Finite Group Databasebeta

← #27304751

Subgroups — #27304751

Order 13568.

284 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×Q32×D53C211yesmaximal, characteristic
2C2×#16810463C211yesmaximal, characteristic, complemented
3#20276803C211yesmaximal, characteristic
4#20272897C211yesmaximal, characteristic, complemented
5#20272896C211yesmaximal, characteristic, complemented
6#20272852C211yesmaximal, characteristic
7C2×C53×Q64C211yesmaximal, characteristic, complemented
8C2×Q16×D53—21no—
9C2×Dic424C2211yescharacteristic
10C2×#13133163—21no—
11Q32×D53—22no—
12Q32×D53—22no—
13C2×C16×D53C2211yescharacteristic
14#16810861—21no—
15#16810463—22no—
16C2×#13133123C2211yescharacteristic
17#16810449C2211yescharacteristic
18#16810427—21no—
19C2×Q32×C53C2211yescharacteristic
20C2×Q32×C53C2211yescharacteristic
21C53×Q64—22no—
22C2×C32×C53C2211yescharacteristic
23C2×Q8×D53—41no—
24C2×Dic212—21no—
25C2×#11719925—41no—
26Q16×D53—24no—
27Q16×D53—42no—
28C2×C8×D53D411yescharacteristic
29#13133193—41no—
30Dic424—21no—
31Dic424D412yes—
32#13133163—24no—
33C2×#11719908D411yescharacteristic
34C16×D53—21no—
35C16×D53—21no—
36#13133151—21no—
37#13133141—41no—
38#13133123—21no—
39C2×Q16×C53—21no—
40C2×Q16×C53—21no—
41Q32×C53—21no—
42Q32×C53—21no—
43Q32×C53—21no—
44Q32×C53D412yes—
45C2×C16×C53C2311yescharacteristic
46C32×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49Q8×D53—44no—
50Q8×D53—82no—
51C2×C4×D53—81no—
52C2×C4×D53D811yescharacteristic
53C2×Dic106—41no—
54C2×Dic106—81no—
55Q16×C53—22no—
56Q16×C53—41no—
57Q16×C53—41no—
58Q16×C53—41no—
59C2×C8×C53C2×D411yescharacteristic
60#11719925—44no—
61#11719921—81no—
62C4⋊Dic53—41no—
63#11719919—81no—
64C2×#88322D811yescharacteristic
65Dic212—22no—
66Dic212—41no—
67C8×D53—21no—
68C8×D53—21no—
69C16×C53C2×D411yescharacteristic
70C16×C53C2×D411yescharacteristic
71#11719908—21no—
72C22×D53D1611yescharacteristic
73Q8×C53—42no—
74Q8×C53—81no—
75Q8×C53—81no—
76Q8×C53—81no—
77C2×C4×C53—81no—
78C2×C4×C53—81no—
79C2×C4×C53C2×D811yescharacteristic
80C2×Dic53—81no—
81C2×Dic53—81no—
82C2×Dic53D1611yescharacteristic
83C4×D53—21no—
84C4×D53—21no—
85C4×D53—84no—
86Dic106—42no—
87Dic106—84no—
88Dic106—81no—
89C8×C53#12011yescharacteristic
90C8×C53AGL(1,ℤ/8)11yescharacteristic
91#88322—21no—
92#13665—531nomaximal, Sylow
93C22×C53C2×D1611yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53Q6412yescomplemented
97C4×C53—82no—
98C4×C53—161no—
99C4×C53#44711yescharacteristic
100C4×C53#50911yescharacteristic
101Dic53—21no—
102Dic53—82no—
103Dic53—161no—
104C22×Q32—531no—
105C2×Q64—531no—
106C2×Q64D5311yescharacteristic, complemented
107#2216—531no—
108#1416—531no—
109#1416—531no—
110C22⋊C32—531no—
111C2×C53#218411yescharacteristic
112C2×C53C2×Q6411yescharacteristic
113C2×C53#226411yescharacteristic
114D53—22no—
115C22×Q16—1061no—
116C2×Q32—531no—
117C2×Q32—1061no—
118C2×Q32—1062no—
119C2×Q32—1062no—
120C2×Q32C2×D5311yescharacteristic
121C2×Q32C2×D5311yescharacteristic
122C22×C16—531no—
123#467—1061no—
124Q64—22no—
125Q64—1062no—
126C2×C32—531no—
127C2×C32C2×D5311yescharacteristic
128#366—531no—
129#358—1061no—
130C53#1366511yescharacteristic, complemented, Sylow
131C22×Q8—2121no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—1064no—
135C2×Q16—1061no—
136C2×Q16—2121no—
137C2×Q16—2122no—
138C22×C8—531no—
139C22⋊Q8—2121no—
140Q32—21no—
141Q32—21no—
142Q32—21no—
143Q32—532no—
144Q32—1064no—
145Q32—1061no—
146Q32C22⋊D5312yes—
147C2×C16—531no—
148C2×C16—1061no—
149C2×C16—1061no—
150C2×C16C22×D5311yescharacteristic
151#107—1061no—
152#103—2121no—
153C32—21no—
154C32—1061no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—4241no—
158C2×Q8—4242no—
159C2×Q8—2124no—
160C2×Q8—2121no—
161C22×C4—531no—
162C22×C4—4241no—
163Q16—22no—
164Q16—41no—
165Q16—41no—
166Q16—41no—
167Q16—1062no—
168Q16—2124no—
169Q16—2121no—
170C2×C8—531no—
171C2×C8—1061no—
172C2×C8—1061no—
173C2×C8D4×D5311yescharacteristic
174C4⋊C4—4241no—
175C4⋊C4—2121no—
176C22⋊C4—4241no—
177C16—1061no—
178C16D4×D5311yescharacteristic
179C16C2×C22⋊D5311yescharacteristic
180C23—531no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184Q8—81no—
185Q8—4244no—
186Q8—4241no—
187Q8—2122no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—531no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—4244no—
194C2×C4—4241no—
195C2×C4—4241no—
196C2×C4D8×D5311yescharacteristic
197C8—1061no—
198C8#1313318511yescharacteristic
199C8#1313324111yescharacteristic
200C22—532no—
201C22—1061no—
202C22—1061no—
203C22D16×D5311yescharacteristic
204C4—82no—
205C4—161no—
206C4—8481no—
207C4—1061no—
208C4—4242no—
209C4#1681075511yescharacteristic
210C4#1681105711yescharacteristic
211C2—1062no—
212C2#2027662511yescharacteristic
213C2Q64×D5311yescharacteristic
214C2#2027703411yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC2×C53×Q64C2
socleC22×C53C2×D16
prime radicalC22×D53D16
hypercenterC2×Q64D53
hypocenterC53#13665
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×Q64

Complements

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

NH ≅ G/Ntype
2. C2×#16810463C2semidirect
4. #20272897C2semidirect
5. #20272896C2semidirect
7. C2×C53×Q64C2semidirect
96. C2×D53Q64semidirect
106. C2×Q64D53semidirect
130. C53#13665semidirect

Maximal subgroups

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

subgroupindex
C2×Q32×D532
C2×#168104632
#202768032
#202728972
#202728962
#202728522
C2×C53×Q642
#1366553