Finite Group Databasebeta

← #27304762

Subgroups — #27304762

Order 13568.

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

normal subgroups only

This group has 288 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
1C2×#16811056C211yesmaximal, characteristic
2C2×#16810463C211yesmaximal, characteristic, complemented
3C2×#16810462C211yesmaximal, characteristic, complemented
4#20276807C211yesmaximal, characteristic, complemented
5#20272892C211yesmaximal, characteristic, complemented
6#20272852C211yesmaximal, characteristic, complemented
7C53×#1416C211yesmaximal, characteristic, complemented
8C2×#13133240—21no—
9#16811056—22no—
10#16811056—22no—
11C2×Dic424C2211yescharacteristic
12C2×#13133160—21no—
13C2×D848C2211yescharacteristic
14C2×C16×D53C2211yescharacteristic
15#16810865—21no—
16#16810463—22no—
17#16810462—22no—
18C2×#13133123C2211yescharacteristic
19#16810426—21no—
20C2×Q32×C53C2211yescharacteristic
21C2×C32×C53C2211yescharacteristic
22C53×#366C2211yescharacteristic
23C2×#11719949—41no—
24#13133240—24no—
25#13133240—42no—
26C2×Dic212—21no—
27C2×#11719924—41no—
28C2×D424—21no—
29C2×C8×D53D411yescharacteristic
30#13133197—41no—
31Dic424—21no—
32Dic424—21no—
33#13133160—24no—
34D848—21no—
35D848D412yes—
36C2×#11719908D411yescharacteristic
37C16×D53—21no—
38C16×D53—21no—
39#13133140—41no—
40#13133123—21no—
41C2×Q16×C53—21no—
42Q32×C53—21no—
43Q32×C53D412yes—
44C2×C16×C53C2311yescharacteristic
45#107×C53—21no—
46C32×C53—21no—
47C2×Q8×C53—41no—
48#11719949—44no—
49#11719949—82no—
50C2×D212—41no—
51C2×D212—81no—
52C2×C4×D53—81no—
53C2×C4×D53D811yescharacteristic
54C2×Dic106—41no—
55Q16×C53—22no—
56Q16×C53—41no—
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—41no—
59#11719924—44no—
60#11719921—81no—
61C4⋊Dic53—81no—
62C2×#88322D811yescharacteristic
63Dic212—41no—
64Dic212—41no—
65D424—22no—
66D424—41no—
67C8×D53—21no—
68C8×D53—21no—
69C16×C53C2×D411yescharacteristic
70C16×C53C2×D411yescharacteristic
71#11719908—21no—
72C22×D53—81no—
73C22×D53D1611yescharacteristic
74Q8×C53—42no—
75Q8×C53—81no—
76C2×C4×C53—81no—
77C2×C4×C53—81no—
78C2×C4×C53C2×D811yescharacteristic
79C2×Dic53—81no—
80C2×Dic53D1611yescharacteristic
81D212—42no—
82D212—84no—
83D212—81no—
84C4×D53—21no—
85C4×D53—21no—
86C4×D53—84no—
87Dic106—81no—
88Dic106—81no—
89C8×C53#12011yescharacteristic
90C8×C53AGL(1,ℤ/8)11yescharacteristic
91#88322—21no—
92#13666—531nomaximal, Sylow
93C22×C53C2×D1611yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—82no—
98C2×D53—161no—
99C2×D53—161no—
100C4×C53—82no—
101C4×C53—161no—
102C4×C53#44711yescharacteristic
103C4×C53#50911yescharacteristic
104Dic53—21no—
105Dic53—161no—
106C2×#508—531no—
107C2×Q64—531no—
108C2×SD64—531no—
109#2216—531no—
110#1416D5311yescharacteristic, complemented
111#1415—531no—
112C22⋊C32—531no—
113C2×C53#218411yescharacteristic
114C2×C53#226211yescharacteristic
115C2×C53#226411yescharacteristic
116D53—41no—
117D53—162no—
118C2×#135—1061no—
119#508—1062no—
120#508—1062no—
121C2×Q32—531no—
122C2×Q32C2×D5311yescharacteristic
123C2×SD32—1061no—
124C2×D16—531no—
125C22×C16—531no—
126#467—1061no—
127Q64—1062no—
128SD64—1062no—
129C2×C32—531no—
130C2×C32C2×D5311yescharacteristic
131#366C2×D5311yescharacteristic
132#358—1061no—
133C53#1366611yescharacteristic, complemented, Sylow
134C2×#41—2121no—
135#135—1064no—
136#135—2122no—
137C2×Q16—21no—
138C2×Q16—1061no—
139C2×SD16—2121no—
140C2×D8—1061no—
141C22×C8—531no—
142C22⋊Q8—2121no—
143Q32—21no—
144Q32—1061no—
145Q32—1061no—
146Q32D21212yes—
147SD32—1064no—
148D16—532no—
149D16—1061no—
150C2×C16—531no—
151C2×C16—1061no—
152C2×C16—1061no—
153C2×C16C22×D5311yescharacteristic
154#107—21no—
155#103—2121no—
156C32—21no—
157C32—1061no—
158#41—4242no—
159#41—2124no—
160C2×Q8—41no—
161C2×Q8—2121no—
162C2×D4—4241no—
163C2×D4—2121no—
164C22×C4—531no—
165C22×C4—4241no—
166Q16—22no—
167Q16—41no—
168Q16—2121no—
169Q16—2121no—
170SD16—2124no—
171D8—1062no—
172D8—2121no—
173C2×C8—531no—
174C2×C8—1061no—
175C2×C8—1061no—
176C2×C8D4×D5311yescharacteristic
177C4⋊C4—41no—
178C4⋊C4—4241no—
179C22⋊C4—4241no—
180C16—1061no—
181C16C2×D21211yescharacteristic
182C16D4×D5311yescharacteristic
183C23—531no—
184C23—4241no—
185Q8—42no—
186Q8—81no—
187Q8—4241no—
188Q8—4241no—
189D4—4244no—
190D4—4241no—
191D4—2122no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—531no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—4244no—
198C2×C4—4241no—
199C2×C4D8×D5311yescharacteristic
200C8—1061no—
201C8#1313318711yescharacteristic
202C8#1313324411yescharacteristic
203C22—8481no—
204C22—8481no—
205C22—1061no—
206C22—1061no—
207C22—1061no—
208C22—4242no—
209C22D16×D5311yescharacteristic
210C4—82no—
211C4—161no—
212C4—8481no—
213C4—1061no—
214C4#1681075811yescharacteristic
215C4#1681106011yescharacteristic
216C2—8482no—
217C2—2121no—
218C2#2027662811yescharacteristic
219C2#2027702311yescharacteristic
220C2#2027703411yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1416C2
socleC22×C53C2×D16
prime radicalC2×#16811056C2
hypercenter#1416D53
hypocenterC53#13666
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1416

Complements

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

NH ≅ G/Ntype
2. C2×#16810463C2semidirect
3. C2×#16810462C2semidirect
4. #20276807C2semidirect
5. #20272892C2semidirect
6. #20272852C2semidirect
7. C53×#1416C2semidirect
110. #1416D53semidirect
133. C53#13666semidirect

Maximal subgroups

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

subgroupindex
C2×#168110562
C2×#168104632
C2×#168104622
#202768072
#202728922
#202728522
C53×#14162
#1366653