semidirect
5. #20272754C2semidirect
6. #20272751C2semidirect
7. C53×#2214C2semidirect
105. #2214D53semidirect
130. C53#12438semidirect

Maximal subgroups

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

subgroupindex
#202769722
#202767912
#202765522
#202727902
#202727542
#202727512
C53×#22142
#1243853