Finite Group Databasebeta

← #20275207

Subgroups — #20275207

Order 6784.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#13133248C211yesmaximal, characteristic
2#16811101C211yesmaximal, characteristic, complemented
3#16810639C211yesmaximal, characteristic
4#16810596C211yesmaximal, characteristic
5#16810395C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#419C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—22no—
10#13133248—22no—
11C2×#11719925C412yescomplemented
12C2×#11719924C412yescomplemented
13C2×#11719917C2211yescharacteristic
14Q8×Dic53—22no—
15Q8×Dic53—22no—
16#13133176—22no—
17#13133176C2211yescharacteristic
18#13133176C2211yescharacteristic
19#13133143—22no—
20#13133131—22no—
21C22×Q8×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23#117×C53C2211yescharacteristic
24#103×C53—22no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53C2×C412yes—
29C2×Q8×C53D412yes—
30C2×Q8×C53D412yes—
31C22×C4×C53—21no—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34#11719945—21no—
35#11719945—21no—
36#11719945—21no—
37#11719945—21no—
38C2×D212C2×C411yescharacteristic
39C2×C4×D53—21no—
40C2×Dic106C2×C411yescharacteristic
41M16×C53—21no—
42M16×C53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C42×C53—21no—
47C22⋊Dic53—21no—
48C22⋊Dic53D412yes—
49#11719925—24no—
50#11719924—24no—
51C4⋊Dic53—24no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C4×Dic53—24no—
55C4×Dic53—21no—
56C4×Dic53D412yes—
57#11719917—21no—
58#11719917—21no—
59C2×#88322C2×C412yes—
60C23×C53C2×D411yescharacteristic
61C22×D53—21no—
62Q8×C53—22no—
63Q8×C53—22no—
64Q8×C53—22no—
65Q8×C53C22⋊C414yes—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—41no—
70C2×C4×C53C22×C411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53#4111yescharacteristic
75C2×C4×C53#4111yescharacteristic
76C22⋊D53—41no—
77C22⋊D53—41no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81D212—21no—
82D212—21no—
83C4×D53—41no—
84C4×D53—41no—
85Dic106—21no—
86Dic106—21no—
87C8×C53—41no—
88#88322—22no—
89C22×C53—41no—
90C22×C53C4×D411yescharacteristic
91C22×C53#12011yescharacteristic
92C22×C53#12111yescharacteristic
93C2×D53—41no—
94C2×D53—81no—
95C4×C53—24no—
96C4×C53—81no—
97C4×C53C2×C22⋊C411yescharacteristic
98C4×C53C4×D411yescharacteristic
99C4×C53#12111yescharacteristic
100C4×C53#12311yescharacteristic
101Dic53—42no—
102Dic53—42no—
103Dic53—41no—
104#1905—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—21no—
107C2×C53—21no—
108C2×C53#38611yescharacteristic
109D53—81no—
110C2×#137—531no—
111#519—531no—
112#421—531no—
113#419D5311yescharacteristic, complemented
114#410—531no—
115#344—531no—
116H(3,ℤ/4)—531no—
117C53#190511yescharacteristic, complemented, Sylow
118C2×#41—531no—
119C22×Q8C2×D5311yescharacteristic
120#137—1062no—
121#137—1062no—
122C2×Q16—532no—
123C2×SD16—532no—
124C2×M16—531no—
125C2×M16C2×D5311yescharacteristic
126C4×Q8—1062no—
127C4×Q8—1062no—
128#117—531no—
129#117—531no—
130#117—1062no—
131#117C2×D5311yescharacteristic
132#104—1062no—
133#103—22no—
134#99—1062no—
135#41—1061no—
136#41—1061no—
137#41—1061no—
138#41—1061no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—531no—
143C2×Q8C4×D5312yes—
144C2×Q8D21212yes—
145C2×Q8C22⋊D5312yes—
146C2×D4—531no—
147C2×D4—1061no—
148C22×C4—21no—
149C22×C4—1061no—
150C22×C4C22×D5311yescharacteristic
151Q16—1064no—
152SD16—1064no—
153M16—21no—
154M16—21no—
155M16—1061no—
156M16—1061no—
157C2×C8—21no—
158C2×C8—532no—
159C4⋊C4—21no—
160C4⋊C4—1064no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C22⋊C4—21no—
164C22⋊C4—532no—
165C22⋊C4—1061no—
166C42—21no—
167C42—532no—
168C42—1064no—
169C42—1061no—
170C23—1061no—
171C23D4×D5311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8—1061no—
176Q8—1061no—
177Q8#1171992114yes—
178D4—1061no—
179D4—1061no—
180D4—2121no—
181D4—2121no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—1061no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4C2×C4×D5311yescharacteristic
192C2×C4C2×D21211yescharacteristic
193C2×C4D4×D5311yescharacteristic
194C2×C4#1171994711yescharacteristic
195C2×C4#1171994711yescharacteristic
196C2×C4C2×C22⋊D5311yescharacteristic
197C8—41no—
198C8—1062no—
199C22—41no—
200C22—4241no—
201C22—2121no—
202C22#1313317711yescharacteristic
203C22#1313318711yescharacteristic
204C22#1313318911yescharacteristic
205C4—24no—
206C4—81no—
207C4—2122no—
208C4—2122no—
209C4—2121no—
210C4C2×#1171992111yescharacteristic
211C4#1313317711yescharacteristic
212C4#1313318911yescharacteristic
213C4#1313320111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—4241no—
218C2#1681050011yescharacteristic
subgroupG/H
centerC2#16810500
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#419C2
socleC2×C53#386
hypercenter#419D53
hypocenterC53#1905
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #16811101C2semidirect
5. #16810395C2semidirect
6. #16810379C2semidirect
7. C53×#419C2semidirect
11. C2×#11719925C4semidirect
12. C2×#11719924C4semidirect
113. #419D53semidirect
117. C53#1905semidirect