Finite Group Databasebeta

← #20275095

Subgroups — #20275095

Order 6784.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811389C211yesmaximal, characteristic, complemented
2#16811169C211yesmaximal, characteristic
3#16810640C211yesmaximal, characteristic, complemented
4#16810596C211yesmaximal, characteristic, complemented
5#16810384C211yesmaximal, characteristic, complemented
6#16810382C211yesmaximal, characteristic, complemented
7C53×#421C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133225—22no—
10#13133223—22no—
11C2×#11719917C2211yescharacteristic
12M16×D53—22no—
13#13133215C411yescharacteristic, complemented
14#13133213C411yescharacteristic, complemented
15#13133206C411yescharacteristic, complemented
16#13133204C411yescharacteristic, complemented
17#13133189—22no—
18C4×C22⋊D53—22no—
19#13133176C2211yescharacteristic
20C2×C4×Dic53C2211yescharacteristic
21#13133144—22no—
22#13133131—22no—
23C2×#41×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25#117×C53C2211yescharacteristic
26#104×C53—22no—
27#41×C53—21no—
28#41×C53—21no—
29#41×C53—21no—
30#41×C53—21no—
31C2×Q8×C53C2×C411yescharacteristic
32C2×D4×C53—21no—
33C2×D4×C53C2×C411yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—22no—
37C2×C22⋊D53D412yes—
38C22×Dic53—21no—
39#11719945—21no—
40#11719945—21no—
41#11719945—21no—
42#11719945—21no—
43C2×D212C2×C411yescharacteristic
44C2×C4×D53D412yes—
45C2×Dic106C2×C411yescharacteristic
46M16×C53—21no—
47M16×C53D412yescomplemented
48C2×C8×C53—21no—
49C4⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C42×C53—21no—
52C22⋊Dic53—22no—
53C22⋊Dic53—21no—
54#11719921—22no—
55C4⋊Dic53—21no—
56#11719919—22no—
57C4×Dic53—21no—
58C4×Dic53—21no—
59C4×Dic53C2×C411yescharacteristic
60C4×Dic53C2×C411yescharacteristic
61#11719917—21no—
62#11719917D412yescomplemented
63C2×#88322—21no—
64#11719912—24no—
65C8×D53—24no—
66C23×C53—21no—
67C23×C53C2×D411yescharacteristic
68C22×D53C22⋊C412yes—
69Q8×C53—21no—
70Q8×C53—21no—
71D4×C53—21no—
72D4×C53—21no—
73D4×C53—41no—
74D4×C53—41no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53C22×C411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53#4111yescharacteristic
84C2×C4×C53#4111yescharacteristic
85C22⋊D53—22no—
86C22⋊D53—22no—
87C22⋊D53—44no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—42no—
91C2×Dic53—41no—
92C2×Dic53C22⋊C412yes—
93D212—21no—
94D212—21no—
95C4×D53—22no—
96C4×D53—22no—
97Dic106—21no—
98Dic106—21no—
99C8×C53—22no—
100#88322—22no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—41no—
104C22×C53C2×C22⋊C411yescharacteristic
105C22×C53#12111yescharacteristic
106C22×C53#12111yescharacteristic
107C2×D53—22no—
108C2×D53—42no—
109C4×C53—41no—
110C4×C53—81no—
111C4×C53C4×D411yescharacteristic
112C4×C53C4×D411yescharacteristic
113C4×C53#12011yescharacteristic
114C4×C53#12311yescharacteristic
115Dic53—22no—
116Dic53—22no—
117Dic53—41no—
118Dic53—81no—
119#1884—531nomaximal, Sylow
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—81no—
124C2×C53#38611yescharacteristic
125D53—42no—
126#568—531no—
127#532—531no—
128#421—531no—
129#421D5311yescharacteristic, complemented
130#410—531no—
131#339—531no—
132#339—531no—
133C53#188411yescharacteristic, complemented, Sylow
134C2×#41—531no—
135C2×#41C2×D5311yescharacteristic
136#131—1062no—
137#131—1062no—
138C2×M16—531no—
139C2×M16—1062no—
140C2×M16C2×D5311yescharacteristic
141C4⋊Q8—531no—
142#127—531no—
143#124—531no—
144#124—531no—
145#121—1062no—
146C4×D4—1062no—
147#117—531no—
148#117C2×D5311yescharacteristic
149C2×C42—531no—
150#104—22no—
151#104—1062no—
152#99—1062no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—1061no—
158#41—1061no—
159#41—1061no—
160#41—1061no—
161C2×Q8—531no—
162C2×Q8C4×D5311yescharacteristic
163C2×D4—21no—
164C2×D4—532no—
165C2×D4—531no—
166C2×D4—1062no—
167C2×D4C4×D5311yescharacteristic
168C22×C4—21no—
169C22×C4—532no—
170C22×C4—1061no—
171C22×C4C22×D5311yescharacteristic
172M16—21no—
173M16—532no—
174M16—1064no—
175M16—1061no—
176M16C22⋊D5312yescomplemented
177C2×C8—21no—
178C2×C8—1064no—
179C2×C8—1061no—
180C4⋊C4—21no—
181C4⋊C4—1062no—
182C4⋊C4—1061no—
183C22⋊C4—21no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—1061no—
187C42—21no—
188C42—531no—
189C42—531no—
190C42—1061no—
191C42—1061no—
192C23—21no—
193C23—532no—
194C23D4×D5311yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—1061no—
198Q8—1061no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—1062no—
204D4—1062no—
205D4—1061no—
206D4—1061no—
207D4—2124no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—532no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—2122no—
218C2×C4—2121no—
219C2×C4C2×C4×D5311yescharacteristic
220C2×C4#1171994511yescharacteristic
221C2×C4D4×D5311yescharacteristic
222C2×C4D4×D5311yescharacteristic
223C2×C4#1171994711yescharacteristic
224C2×C4C2×C22⋊D5311yescharacteristic
225C8—22no—
226C8—1062no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—1062no—
231C22—2122no—
232C22C22⋊C4×D5311yescharacteristic
233C22#1313318811yescharacteristic
234C22#1313319011yescharacteristic
235C4—41no—
236C4—81no—
237C4—1062no—
238C4—1062no—
239C4—4241no—
240C4—2121no—
241C4#1313317711yescharacteristic
242C4C4×C22⋊D5311yescharacteristic
243C4#1313318511yescharacteristic
244C4#1313319911yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—2122no—
250C2#1681049911yescharacteristic
subgroupG/H
centerC4#13133185
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#421C2
socleC2×C53#386
hypercenter#421D53
hypocenterC53#1884
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #16811389C2semidirect
3. #16810640C2semidirect
4. #16810596C2semidirect
5. #16810384C2semidirect
6. #16810382C2semidirect
7. C53×#421C2semidirect
13. #13133215C4semidirect
14. #13133213C4semidirect
15. #13133206C4semidirect
16. #13133204C4semidirect
47. M16×C53D4semidirect
62. #11719917D4semidirect
129. #421D53semidirect
133. C53#1884semidirect
176. M16C22⋊D53semidirect