Finite Group Databasebeta

← #27304767

Subgroups — #27304767

Order 13568.

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

normal subgroups only

This group has 294 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
1#20287851C211yesmaximal, characteristic, complemented
2#20277030C212yesmaximal, complemented
3#20276818C211yesmaximal, characteristic, complemented
4#20272903C212yesmaximal, complemented
5C53×#1400C211yesmaximal, characteristic, complemented
6C2×#13133239—21no—
7#16811066—22no—
8#16811061—22no—
9#16811055—21no—
10#16811055C2212yes—
11C2×D848C2211yescharacteristic, complemented
12#16810859—21no—
13#16810462—22no—
14D1696—22no—
15#16810451C2211yescharacteristic, complemented
16#16810425—21no—
17C2×M32×C53C2211yescharacteristic
18C53×M64C2212yescomplemented
19C2×#11719945—41no—
20#13133239—22no—
21#13133239—22no—
22#13133239—41no—
23#13133239—41no—
24C2×Dic212—21no—
25C2×#11719913—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133191—41no—
29Dic424—21no—
30Dic424D412yescomplemented
31#13133162—22no—
32#13133162—22no—
33D848—21no—
34D848—22no—
35D848D412yescomplemented
36#13133151—21no—
37#13133139—41no—
38C22×C8×C53D411yescharacteristic
39M32×C53—22no—
40C2×C16×C53D411yescharacteristic
41C2×C16×C53C2311yescharacteristic
42C32×C53—22no—
43C22×C4×C53D811yescharacteristic
44C2×C22⋊D53—81no—
45C2×C22⋊D53—81no—
46#11719945—42no—
47#11719945—42no—
48#11719945—81no—
49#11719945—81no—
50C2×D212—41no—
51C2×D212—41no—
52C2×C4×D53—81no—
53C2×Dic106—41no—
54C2×C8×C53—21no—
55C2×C8×C53—21no—
56C2×C8×C53D811yescharacteristic
57C2×C8×C53C2×D411yescharacteristic
58#11719921—81no—
59C4⋊Dic53—41no—
60Dic212—22no—
61Dic212—41no—
62D424—22no—
63D424—41no—
64D424—41no—
65D424—41no—
66#11719913—42no—
67#11719913—42no—
68C16×C53—21no—
69C16×C53C2×D412yes—
70C23×C53D1611yescharacteristic
71C22×D53—81no—
72C22×D53—81no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53D1611yescharacteristic
76C2×C4×C53C2×D811yescharacteristic
77C22⋊D53—82no—
78C22⋊D53—82no—
79C22⋊D53—162no—
80C2×Dic53—81no—
81C2×Dic53—81no—
82D212—42no—
83D212—81no—
84D212—81no—
85D212—81no—
86C4×D53—82no—
87C4×D53—82no—
88Dic106—42no—
89Dic106—81no—
90C8×C53—21no—
91C8×C53#12011yescharacteristic
92C8×C53AGL(1,ℤ/8)11yescharacteristic
93#13667—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—41no—
96C22×C53C2×D1611yescharacteristic
97C2×D53—82no—
98C2×D53—162no—
99C2×D53—161no—
100C2×D53—161no—
101C2×D53—161no—
102C4×C53—21no—
103C4×C53#44711yescharacteristic
104C4×C53#50911yescharacteristic
105Dic53—82no—
106Dic53—161no—
107#3414—531no—
108#2263—532no—
109#2218—531no—
110#1418—532no—
111#1400D5311yescharacteristic, complemented
112C2×C53—21no—
113C2×C53—41no—
114C2×C53#218411yescharacteristic
115D53—162no—
116D53—321no—
117C2×#135—1061no—
118#510—1062no—
119#509—1062no—
120#508—532no—
121#508—1061no—
122C2×D16—531no—
123C2×M32C2×D5311yescharacteristic
124#466—1061no—
125SD64—1062no—
126D32—1062no—
127M64C2×D5312yescomplemented
128#367—531no—
129#357—1061no—
130C53#1366711yescharacteristic, complemented, Sylow
131C2×#41—2121no—
132#135—1062no—
133#135—1062no—
134#135—2121no—
135#135—2121no—
136C2×Q16—1061no—
137C2×SD16—2121no—
138C2×D8—1061no—
139C2×D8—1061no—
140C22×C8D21211yescharacteristic
141#121—2121no—
142Q32—532no—
143Q32—1061no—
144SD32—1062no—
145SD32—1062no—
146D16—532no—
147D16—1061no—
148D16—1062no—
149M32—22no—
150C2×C16D21211yescharacteristic
151C2×C16C22×D5311yescharacteristic
152#107—1061no—
153#102—2121no—
154C32—22no—
155#41—4241no—
156#41—4241no—
157#41—2122no—
158#41—2122no—
159C2×Q8—2121no—
160C2×D4—4241no—
161C2×D4—4241no—
162C2×D4—2121no—
163C2×D4—2121no—
164C22×C4—4241no—
165C22×C4D42411yescharacteristic
166Q16—1062no—
167Q16—2121no—
168SD16—2122no—
169SD16—2122no—
170D8—1062no—
171D8—2121no—
172D8—2121no—
173D8—2121no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8D42411yescharacteristic
177C2×C8C2×D21211yescharacteristic
178C4⋊C4—2121no—
179C22⋊C4—4241no—
180C16—21no—
181C16D4×D5312yes—
182C23—4241no—
183C23—4241no—
184C23D84811yescharacteristic
185Q8—4241no—
186Q8—2122no—
187D4—8482no—
188D4—4241no—
189D4—4242no—
190D4—4242no—
191D4—4241no—
192D4—4241no—
193D4—2122no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—4242no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—4242no—
200C2×C4D84811yescharacteristic
201C2×C4C2×D42411yescharacteristic
202C8—21no—
203C8#1313318711yescharacteristic
204C8#1313324511yescharacteristic
205C22—21no—
206C22—41no—
207C22—8481no—
208C22—8481no—
209C22—8482no—
210C22—8481no—
211C22—4242no—
212C22C2×D84811yescharacteristic
213C4—21no—
214C4—8481no—
215C4—4242no—
216C4#1681075911yescharacteristic
217C4#1681106111yescharacteristic
218C2—21no—
219C2—41no—
220C2—8482no—
221C2—16961no—
222C2#2027662911yescharacteristic
subgroupG/H
centerC2#20276629
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#1400C2
socleC2×C53#2184
hypercenter#1400D53
hypocenterC53#13667
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ #1400

Complements

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

NH ≅ G/Ntype
1. #20287851C2semidirect
2. #20277030C2semidirect
3. #20276818C2semidirect
4. #20272903C2semidirect
5. C53×#1400C2semidirect
11. C2×D848C22semidirect
15. #16810451C22semidirect
18. C53×M64C22semidirect
30. Dic424D4semidirect
35. D848D4semidirect
111. #1400D53semidirect
127. M64C2×D53semidirect
130. C53#13667semidirect

Maximal subgroups

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

subgroupindex
#202878512
#202770302
#202768182
#202729032
C53×#14002
#1366753