Finite Group Databasebeta

← #20287082

Subgroups — #20287082

Order 6784.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811311C212yesmaximal
2#16810967C212yesmaximal, complemented
3#16810955C212yesmaximal, complemented
4#16810927C212yesmaximal, complemented
5#16810865C211yesmaximal, characteristic, complemented
6#16810865C211yesmaximal, characteristic, complemented
7C4×Dic212C211yesmaximal, characteristic, complemented
8C8×D212C211yesmaximal, characteristic
9#16810660C211yesmaximal, characteristic, complemented
10#107×D53C211yesmaximal, characteristic, complemented
11C53×#499C211yesmaximal, characteristic, complemented
12C2×Dic212C2211yescharacteristic
13C2×C8×D53C2211yescharacteristic
14C2×C8×D53C2211yescharacteristic
15#13133212—22no—
16#13133208—22no—
17#13133202—22no—
18#13133197C2212yes—
19#13133197C2212yes—
20C4×Dic106C2212yes—
21C4×D212C2211yescharacteristic
22#13133175—22no—
23#13133175C2212yes—
24C4⋊C4×D53C2212yes—
25#13133151C2211yescharacteristic
26#13133150C2212yes—
27#13133148C2212yes—
28C4⋊#88322C2211yescharacteristic
29#13133142C2212yes—
30#13133140C2212yes—
31#13133140C2212yes—
32#13133129C2211yescharacteristic
33#13133129C2211yescharacteristic
34#125×C53C2212yes—
35#107×C53C2211yescharacteristic
36#107×C53C2211yescharacteristic
37#106×C53C2212yes—
38C4×C8×C53C2211yescharacteristic
39C2×D212D411yescharacteristic
40C2×C4×D53—22no—
41C2×C4×D53C2311yescharacteristic
42C2×C4×D53C2311yescharacteristic
43C2×Dic106C2312yes—
44C2×C8×C53C2311yescharacteristic
45C2×C8×C53C2311yescharacteristic
46C4⋊C4×C53—22no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53C2312yes—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51#11719921—22no—
52#11719921—22no—
53#11719921—22no—
54#11719921D411yescharacteristic
55#11719921D411yescharacteristic
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53D411yescharacteristic
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63C4×Dic53—22no—
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66Dic212—22no—
67C8×D53—22no—
68C8×D53—22no—
69C22×D53C2×D411yescharacteristic
70C22×D53C2×D411yescharacteristic
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D411yescharacteristic
80D212—22no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—21no—
84C4×D53—21no—
85C4×D53—44no—
86Dic106—22no—
87Dic106—22no—
88C8×C53—21no—
89C8×C53#4112yes—
90#88322—21no—
91#88322—21no—
92C22×C53C22×D411yescharacteristic
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—21no—
98C2×D53#13512yes—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—42no—
102C4×C53—42no—
103C4×C53C2×#4111yescharacteristic
104C4×C532−1+411yescharacteristic
105Dic53—21no—
106Dic53—21no—
107Dic53—42no—
108Dic53—42no—
109#3336—531nomaximal, Sylow
110C2×C53#54711yescharacteristic
111C2×C53C2×#13511yescharacteristic
112C2×C53#57811yescharacteristic
113D53—22no—
114D53—41no—
115#552—532no—
116#499D5311yescharacteristic, complemented
117#485—532no—
118#483—532no—
119#479—532no—
120#467—531no—
121#467—531no—
122C4×Q16—531no—
123C8×D4—531no—
124#427—531no—
125C2×#107—531no—
126C53#333611yescharacteristic, complemented, Sylow
127C2×Q16—531no—
128C22×C8—531no—
129C22×C8—531no—
130#126—1062no—
131#125—1062no—
132#125C2×D5312yes—
133#123—1062no—
134C22⋊Q8—532no—
135C22⋊Q8—532no—
136C4×Q8—532no—
137C4×D4—531no—
138#117—532no—
139#117—1062no—
140C2×C4⋊C4—532no—
141#107—531no—
142#107—532no—
143#107C2×D5311yescharacteristic
144#107C2×D5311yescharacteristic
145#106—532no—
146#106C2×D5312yes—
147C4⋊C8—531no—
148#103—532no—
149#103—532no—
150#103—532no—
151C22⋊C8—531no—
152C22⋊C8—531no—
153C4×C8C2×D5311yescharacteristic
154C2×Q8—532no—
155C2×D4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—1062no—
159Q16—1062no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8—1062no—
163C2×C8—1062no—
164C2×C8C22×D5311yescharacteristic
165C2×C8C22×D5311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—531no—
169C4⋊C4—532no—
170C4⋊C4—1062no—
171C4⋊C4—1062no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4C22×D5312yes—
175C4⋊C4C22×D5312yes—
176C22⋊C4—531no—
177C22⋊C4—531no—
178C22⋊C4—1062no—
179C22⋊C4—1062no—
180C22⋊C4—1062no—
181C42—1062no—
182C42—1062no—
183C42C22×D5311yescharacteristic
184C23—531no—
185C23—531no—
186Q8—1062no—
187Q8—1062no—
188D4—1062no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—531no—
192C2×C4—531no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—2124no—
200C2×C4D4×D5311yescharacteristic
201C2×C4D4×D5311yescharacteristic
202C2×C4C23×D5311yescharacteristic
203C8—21no—
204C8—1061no—
205C8—1061no—
206C8#1171994912yes—
207C22—532no—
208C22—1061no—
209C22—1061no—
210C22—1061no—
211C22—1061no—
212C22—1061no—
213C22C2×D4×D5311yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—42no—
218C4—1061no—
219C4—1061no—
220C4—2122no—
221C4—2122no—
222C4C2×#1171994911yescharacteristic
223C4#1313326511yescharacteristic
224C2—1062no—
225C2—2121no—
226C2#1681127611yescharacteristic
227C2#135×D5311yescharacteristic
228C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#499C2
socleC22×C53C22×D4
hypercenter#499D53
hypocenterC53#3336
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. #16810967C2semidirect
3. #16810955C2semidirect
4. #16810927C2semidirect
5. #16810865C2semidirect
6. #16810865C2semidirect
7. C4×Dic212C2semidirect
9. #16810660C2semidirect
10. #107×D53C2semidirect
11. C53×#499C2semidirect
116. #499D53semidirect
126. C53#3336semidirect