Finite Group Databasebeta

← #27298342

Subgroups — #27298342

Order 13568.

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276971C211yesmaximal, characteristic, complemented
2#20276796C211yesmaximal, characteristic, complemented
3#20276791C211yesmaximal, characteristic
4#20272791C211yesmaximal, characteristic, complemented
5#20272753C211yesmaximal, characteristic, complemented
6#20272751C211yesmaximal, characteristic, complemented
7C53×#2170C211yesmaximal, characteristic, complemented
8#16811063—21no—
9#16811058—21no—
10C2×#13133162—21no—
11C2×#13133162—21no—
12#16811003C2211yescharacteristic
13#16811000C2211yescharacteristic
14#16810444C2211yescharacteristic
15#16810444C2211yescharacteristic
16#16810438—21no—
17#16810435—21no—
18C53×#504—21no—
19C53×#443C2211yescharacteristic
20C53×#364C2211yescharacteristic
21C53×#350—21no—
22C4×C16×C53C2211yescharacteristic
23C2×Dic212—21no—
24C2×Dic212—21no—
25C2×D424—21no—
26C2×D424—21no—
27#13133217—21no—
28#13133214—21no—
29#13133163—41no—
30#13133162—42no—
31#13133162—41no—
32#13133162—41no—
33#13133161—41no—
34#13133160—41no—
35#13133158—41no—
36#13133157—21no—
37#13133157—21no—
38#135×C53D411yescharacteristic
39#131×C53—21no—
40M32×C53—21no—
41M32×C53—41no—
42C2×C16×C53—41no—
43C2×C16×C53D411yescharacteristic
44C2×C16×C53D411yescharacteristic
45#108×C53D411yescharacteristic
46#104×C53—21no—
47C4×C8×C53C2311yescharacteristic
48#41×C53—21no—
49C2×D212—41no—
50C2×D212—81no—
51C2×Dic106—41no—
52Q16×C53D811yescharacteristic
53SD16×C53—21no—
54D8×C53D811yescharacteristic
55M16×C53—21no—
56M16×C53—41no—
57C2×C8×C53—41no—
58C2×C8×C53C2×D411yescharacteristic
59C2×C8×C53C2×D411yescharacteristic
60C42×C53#4111yescharacteristic
61C4⋊Dic53—81no—
62Dic212—41no—
63Dic212—41no—
64Dic212—41no—
65Dic212—41no—
66D424—41no—
67D424—41no—
68D424—41no—
69D424—41no—
70C16×C53—21no—
71C16×C53—41no—
72C16×C53D811yescharacteristic
73C16×C53D811yescharacteristic
74#11719908—41no—
75#11719908—41no—
76C22×D53—81no—
77Q8×C53—21no—
78D4×C53—21no—
79D4×C53—41no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53#12111yescharacteristic
83C2×Dic53—81no—
84D212—81no—
85D212—81no—
86D212—162no—
87Dic106—81no—
88Dic106—81no—
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#45911yescharacteristic
102C4×C53#46611yescharacteristic
103Dic53—161no—
104#2248—531no—
105#2214—531no—
106#2213—531no—
107#2170D5311yescharacteristic, complemented
108#1373—531no—
109#1358—531no—
110#1357—531no—
111C2×C53—21no—
112C2×C53—81no—
113C2×C53#166711yescharacteristic
114D53—321no—
115#510—1061no—
116#509—1061no—
117C2×SD32—1061no—
118C2×SD32—1061no—
119#504—21no—
120#494—531no—
121#493—531no—
122#443C2×D5311yescharacteristic
123#364—531no—
124#364—531no—
125#364C2×D5311yescharacteristic
126#362—1061no—
127#361—1061no—
128#350—21no—
129C4×C16C2×D5311yescharacteristic
130C53#1243811yescharacteristic, complemented, Sylow
131#135D21211yescharacteristic
132C2×Q16—1061no—
133C2×Q16—1061no—
134C2×D8—1061no—
135C2×D8—1061no—
136#131—21no—
137C4⋊Q8—1061no—
138#127—1061no—
139Q32—2121no—
140SD32—2122no—
141SD32—2121no—
142SD32—2121no—
143SD32—2121no—
144SD32—2121no—
145D16—2121no—
146M32—21no—
147M32—41no—
148M32—1061no—
149M32—1061no—
150C2×C16—41no—
151C2×C16C22⋊D5311yescharacteristic
152C2×C16C22⋊D5311yescharacteristic
153#108D21211yescharacteristic
154#104—21no—
155C4×C8C22×D5311yescharacteristic
156#41—21no—
157C2×Q8—2121no—
158C2×D4—4241no—
159C2×D4—2121no—
160Q16—2121no—
161Q16—2121no—
162Q16—2121no—
163Q16—2121no—
164Q16D42411yescharacteristic
165SD16—21no—
166D8—2121no—
167D8—2121no—
168D8—2121no—
169D8—2121no—
170D8D42411yescharacteristic
171M16—21no—
172M16—41no—
173C2×C8—41no—
174C2×C8C2×D21211yescharacteristic
175C2×C8C2×C22⋊D5311yescharacteristic
176C4⋊C4—4241no—
177C42#1171994511yescharacteristic
178C16—21no—
179C16—41no—
180C16—2121no—
181C16—2121no—
182C16#1171992211yescharacteristic
183C16#1171992211yescharacteristic
184C23—4241no—
185Q8—21no—
186Q8—4241no—
187Q8—4241no—
188D4—21no—
189D4—41no—
190D4—8482no—
191D4—4241no—
192D4—4241no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—4241no—
196C2×C4#1313319111yescharacteristic
197C8—41no—
198C8C2×#1171992211yescharacteristic
199C8C2×D42411yescharacteristic
200C8#1313323911yescharacteristic
201C8#1313324211yescharacteristic
202C22—41no—
203C22—8482no—
204C22—8481no—
205C22#1681089311yescharacteristic
206C4—22no—
207C4—41no—
208C4—8481no—
209C4#1681081811yescharacteristic
210C4#1681085911yescharacteristic
211C2—21no—
212C2—81no—
213C2—16961no—
214C2#2027407011yescharacteristic
subgroupG/H
centerC2#20274070
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2170C2
socleC2×C53#1667
prime radical#20276791C2
hypercenter#2170D53
hypocenterC53#12438
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2170

Complements

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

NH ≅ G/Ntype
1. #20276971C2semidirect
2. #20276796C2semidirect
4. #20272791C2semidirect
5. #20272753C2semidirect
6. #20272751C2semidirect
7. C53×#2170C2semidirect
107. #2170D53semidirect
130. C53#12438semidirect

Maximal subgroups

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

subgroupindex
#202769712
#202767962
#202767912
#202727912
#202727532
#202727512
C53×#21702
#1243853