Finite Group Databasebeta

← #28310122

Subgroups — #28310122

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#20290365C211yesmaximal, characteristic, complemented
2#20290311C211yesmaximal, characteristic, complemented
3#20289741C211yesmaximal, characteristic, complemented
4#20289673C211yesmaximal, characteristic, complemented
5#20289673C211yesmaximal, characteristic, complemented
6#20289634C211yesmaximal, characteristic, complemented
7#20276591C211yesmaximal, characteristic
8#16811658—21no—
9D4×#88324—21no—
10#16811512C2211yescharacteristic
11#16811484C2211yescharacteristic
12#16811484C2211yescharacteristic
13D4⋊#88324—21no—
14D4⋊#88324—21no—
15#16811464C2211yescharacteristic
16D8×Dic53C2211yescharacteristic
17D8×Dic53C2211yescharacteristic
18#16810451C411yescharacteristic
19#16810424C411yescharacteristic
20#16810424C411yescharacteristic
21#16810398C2211yescharacteristic
22C2×D16×C53C411yescharacteristic
23C2×#11719940—41no—
24C22×#88324—41no—
25C2×#11719910—21no—
26C4⋊#88324—21no—
27C4⋊#88324—21no—
28C22⋊#88324—41no—
29C22⋊#88324—41no—
30C4×#88324D411yescharacteristic
31C4×#88324D411yescharacteristic
32D4×Dic53—21no—
33D4×Dic53—21no—
34C2×#11719908C2×C411yescharacteristic
35#13133151C2×C411yescharacteristic
36#13133151C2×C411yescharacteristic
37#13133138—21no—
38#13133138—21no—
39C8×Dic53C2311yescharacteristic
40C2×D8×C53C2×C411yescharacteristic
41C2×D8×C53C2×C411yescharacteristic
42D16×C53—22no—
43C2×C16×C53C2×C411yescharacteristic
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C22×Dic53—41no—
47C22×Dic53—41no—
48#11719940—82no—
49C2×#88324—21no—
50C2×#88324—21no—
51C2×#88324—41no—
52C2×#88324—41no—
53C2×#88324—82no—
54D8×C53—21no—
55D8×C53—21no—
56D8×C53—21no—
57D8×C53M1612yescomplemented
58C2×C8×C53C22×C411yescharacteristic
59C22⋊Dic53—41no—
60C22⋊Dic53—41no—
61C4⋊Dic53—21no—
62C4⋊Dic53—21no—
63C4×Dic53C2×D411yescharacteristic
64C2×#88322#4111yescharacteristic
65#11719910—22no—
66C16×C53—21no—
67#11719908—21no—
68C23×C53—41no—
69C23×C53—41no—
70D4×C53—22no—
71D4×C53—41no—
72D4×C53—41no—
73D4×C53—41no—
74C2×C4×C53C4×D411yescharacteristic
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—81no—
78C2×Dic53—81no—
79C2×Dic53—81no—
80C2×Dic53—81no—
81C2×Dic53AGL(1,ℤ/8)11yescharacteristic
82C2×Dic53AGL(1,ℤ/8)11yescharacteristic
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—81no—
95C22×C53—81no—
96C22×C53—81no—
97C22×C53#44211yescharacteristic
98C4×C53#43511yescharacteristic
99C4×C53#44311yescharacteristic
100Dic53—21no—
101Dic53—21no—
102Dic53—81no—
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—161no—
113C2×C53#158611yescharacteristic
114C2×C53AGL(1,ℤ/16)11yescharacteristic
115C2×C53AGL(1,ℤ/16)11yescharacteristic
116C2×D16#615711yescharacteristic
117C4×D8—531no—
118C4×D8—531no—
119#436—1061no—
120C8×D4—1061no—
121#367—531no—
122#357—531no—
123#357—531no—
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×D8C2×#615711yescharacteristic
133C2×D8C2×#615711yescharacteristic
134C2×M16—2121no—
135C22×C8—2121no—
136C4×D4—1061no—
137C4×D4—1061no—
138D16—22no—
139C2×C16—531no—
140C2×C16—1061no—
141C2×C16C2×#615711yescharacteristic
142#107—531no—
143#107—531no—
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—21no—
155C22×C4—2121no—
156C22×C4—2121no—
157D8—21no—
158D8—21no—
159D8—21no—
160D8#1171994012yescomplemented
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—1061no—
170C4⋊C4—1061no—
171C22⋊C4—2121no—
172C22⋊C4—2121no—
173C42—531no—
174C16—21no—
175C16—1062no—
176C16—1061no—
177C23—41no—
178C23—41no—
179D4—22no—
180D4—41no—
181D4—41no—
182D4—41no—
183C2×C4—531no—
184C2×C4—531no—
185C2×C4—4241no—
186C2×C4—4241no—
187C2×C4—4241no—
188C2×C4—4241no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4D4×#615711yescharacteristic
192C8—1061no—
193C8—4241no—
194C8—4241no—
195C8—2121no—
196C8—2121no—
197C8C2×#1171994011yescharacteristic
198C8#1313330411yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22#1681166411yescharacteristic
206C4—1061no—
207C4—1061no—
208C4—4241no—
209C4—4241no—
210C4#1681165411yescharacteristic
211C4#1681166511yescharacteristic
212C2—82no—
213C2—161no—
214C2#2029033611yescharacteristic
215C2#2029112411yescharacteristic
216C2#2029112511yescharacteristic
subgroupG/H
centerC22#16811664
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
Fitting, prime radicalC2×D16×C53C4
socleC22×C53#442
cosocleC8×Dic53C23
hypercenterC2×D16#6157
hypocenterC53#12078
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #20290365C2semidirect
2. #20290311C2semidirect
3. #20289741C2semidirect
4. #20289673C2semidirect
5. #20289673C2semidirect
6. #20289634C2semidirect
57. D8×C53M16semidirect
131. C53#12078semidirect
160. D8#11719940semidirect

Maximal subgroups

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

subgroupindex
#202903652
#202903112
#202897412
#202896732
#202896342
#202765912
#1207853