Finite Group Databasebeta

← #27295607

Subgroups — #27295607

Order 13568.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#20285365C211yesmaximal, characteristic
2#20276531C211yesmaximal, characteristic, complemented
3#20272777C212yesmaximal, complemented
4#20272751C212yesmaximal, complemented
5C53×#2153C211yesmaximal, characteristic, complemented
6#16811228C412yes—
7#16811025C2211yescharacteristic
8#16811000C2211yescharacteristic
9#16810988C412yes—
10#16810876—21no—
11#16810859—21no—
12#16810736—21no—
13C4×D424—21no—
14#16810444C2212yes—
15#16810405—21no—
16#16810401—21no—
17C2×M32×C53—21no—
18C53×#405C2211yescharacteristic
19C53×#364—21no—
20C53×#364C2212yes—
21C2×D424—21no—
22C2×D424—21no—
23#13133214C2×C412yes—
24#13133209C2×C412yes—
25#13133202—42no—
26#13133191—22no—
27#13133190—42no—
28C4×D212—22no—
29#13133157—22no—
30#13133151—21no—
31#13133149—21no—
32#13133139—22no—
33#13133139—22no—
34C2×M16×C53D411yescharacteristic
35C22×C8×C53D411yescharacteristic
36#117×C53C2×C411yescharacteristic
37M32×C53—22no—
38M32×C53—41no—
39C2×C16×C53—41no—
40#97×C53C2×C411yescharacteristic
41#97×C53D411yescharacteristic
42C4×C8×C53D411yescharacteristic
43C4×C8×C53C2311yescharacteristic
44C22×C4×C53C22⋊C411yescharacteristic
45C2×C22⋊D53—42no—
46C2×D212—22no—
47C2×D212—42no—
48C2×C4×D53—42no—
49M16×C53—22no—
50C2×C8×C53—21no—
51C2×C8×C53—21no—
52C2×C8×C53C22⋊C411yescharacteristic
53C2×C8×C53C22⋊C411yescharacteristic
54C2×C8×C53C2×D411yescharacteristic
55C2×C8×C53C2×D411yescharacteristic
56C4⋊C4×C53—21no—
57C22⋊C4×C53—21no—
58C42×C53C22⋊C411yescharacteristic
59C42×C53C22×C411yescharacteristic
60#11719921—42no—
61#11719921—42no—
62C4⋊Dic53—22no—
63#11719919—42no—
64D424—42no—
65D424—41no—
66D424—41no—
67C16×C53—42no—
68#11719908—42no—
69C23×C53#10011yescharacteristic
70C22×D53—42no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53#10011yescharacteristic
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C22⋊D53—162no—
78C2×Dic53—42no—
79D212—42no—
80D212—42no—
81D212—162no—
82C4×D53—162no—
83C8×C53—21no—
84C8×C53—21no—
85C8×C53#13512yes—
86C8×C53AGL(1,ℤ/8)11yescharacteristic
87C8×C53#13711yescharacteristic
88#11989—531nomaximal, Sylow
89C22×C53—21no—
90C22×C53—41no—
91C22×C53C2×#10011yescharacteristic
92C2×D53—82no—
93C2×D53—82no—
94C2×D53—82no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53—41no—
98C4×C53#41611yescharacteristic
99C4×C53#41711yescharacteristic
100Dic53—82no—
101#3176—531no—
102#2166—531no—
103#2153D5311yescharacteristic, complemented
104#1366—532no—
105#1357—532no—
106C2×C53—21no—
107C2×C53—41no—
108C2×C53#155111yescharacteristic
109D53—162no—
110#540—532no—
111C2×M32—21no—
112#499—531no—
113#493—531no—
114#490—532no—
115#469—1061no—
116#466—1061no—
117#442—1061no—
118C4×D8—1061no—
119#405C2×D5311yescharacteristic
120#364—21no—
121#364—532no—
122#364C2×D5312yes—
123#349—1061no—
124#347—1061no—
125C53#1198911yescharacteristic, complemented, Sylow
126C2×D8—1061no—
127C2×D8—1061no—
128C2×M16D21211yescharacteristic
129C22×C8C22⋊D5311yescharacteristic
130#127—532no—
131#125—532no—
132#123—2122no—
133#121—1062no—
134#121—2122no—
135C4×D4—1062no—
136#117C4×D5311yescharacteristic
137M32—22no—
138M32—41no—
139M32—1062no—
140C2×C16—41no—
141#107—1061no—
142#106—1061no—
143#102—1062no—
144#102—1062no—
145#97C4×D5311yescharacteristic
146#97D21211yescharacteristic
147C4×C8C22⋊D5311yescharacteristic
148C4×C8C22×D5311yescharacteristic
149C2×D4—1062no—
150C2×D4—2122no—
151C2×D4—2122no—
152C22×C4—2122no—
153C22×C4#1171992111yescharacteristic
154D8—2122no—
155D8—2121no—
156D8—2121no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8#1171992111yescharacteristic
161C2×C8#1171992111yescharacteristic
162C2×C8C2×D21211yescharacteristic
163C2×C8C2×C22⋊D5311yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—1062no—
166C4⋊C4—2122no—
167C22⋊C4—21no—
168C22⋊C4—2122no—
169C22⋊C4—2122no—
170C42#1171992111yescharacteristic
171C42C2×C4×D5311yescharacteristic
172C16—42no—
173C16—2122no—
174C23—2122no—
175C23#1313313311yescharacteristic
176D4—8482no—
177D4—8482no—
178D4—2122no—
179D4—2122no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—8482no—
185C2×C4—2122no—
186C2×C4#1313313311yescharacteristic
187C2×C4C2×#1171992111yescharacteristic
188C8—21no—
189C8—21no—
190C8#1313323912yes—
191C8#1313324311yescharacteristic
192C8#1313324811yescharacteristic
193C22—21no—
194C22—41no—
195C22—4242no—
196C22—4242no—
197C22—4242no—
198C22C2×#1313313311yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—4242no—
203C4#1681062011yescharacteristic
204C4#1681062211yescharacteristic
205C2—21no—
206C2—41no—
207C2—8482no—
208C2#2027340811yescharacteristic
subgroupG/H
centerC2#20273408
commutatorC42×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#2153C2
socleC2×C53#1551
prime radical#20285365C2
cosocleC4×C8×C53C23
hypercenter#2153D53
hypocenterC53#11989
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #20276531C2semidirect
3. #20272777C2semidirect
4. #20272751C2semidirect
5. C53×#2153C2semidirect
103. #2153D53semidirect
125. C53#11989semidirect

Maximal subgroups

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

subgroupindex
#202853652
#202765312
#202727772
#202727512
C53×#21532
#1198953