Finite Group Databasebeta

← #28299273

Subgroups — #28299273

Order 13568.

348 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 50 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×#16811528C212yesmaximal
2#20275521C211yesmaximal, characteristic
3C2×#13133276—22no—
4#16811528—22no—
5#16811528C414yescomplemented
6C2×#13133214C2211yescharacteristic
7C2×#13133139—22no—
8#16810493C411yescharacteristic, complemented
9C2×C4×C8×C53C411yescharacteristic, complemented
10C2×C22⋊#6157—42no—
11#13133276—24no—
12#13133276—42no—
13C22×D212—41no—
14C22×D212D412yes—
15#13133214—41no—
16#13133214C2×C412yes—
17#13133214D411yescharacteristic
18#13133214Q811yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—41no—
21#13133139—44no—
22C22×C8×C53—21no—
23C2×C42×C53C2×C411yescharacteristic
24C4×C8×C53—41no—
25C23×D53—22no—
26C22×#6157—82no—
27C22×C4×C53—21no—
28C22×C4×C53C22⋊C411yescharacteristic
29C22×Dic53—41no—
30C2×D212—22no—
31C2×D212—42no—
32C2×D212—42no—
33C2×D212—42no—
34C2×D212—41no—
35C2×D212—41no—
36C2×D212—82no—
37C2×D212—82no—
38C2×D212C22⋊C412yes—
39C2×D212C4⋊C412yes—
40C22⋊#6157—44no—
41C22⋊#6157—82no—
42C2×C8×C53—22no—
43C2×C8×C53—41no—
44C2×C8×C53—41no—
45C42×C53—21no—
46C42×C53C4211yescharacteristic
47C42×C53C4⋊C411yescharacteristic
48C4⋊Dic53—42no—
49C4⋊Dic53—82no—
50C23×C53#9911yescharacteristic
51C22×D53—22no—
52C22×D53—22no—
53C22×D53—42no—
54C22×D53—42no—
55C22×D53—82no—
56C22×D53—82no—
57C2×#6157—84no—
58C2×#6157—162no—
59C2×#6157—162no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53#9511yescharacteristic
67C2×C4×C53#9911yescharacteristic
68C2×Dic53—82no—
69C2×Dic53—81no—
70D212—42no—
71D212—42no—
72D212—82no—
73D212—84no—
74D212—82no—
75D212—82no—
76C8×C53—42no—
77#7498—531nomaximal, Sylow
78C22×C53—21no—
79C22×C53—21no—
80C22×C53#34211yescharacteristic
81C22×C53#35311yescharacteristic
82C22×C53#35411yescharacteristic
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—82no—
86C2×D53—82no—
87C2×D53—82no—
88C2×D53—82no—
89C2×D53—162no—
90#6157—164no—
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—21no—
94C4×C53—21no—
95Dic53—161no—
96C2×#353—532no—
97#1957—531no—
98C2×C53—21no—
99C2×C53—21no—
100C2×C53#139311yescharacteristic
101C2×C53#140811yescharacteristic
102C2×C53#140911yescharacteristic
103D53—82no—
104D53—82no—
105C2×#127—531no—
106C2×#102—1062no—
107C2×#99—1062no—
108C2×C4×C8#615711yescharacteristic, complemented
109#384—531no—
110#353—534no—
111#353—1062no—
112C53#749811yescharacteristic, complemented, Sylow
113C22×D4—532no—
114C22×D4—2121no—
115C22×C8—21no—
116#127—532no—
117#127—531no—
118#127—531no—
119#127—2121no—
120C2×C4⋊C4—1061no—
121C2×C4⋊C4—2121no—
122C2×C22⋊C4—2122no—
123C2×C42C2×#615711yescharacteristic
124#102—2124no—
125#99—1064no—
126#99—2122no—
127C4×C8—41no—
128C24—1062no—
129C2×D4—532no—
130C2×D4—532no—
131C2×D4—1062no—
132C2×D4—4242no—
133C2×D4—4242no—
134C2×D4—2122no—
135C2×D4—2121no—
136C2×D4—2121no—
137C2×D4—2122no—
138C2×D4—2122no—
139C22×C4—21no—
140C22×C4—4242no—
141C22×C4—2121no—
142C22×C4C22⋊#615711yescharacteristic
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C4⋊C4—4242no—
147C4⋊C4—2122no—
148C22⋊C4—4242no—
149C22⋊C4—2124no—
150C42—21no—
151C42C4×#615711yescharacteristic
152C42C4⋊#615711yescharacteristic
153C23—1062no—
154C23—1062no—
155C23—4242no—
156C23—4242no—
157C23—2122no—
158C23—2122no—
159C23#1313327611yescharacteristic
160D4—4244no—
161D4—4242no—
162D4—4242no—
163D4—4242no—
164D4—2122no—
165D4—2122no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—8482no—
173C2×C4—8482no—
174C2×C4—4242no—
175C2×C4—4244no—
176C2×C4—4241no—
177C2×C4(C2×#6157)⋊C411yescharacteristic
178C2×C4#1313327511yescharacteristic
179C8—42no—
180C22—21no—
181C22—21no—
182C22—8482no—
183C22—4242no—
184C22—4242no—
185C22—4242no—
186C22—4242no—
187C22—2122no—
188C22—2122no—
189C22#1681150111yescharacteristic
190C22#1681152811yescharacteristic
191C22#1681153011yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—8484no—
197C4—8481no—
198C2—21no—
199C2—21no—
200C2—4242no—
201C2—4242no—
202C2#2028980911yescharacteristic
203C2#2028984711yescharacteristic
204C2#2028984911yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC42×C53C42
FrattiniC2×C42C2×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#353
prime radical, cosocleC2×#13133214C22
hypercenterC2×C4×C8#6157
hypocenterC53#7498
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
5. #16811528C4semidirect
8. #16810493C4semidirect
9. C2×C4×C8×C53C4semidirect
108. C2×C4×C8#6157semidirect
112. C53#7498semidirect

Maximal subgroups

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

subgroupindex
C2×#168115282
#202755212
#749853