Finite Group Databasebeta

← #20275239

Subgroups — #20275239

Order 6784.

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

normal subgroups only

This group has 344 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#16811190C211yesmaximal, characteristic
3#16810628C211yesmaximal, characteristic, complemented
4#16810596C211yesmaximal, characteristic, complemented
5#16810380C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#418C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133225—22no—
10#13133223—22no—
11C2×#11719917C2211yescharacteristic
12M16×D53—22no—
13#13133213C412yescomplemented
14#13133204C412yescomplemented
15#13133188—22no—
16#13133186—22no—
17#13133176C2211yescharacteristic
18#13133176C2211yescharacteristic
19#13133138—22no—
20#13133131—22no—
21C22×D4×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23#117×C53C2211yescharacteristic
24#102×C53—22no—
25C24×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53C2×C412yes—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—24no—
34C2×C22⋊D53D412yes—
35#11719945—21no—
36#11719945—21no—
37#11719945—21no—
38#11719945—21no—
39C2×D212C2×C411yescharacteristic
40C2×C4×D53D412yes—
41C2×Dic106C2×C411yescharacteristic
42M16×C53—21no—
43M16×C53—21no—
44C2×C8×C53D412yescomplemented
45C4⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C42×C53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53—24no—
50C22⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53C4×Dic53—21no—
54C4×Dic53C2×C412yes—
55#11719917—21no—
56#11719917—21no—
57C2×#88322D412yescomplemented
58#11719912—24no—
59C8×D53—24no—
60C23×C53—22no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53—82no—
64C23×C53C2×D411yescharacteristic
65C22×D53C22⋊C412yes—
66D4×C53—22no—
67D4×C53—22no—
68D4×C53—41no—
69D4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C22×C411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53#4111yescharacteristic
76C2×C4×C53#4111yescharacteristic
77C22⋊D53—22no—
78C22⋊D53—22no—
79C22⋊D53—84no—
80C2×Dic53—22no—
81C2×Dic53—41no—
82C2×Dic53C22⋊C412yes—
83D212—21no—
84D212—21no—
85C4×D53—22no—
86C4×D53—22no—
87Dic106—21no—
88Dic106—21no—
89C8×C53—22no—
90#88322—22no—
91C22×C53—41no—
92C22×C53—44no—
93C22×C53—42no—
94C22×C53—82no—
95C22×C53—82no—
96C22×C53C2×C22⋊C411yescharacteristic
97C22×C53#12111yescharacteristic
98C22×C53#12111yescharacteristic
99C2×D53—22no—
100C2×D53—42no—
101C4×C53—81no—
102C4×C53C4×D411yescharacteristic
103C4×C53C4×D411yescharacteristic
104C4×C53#12011yescharacteristic
105C4×C53#12311yescharacteristic
106Dic53—22no—
107Dic53—42no—
108Dic53—81no—
109#1910—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53—82no—
114C2×C53#38611yescharacteristic
115D53—42no—
116#568—531no—
117#535—531no—
118#418—531no—
119#418D5311yescharacteristic, complemented
120#410—531no—
121H(3,ℤ/4)—531no—
122H(3,ℤ/4)—531no—
123C53#191011yescharacteristic, complemented, Sylow
124C2×#41—531no—
125C22×D4C2×D5311yescharacteristic
126#131—1062no—
127#131—1062no—
128C2×M16—531no—
129C2×M16—1062no—
130C2×M16C2×D5311yescharacteristic
131#127—532no—
132#124—532no—
133#121—1062no—
134#120—1062no—
135#117—531no—
136#117—531no—
137#117C2×D5311yescharacteristic
138#102—22no—
139#102—1062no—
140#99—1062no—
141C24—21no—
142#41—1061no—
143#41—1061no—
144#41—1061no—
145#41—1061no—
146C2×Q8—531no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—532no—
153C2×D4—531no—
154C2×D4—1064no—
155C2×D4C4×D5312yes—
156C22×C4—532no—
157C22×C4C22×D5311yescharacteristic
158M16—21no—
159M16—21no—
160M16—1064no—
161M16—1061no—
162M16—1061no—
163C2×C8—532no—
164C2×C8—1064no—
165C2×C8C22⋊D5312yescomplemented
166C4⋊C4—21no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C22⋊C4—21no—
170C22⋊C4—1064no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C42—21no—
174C42—532no—
175C42—1061no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—82no—
180C23—532no—
181C23D4×D5311yescharacteristic
182Q8—1061no—
183Q8—1061no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4—1062no—
189D4—1062no—
190D4—1061no—
191D4—1061no—
192D4—4244no—
193C2×C4—41no—
194C2×C4—532no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—2121no—
199C2×C4C2×C4×D5311yescharacteristic
200C2×C4#1171994511yescharacteristic
201C2×C4D4×D5311yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C2×C4#1171994711yescharacteristic
204C2×C4C2×C22⋊D5311yescharacteristic
205C8—22no—
206C8—1062no—
207C22—41no—
208C22—44no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—1062no—
213C22—2122no—
214C22C22⋊C4×D5311yescharacteristic
215C22#1313318811yescharacteristic
216C22#1313319011yescharacteristic
217C4—81no—
218C4—1062no—
219C4—4241no—
220C4—2122no—
221C4#1313317711yescharacteristic
222C4C4×C22⋊D5311yescharacteristic
223C4#1313318511yescharacteristic
224C4#1313319911yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—82no—
229C2—2122no—
230C2#1681049911yescharacteristic
subgroupG/H
centerC2#16810499
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#418C2
socleC2×C53#386
hypercenter#418D53
hypocenterC53#1910
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #16811389C2semidirect
3. #16810628C2semidirect
4. #16810596C2semidirect
5. #16810380C2semidirect
6. #16810379C2semidirect
7. C53×#418C2semidirect
13. #13133213C4semidirect
14. #13133204C4semidirect
44. C2×C8×C53D4semidirect
57. C2×#88322D4semidirect
119. #418D53semidirect
123. C53#1910semidirect
165. C2×C8C22⋊D53semidirect