Finite Group Databasebeta

← #28310118

Subgroups — #28310118

Order 13568.

234 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 62 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#20290364C211yesmaximal, characteristic, complemented
2#20290311C211yesmaximal, characteristic, complemented
3#20289739C211yesmaximal, characteristic, complemented
4#20289674C211yesmaximal, characteristic, complemented
5#20289673C211yesmaximal, characteristic, complemented
6#20289634C211yesmaximal, characteristic, complemented
7#20276588C211yesmaximal, characteristic
8#16811657—21no—
9D4×#88324—21no—
10#16811512C2211yescharacteristic
11#16811484C2211yescharacteristic
12#16811484C2211yescharacteristic
13#16811471—21no—
14D4⋊#88324—21no—
15#16811464C2211yescharacteristic
16C2×#13133158C411yescharacteristic
17D8×Dic53C2211yescharacteristic
18#16810705C2211yescharacteristic
19#16810450C411yescharacteristic
20#16810425C411yescharacteristic
21#16810398C2211yescharacteristic
22C53×#357C411yescharacteristic
23C2×#11719936—41no—
24C22×#88324—41no—
25C2×#11719910—21no—
26C4⋊#88324—21no—
27C4⋊#88324—21no—
28#13133274—41no—
29C22⋊#88324—41no—
30C4×#88324D411yescharacteristic
31C4×#88324D411yescharacteristic
32C2×D424C2×C411yescharacteristic
33D4×Dic53—21no—
34#13133179—21no—
35#13133158—22no—
36C2×#11719908C2×C411yescharacteristic
37#13133151C2×C411yescharacteristic
38#13133138—21no—
39#13133137—21no—
40C8×Dic53C2311yescharacteristic
41C2×D8×C53C2×C411yescharacteristic
42C2×C16×C53C2×C411yescharacteristic
43#107×C53C2×C411yescharacteristic
44C2×D4×C53—21no—
45C22×Dic53—41no—
46C2×D212—21no—
47C2×C4×D53—41no—
48C2×#88324—21no—
49C2×#88324—21no—
50C2×#88324—41no—
51C2×#88324—41no—
52C2×#88324—82no—
53#11719936—82no—
54D8×C53—21no—
55D8×C53M1612yescomplemented
56C2×C8×C53C22×C411yescharacteristic
57C4⋊C4×C53—21no—
58C22⋊Dic53—41no—
59#11719921—41no—
60C4⋊Dic53—21no—
61C4×Dic53C2×D411yescharacteristic
62C2×#88322#4111yescharacteristic
63D424—21no—
64D424—21no—
65#11719910—22no—
66C16×C53—21no—
67#11719908—21no—
68C23×C53—41no—
69C22×D53—41no—
70D4×C53—22no—
71D4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C4×D411yescharacteristic
74C2×Dic53—41no—
75C2×Dic53—81no—
76C2×Dic53—81no—
77C2×Dic53AGL(1,ℤ/8)11yescharacteristic
78C2×Dic53AGL(1,ℤ/8)11yescharacteristic
79D212—41no—
80D212—41no—
81C4×D53—81no—
82C4×D53—81no—
83#88324—41no—
84#88324—41no—
85#88324—81no—
86#88324—81no—
87C8×C53C2×M1611yescharacteristic
88C8×C53#13111yescharacteristic
89#88322—21no—
90#12078—531nomaximal, Sylow
91C22×C53—42no—
92C22×C53—81no—
93C22×C53—81no—
94C22×C53#44211yescharacteristic
95C2×D53—81no—
96C2×D53—81no—
97C2×D53—81no—
98C4×C53—81no—
99C4×C53#43511yescharacteristic
100C4×C53#44311yescharacteristic
101Dic53—21no—
102Dic53—21no—
103Dic53—81no—
104#2177—531no—
105#1594—531no—
106#1580—531no—
107#1368—531no—
108#1333—531no—
109#1333—531no—
110#1313—531no—
111C2×C53—82no—
112C2×C53#158611yescharacteristic
113C2×C53AGL(1,ℤ/16)11yescharacteristic
114C2×C53AGL(1,ℤ/16)11yescharacteristic
115D53—161no—
116C2×D16—531no—
117C4×D8—531no—
118C4×D8—531no—
119#436—1061no—
120C8×D4—1061no—
121#367—531no—
122#357—531no—
123#357#615711yescharacteristic
124#346—531no—
125#346—531no—
126#334—531no—
127#334—531no—
128D4⋊C8—1061no—
129D4⋊C8—1061no—
130#322—531no—
131C53#1207811yescharacteristic, complemented, Sylow
132C2×D8—531no—
133C2×D8C2×#615711yescharacteristic
134C2×M16—2121no—
135C22×C8—2121no—
136C4×D4—1061no—
137C4×D4—1061no—
138D16—1062no—
139C2×C16—531no—
140C2×C16—1061no—
141C2×C16C2×#615711yescharacteristic
142#107—531no—
143#107C2×#615711yescharacteristic
144C4⋊C8—1061no—
145C4⋊C8—1061no—
146#102—1061no—
147#102—1061no—
148C22⋊C8—2121no—
149C22⋊C8—2121no—
150C4×C8—531no—
151C4×C8—531no—
152C4×C8—531no—
153C2×D4—21no—
154C2×D4—1061no—
155C22×C4—2121no—
156C22×C4—2121no—
157D8—21no—
158D8—1061no—
159D8—1061no—
160D8#1171993612yescomplemented
161M16—4242no—
162C2×C8—531no—
163C2×C8—1061no—
164C2×C8—1061no—
165C2×C8—4242no—
166C2×C8—2121no—
167C2×C8—2121no—
168C2×C8C22×#615711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1061no—
171C22⋊C4—2121no—
172C22⋊C4—2121no—
173C42—531no—
174C16—21no—
175C16—1062no—
176C16—1061no—
177C23—41no—
178C23—2121no—
179D4—22no—
180D4—41no—
181D4—2121no—
182D4—2121no—
183C2×C4—41no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—4241no—
187C2×C4—4241no—
188C2×C4—4241no—
189C2×C4—4241no—
190C2×C4—2121no—
191C2×C4D4×#615711yescharacteristic
192C8—1061no—
193C8—4241no—
194C8—4241no—
195C8—2121no—
196C8—2121no—
197C8C2×#1171993611yescharacteristic
198C8#1313330411yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—4241no—
203C22—4241no—
204C22—4241no—
205C22#1681166411yescharacteristic
206C4—81no—
207C4—1061no—
208C4—1061no—
209C4—4241no—
210C4#1681165211yescharacteristic
211C4#1681166611yescharacteristic
212C2—82no—
213C2—8481no—
214C2#2029033411yescharacteristic
215C2#2029112411yescharacteristic
216C2#2029112711yescharacteristic
subgroupG/H
centerC22#16811664
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#357C4
socleC22×C53#442
prime radicalC2×#13133158C4
cosocleC8×Dic53C23
hypercenter#357#6157
hypocenterC53#12078
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #357

Complements

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

NH ≅ G/Ntype
1. #20290364C2semidirect
2. #20290311C2semidirect
3. #20289739C2semidirect
4. #20289674C2semidirect
5. #20289673C2semidirect
6. #20289634C2semidirect
55. D8×C53M16semidirect
131. C53#12078semidirect
160. D8#11719936semidirect

Maximal subgroups

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

subgroupindex
#202903642
#202903112
#202897392
#202896742
#202896732
#202896342
#202765882
#1207853