Finite Group Databasebeta

← #27304754

Subgroups — #27304754

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×Dic848C211yesmaximal, characteristic, complemented
3C2×#16810463C211yesmaximal, characteristic, complemented
4#20276807C211yesmaximal, characteristic
5#20272895C211yesmaximal, characteristic, complemented
6#20272852C211yesmaximal, characteristic
7C53×#1416C211yesmaximal, characteristic, complemented
8C2×Q16×D53—21no—
9C2×Dic424C2211yescharacteristic
10C2×Dic424C2211yescharacteristic
11C2×#13133163—21no—
12Q32×D53—22no—
13Q32×D53—22no—
14C2×C16×D53C2211yescharacteristic
15#16810865—21no—
16Dic848—22no—
17#16810463—22no—
18C2×#13133123C2211yescharacteristic
19#16810426—21no—
20C2×Q32×C53C2211yescharacteristic
21C2×C32×C53C2211yescharacteristic
22C53×#366C2211yescharacteristic
23C2×Q8×D53—41no—
24C2×Dic212—21no—
25C2×Dic212—21no—
26C2×#11719925—41no—
27Q16×D53—24no—
28Q16×D53—42no—
29C2×C8×D53D411yescharacteristic
30#13133197—41no—
31Dic424—21no—
32Dic424—21no—
33Dic424—21no—
34Dic424D412yes—
35#13133163—24no—
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—
48Q8×D53—44no—
49Q8×D53—82no—
50C2×C4×D53—81no—
51C2×C4×D53D811yescharacteristic
52C2×Dic106—41no—
53C2×Dic106—41no—
54C2×Dic106—81no—
55Q16×C53—22no—
56Q16×C53—41no—
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—41no—
59#11719925—44no—
60#11719921—81no—
61C4⋊Dic53—81no—
62C2×#88322D811yescharacteristic
63Dic212—22no—
64Dic212—41no—
65Dic212—41no—
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—
75C2×C4×C53—81no—
76C2×C4×C53—81no—
77C2×C4×C53C2×D811yescharacteristic
78C2×Dic53—81no—
79C2×Dic53—81no—
80C2×Dic53D1611yescharacteristic
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—84no—
84Dic106—42no—
85Dic106—84no—
86Dic106—81no—
87Dic106—81no—
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×Q64—531no—
107#2216—531no—
108#1416—531no—
109#1416D5311yescharacteristic, complemented
110C22⋊C32—531no—
111C2×C53#218411yescharacteristic
112C2×C53C2×Q6411yescharacteristic
113C2×C53#226411yescharacteristic
114D53—22no—
115C22×Q16—1061no—
116C2×Q32—531no—
117C2×Q32—531no—
118C2×Q32—1061no—
119C2×Q32—1062no—
120C2×Q32—1062no—
121C2×Q32C2×D5311yescharacteristic
122C22×C16—531no—
123#467—1061no—
124Q64—1062no—
125Q64—1062no—
126C2×C32—531no—
127C2×C32C2×D5311yescharacteristic
128#366C2×D5311yescharacteristic
129#358—1061no—
130C53#1366511yescharacteristic, complemented, Sylow
131C22×Q8—2121no—
132C2×Q16—21no—
133C2×Q16—1064no—
134C2×Q16—1061no—
135C2×Q16—1061no—
136C2×Q16—2121no—
137C2×Q16—2122no—
138C22×C8—531no—
139C22⋊Q8—2121no—
140Q32—21no—
141Q32—532no—
142Q32—1064no—
143Q32—1061no—
144Q32—1061no—
145Q32—1061no—
146Q32D21212yes—
147C2×C16—531no—
148C2×C16—1061no—
149C2×C16—1061no—
150C2×C16C22×D5311yescharacteristic
151#107—21no—
152#103—2121no—
153C32—21no—
154C32—1061no—
155C2×Q8—41no—
156C2×Q8—4241no—
157C2×Q8—4242no—
158C2×Q8—2124no—
159C2×Q8—2121no—
160C2×Q8—2121no—
161C22×C4—531no—
162C22×C4—4241no—
163Q16—22no—
164Q16—41no—
165Q16—1062no—
166Q16—2124no—
167Q16—2121no—
168Q16—2121no—
169Q16—2121no—
170C2×C8—531no—
171C2×C8—1061no—
172C2×C8—1061no—
173C2×C8D4×D5311yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—4241no—
176C22⋊C4—4241no—
177C16—1061no—
178C16C2×D21211yescharacteristic
179C16D4×D5311yescharacteristic
180C23—531no—
181Q8—42no—
182Q8—81no—
183Q8—4244no—
184Q8—4241no—
185Q8—4241no—
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#1313318711yescharacteristic
199C8#1313324411yescharacteristic
200C22—532no—
201C22—1061no—
202C22—1061no—
203C22D16×D5311yescharacteristic
204C4—82no—
205C4—161no—
206C4—8481no—
207C4—1061no—
208C4—4242no—
209C4#1681075811yescharacteristic
210C4#1681106011yescharacteristic
211C2—1062no—
212C2#2027662811yescharacteristic
213C2Q64×D5311yescharacteristic
214C2#2027703111yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1416C2
socleC22×C53C2×D16
prime radicalC22×D53D16
hypercenter#1416D53
hypocenterC53#13665
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×Dic848C2semidirect
3. C2×#16810463C2semidirect
5. #20272895C2semidirect
7. C53×#1416C2semidirect
96. C2×D53Q64semidirect
109. #1416D53semidirect
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×Dic8482
C2×#168104632
#202768072
#202728952
#202728522
C53×#14162
#1366553