Finite Group Databasebeta

← #27304951

Subgroups — #27304951

Order 13568.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16810462C211yesmaximal, characteristic, complemented
2C2×#16810456C211yesmaximal, characteristic, complemented
3#20276807C211yesmaximal, characteristic, complemented
4#20276802C211yesmaximal, characteristic
5#20272895C211yesmaximal, characteristic, complemented
6#20272892C211yesmaximal, characteristic, complemented
7C53×#1423C211yesmaximal, characteristic, complemented
8C2×Dic424C2211yescharacteristic
9C2×D848C2211yescharacteristic
10C2×C16×D53C2211yescharacteristic
11#16810865—21no—
12#16810860—21no—
13#16810462—22no—
14#16810456—22no—
15C2×#13133123C2211yescharacteristic
16#16810426—21no—
17#16810423—21no—
18C2×C32×C53C2211yescharacteristic
19C53×#366C2211yescharacteristic
20C53×#366C2211yescharacteristic
21C2×Dic212—21no—
22C2×D424—21no—
23C2×C8×D53D411yescharacteristic
24#13133197—41no—
25#13133192—41no—
26Dic424—21no—
27Dic424—21no—
28D848—21no—
29D848—21no—
30C2×#11719908D411yescharacteristic
31C16×D53—21no—
32C16×D53—21no—
33#13133140—41no—
34#13133137—41no—
35#13133123—21no—
36C2×C16×C53C2311yescharacteristic
37#107×C53—21no—
38#107×C53—21no—
39C32×C53D412yescomplemented
40C2×D212—41no—
41C2×D212—81no—
42C2×C4×D53D811yescharacteristic
43C2×Dic106—41no—
44C2×C8×C53C2×D411yescharacteristic
45C4⋊C4×C53—41no—
46C4⋊C4×C53—41no—
47#11719921—81no—
48#11719921—81no—
49C4⋊Dic53—81no—
50C2×#88322D811yescharacteristic
51Dic212—41no—
52Dic212—41no—
53D424—41no—
54D424—41no—
55C8×D53—21no—
56C8×D53—21no—
57C16×C53C2×D411yescharacteristic
58C16×C53#4111yescharacteristic
59#11719908—21no—
60C22×D53—81no—
61C22×D53D1611yescharacteristic
62C2×C4×C53—81no—
63C2×C4×C53—81no—
64C2×C4×C53C2×D811yescharacteristic
65C2×Dic53—81no—
66C2×Dic53D1611yescharacteristic
67D212—81no—
68D212—81no—
69D212—162no—
70C4×D53—21no—
71C4×D53—21no—
72Dic106—81no—
73Dic106—81no—
74C8×C53#12111yescharacteristic
75C8×C53#13511yescharacteristic
76#88322—21no—
77#13699—531nomaximal, Sylow
78C22×C53C2×D1611yescharacteristic
79C2×D53—21no—
80C2×D53—21no—
81C2×D53—21no—
82C2×D53—161no—
83C2×D53—161no—
84C2×D53—161no—
85C4×C53—161no—
86C4×C53—161no—
87C4×C53#46611yescharacteristic
88C4×C53#50811yescharacteristic
89Dic53—21no—
90Dic53—161no—
91C2×SD64—531no—
92C2×M64—531no—
93#2216—531no—
94#2215—531no—
95#1423D5311yescharacteristic, complemented
96#1416—531no—
97#1415—531no—
98C2×C53#221511yescharacteristic
99C2×C53#226311yescharacteristic
100C2×C53#226411yescharacteristic
101D53—41no—
102D53—321no—
103C2×Q32—531no—
104C2×D16—531no—
105C22×C16—531no—
106#467—1061no—
107#466—1061no—
108SD64—1062no—
109M64—1062no—
110C2×C32—531no—
111C2×C32C2×D5311yescharacteristic
112#366C2×D5311yescharacteristic
113#366C2×D5311yescharacteristic
114#358—1061no—
115#357—1061no—
116C53#1369911yescharacteristic, complemented, Sylow
117C2×Q16—1061no—
118C2×D8—1061no—
119C22×C8—531no—
120C22⋊Q8—2121no—
121#121—2121no—
122Q32—1061no—
123Q32—1061no—
124D16—1061no—
125D16—1061no—
126C2×C16—531no—
127C2×C16—1061no—
128C2×C16—1061no—
129C2×C16C22×D5311yescharacteristic
130#107—21no—
131#107—21no—
132#103—2121no—
133#102—2121no—
134C32—1061no—
135C32D21212yescomplemented
136C2×Q8—2121no—
137C2×D4—4241no—
138C2×D4—2121no—
139C22×C4—531no—
140Q16—2121no—
141Q16—2121no—
142D8—2121no—
143D8—2121no—
144C2×C8—531no—
145C2×C8—1061no—
146C2×C8—1061no—
147C2×C8D4×D5311yescharacteristic
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—4241no—
151C22⋊C4—4241no—
152C22⋊C4—4241no—
153C16—1061no—
154C16C2×D21211yescharacteristic
155C16#1171994911yescharacteristic
156C23—531no—
157C23—4241no—
158Q8—4241no—
159Q8—4241no—
160D4—8482no—
161D4—4241no—
162D4—4241no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—531no—
166C2×C4—1061no—
167C2×C4—1061no—
168C2×C4—4241no—
169C2×C4D8×D5311yescharacteristic
170C8—1061no—
171C8#1313319211yescharacteristic
172C8#1313324011yescharacteristic
173C22—8481no—
174C22—8481no—
175C22—8481no—
176C22—1061no—
177C22—1061no—
178C22—1061no—
179C22D16×D5311yescharacteristic
180C4—161no—
181C4—161no—
182C4—8481no—
183C4—1061no—
184C4#1681086011yescharacteristic
185C4#1681105611yescharacteristic
186C2—16961no—
187C2—2121no—
188C2#2027680211yescharacteristic
189C2#2027702611yescharacteristic
190C2#2027703411yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1423C2
socleC22×C53C2×D16
prime radical#20276802C2
hypercenter#1423D53
hypocenterC53#13699
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1423

Complements

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

NH ≅ G/Ntype
1. C2×#16810462C2semidirect
2. C2×#16810456C2semidirect
3. #20276807C2semidirect
5. #20272895C2semidirect
6. #20272892C2semidirect
7. C53×#1423C2semidirect
39. C32×C53D4semidirect
95. #1423D53semidirect
116. C53#13699semidirect
135. C32D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168104622
C2×#168104562
#202768072
#202768022
#202728952
#202728922
C53×#14232
#1369953