Finite Group Databasebeta

← #27304876

Subgroups — #27304876

Order 13568.

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

normal subgroups only

This group has 220 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×#16810461C211yesmaximal, characteristic, complemented
3#20276963C211yesmaximal, characteristic
4#20276573C211yesmaximal, characteristic, complemented
5#20272910C211yesmaximal, characteristic, complemented
6#20272892C211yesmaximal, characteristic, complemented
7C53×#1415C211yesmaximal, characteristic, complemented
8C2×Dic424C2211yescharacteristic
9C2×D848C2211yescharacteristic
10C2×#13133158—21no—
11#16810998—21no—
12#16810717—21no—
13#16810462—22no—
14#16810461—22no—
15C2×#13133123C2211yescharacteristic
16#16810426—21no—
17C16×Dic53C2211yescharacteristic
18C2×D16×C53C2211yescharacteristic
19C2×C32×C53C2211yescharacteristic
20C53×#366C2211yescharacteristic
21C2×Dic212—21no—
22C2×D424—21no—
23C2×#11719922—41no—
24#13133213—41no—
25#13133182—41no—
26Dic424—21no—
27Dic424D412yescomplemented
28D848—21no—
29D848—21no—
30#13133158—42no—
31C2×#11719908D411yescharacteristic
32#13133140—41no—
33C8×Dic53D411yescharacteristic
34#13133123—21no—
35C2×D8×C53—21no—
36D16×C53—21no—
37D16×C53—21no—
38C2×C16×C53C2311yescharacteristic
39#107×C53—21no—
40C32×C53—21no—
41C2×D4×C53—41no—
42C2×C22⋊D53—81no—
43C2×C22⋊D53—81no—
44C2×D212—41no—
45C2×Dic106—41no—
46D8×C53—41no—
47D8×C53—41no—
48C2×C8×C53C2×D411yescharacteristic
49C4⋊C4×C53—41no—
50#11719922—82no—
51#11719919—81no—
52C4×Dic53—81no—
53C4×Dic53D811yescharacteristic
54C2×#88322D811yescharacteristic
55Dic212—22no—
56Dic212—41no—
57D424—41no—
58D424—41no—
59C16×C53C2×D411yescharacteristic
60C16×C53#4111yescharacteristic
61#11719908—21no—
62C23×C53—81no—
63C22×D53—81no—
64D4×C53—81no—
65D4×C53—81no—
66C2×C4×C53—81no—
67C2×C4×C53C2×D811yescharacteristic
68C22⋊D53—162no—
69C22⋊D53—162no—
70C2×Dic53—81no—
71C2×Dic53D1611yescharacteristic
72C2×Dic53D1611yescharacteristic
73D212—81no—
74D212—81no—
75Dic106—42no—
76Dic106—81no—
77C8×C53#12111yescharacteristic
78C8×C53AGL(1,ℤ/8)11yescharacteristic
79#88322—21no—
80#13686—531nomaximal, Sylow
81C22×C53—161no—
82C22×C53—161no—
83C22×C53—161no—
84C22×C53C2×D1611yescharacteristic
85C2×D53—161no—
86C2×D53—161no—
87C2×D53—161no—
88C4×C53—161no—
89C4×C53#45911yescharacteristic
90C4×C53#50911yescharacteristic
91Dic53—21no—
92Dic53—82no—
93Dic53SD6412yescomplemented
94C2×SD64—531no—
95C2×SD64—531no—
96#2246—531no—
97C4×Q32—531no—
98C4⋊C32—531no—
99#1415—531no—
100#1415D5311yescharacteristic, complemented
101C2×C53—321no—
102C2×C53#220611yescharacteristic
103C2×C53C2×SD6411yescharacteristic
104C2×C53#226311yescharacteristic
105D53—321no—
106C2×Q32—531no—
107C2×D16—531no—
108C2×D16—1061no—
109C2×D16C2×D5311yescharacteristic
110#493—1061no—
111C4×Q16—1061no—
112SD64—1062no—
113SD64—1062no—
114C2×C32—531no—
115C2×C32C2×D5311yescharacteristic
116#366C2×D5311yescharacteristic
117#358—1061no—
118C4×C16—531no—
119C53#1368611yescharacteristic, complemented, Sylow
120C2×Q16—1061no—
121C2×D8—21no—
122C2×D8—1061no—
123C2×D8—2121no—
124#127—2121no—
125C4×Q8—2121no—
126Q32—532no—
127Q32—1061no—
128D16—21no—
129D16—21no—
130D16—1061no—
131D16—1061no—
132D16—2122no—
133C2×C16—531no—
134C2×C16C22×D5311yescharacteristic
135#107—21no—
136#103—2121no—
137C4×C8—531no—
138C32—21no—
139C32—1061no—
140C2×Q8—2121no—
141C2×D4—41no—
142C2×D4—4241no—
143C2×D4—4241no—
144C2×D4—2121no—
145Q16—1062no—
146Q16—2121no—
147D8—41no—
148D8—41no—
149D8—4242no—
150D8—2121no—
151D8—2121no—
152C2×C8—531no—
153C2×C8D4×D5311yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—4241no—
156C42—531no—
157C42—4241no—
158C16—1061no—
159C16#1171994511yescharacteristic
160C16D4×D5311yescharacteristic
161C23—81no—
162C23—4241no—
163Q8—4241no—
164Q8—2122no—
165D4—81no—
166D4—81no—
167D4—8482no—
168D4—8482no—
169D4—4241no—
170D4—4241no—
171C2×C4—81no—
172C2×C4—531no—
173C2×C4—531no—
174C2×C4—4241no—
175C2×C4D8×D5311yescharacteristic
176C8—1061no—
177C8#1313319011yescharacteristic
178C8#1313324411yescharacteristic
179C22—161no—
180C22—161no—
181C22—161no—
182C22—8481no—
183C22—8481no—
184C22—8481no—
185C22D16×D5311yescharacteristic
186C4—161no—
187C4—532no—
188C4—1061no—
189C4—4242no—
190C4#1681081711yescharacteristic
191C4#1681106011yescharacteristic
192C2—321no—
193C2—16961no—
194C2#2027674911yescharacteristic
195C2SD64×D5311yescharacteristic
196C2#2027702611yescharacteristic
subgroupG/H
centerC22D16×D53
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1415C2
socleC22×C53C2×D16
prime radical#20276963C2
hypercenter#1415D53
hypocenterC53#13686
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
1. C2×#16810462C2semidirect
2. C2×#16810461C2semidirect
4. #20276573C2semidirect
5. #20272910C2semidirect
6. #20272892C2semidirect
7. C53×#1415C2semidirect
27. Dic424D4semidirect
93. Dic53SD64semidirect
100. #1415D53semidirect
119. C53#13686semidirect

Maximal subgroups

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

subgroupindex
C2×#168104622
C2×#168104612
#202769632
#202765732
#202729102
#202728922
C53×#14152
#1368653