Finite Group Databasebeta

← #20287806

Subgroups — #20287806

Order 6784.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811355C212yesmaximal, complemented
2#16811009C212yesmaximal, complemented
3#16810847C212yesmaximal, complemented
4#16810835C212yesmaximal, complemented
5#16810816C212yesmaximal, complemented
6#16810753C211yesmaximal, characteristic, complemented
7#16810724C212yesmaximal, complemented
8C4×D424C211yesmaximal, characteristic, complemented
9C53×#495C211yesmaximal, characteristic, complemented
10C2×#11719924C2212yes—
11C2×D424C2211yescharacteristic
12C2×#11719922C2212yes—
13#13133213C2212yes—
14#13133206C2212yes—
15#13133203—22no—
16#13133201—22no—
17#13133199—22no—
18Q8×Dic53C2212yes—
19C4×Dic106C2211yescharacteristic
20C4×D212C2212yes—
21D4×Dic53C2212yes—
22#13133177—22no—
23#13133151C2211yescharacteristic
24C4⋊#88322C2211yescharacteristic
25#13133146C2212yes—
26#13133141C2212yes—
27#13133139C2212yes—
28#13133138C2212yes—
29#13133127C2212yes—
30C2×Q16×C53C2211yescharacteristic
31C2×SD16×C53C2212yes—
32C2×D8×C53C2211yescharacteristic
33#124×C53C2212yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53C2312yes—
36C2×D4×C53C2312yes—
37C2×C22⋊D53—22no—
38C2×C22⋊D53—22no—
39C22×Dic53—22no—
40C2×D212C2312yes—
41C2×C4×D53—22no—
42C2×Dic106D411yescharacteristic
43Q16×C53—41no—
44SD16×C53—24no—
45D8×C53—41no—
46C2×C8×C53C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C22⋊C4×C53—22no—
49C22⋊C4×C53—22no—
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52#11719924—42no—
53#11719922—42no—
54#11719921—22no—
55#11719921—22no—
56#11719921—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53D411yescharacteristic
59C4⋊Dic53C2312yes—
60#11719919D412yes—
61C4×Dic53—22no—
62C4×Dic53C2312yes—
63C2×#88322C2312yes—
64D424—22no—
65C23×C53—22no—
66C22×D53—22no—
67Q8×C53—22no—
68Q8×C53—22no—
69D4×C53—22no—
70D4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2411yescharacteristic
75C22⋊D53—44no—
76C22⋊D53—82no—
77C2×Dic53—22no—
78C2×Dic53—82no—
79C2×Dic53C2×D412yes—
80C2×Dic53C2×D412yes—
81D212—22no—
82D212—22no—
83C4×D53—82no—
84Dic106—41no—
85C8×C53—21no—
86C8×C53#4112yes—
87#88322—22no—
88C22×C53—42no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53C22×D411yescharacteristic
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—42no—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—42no—
98C4×C53C2×#4111yescharacteristic
99C4×C53H(5,𝔽2)11yescharacteristic
100Dic53—22no—
101Dic53—22no—
102Dic53—42no—
103#3403—531nomaximal, Sylow
104C2×C53—82no—
105C2×C53#55011yescharacteristic
106C2×C53#57511yescharacteristic
107C2×C53#57611yescharacteristic
108D53—82no—
109#559—532no—
110#496—532no—
111#495D5311yescharacteristic, complemented
112#464—532no—
113#463—532no—
114#459—532no—
115#446—531no—
116#440—532no—
117C4×D8—531no—
118C53#340311yescharacteristic, complemented, Sylow
119C2×Q16C2×D5311yescharacteristic
120C2×SD16—532no—
121C2×SD16C2×D5312yes—
122C2×D8—532no—
123C2×D8—531no—
124C2×D8C2×D5311yescharacteristic
125#127—532no—
126#124—532no—
127#124—1062no—
128#124C2×D5312yes—
129#123—1062no—
130#123—1062no—
131C4×Q8—531no—
132C4×Q8—532no—
133C4×D4—532no—
134C4×D4—532no—
135C4×D4—1062no—
136#107—531no—
137C4⋊C8—532no—
138C4⋊C8—531no—
139#103—532no—
140#102—532no—
141#102—532no—
142#97—532no—
143C4×C8C2×D5311yescharacteristic
144C2×Q8—531no—
145C2×Q8C22×D5312yes—
146C2×D4—532no—
147C2×D4—1062no—
148C2×D4—1062no—
149C2×D4C22×D5312yes—
150C22×C4—1062no—
151C22×C4—1062no—
152Q16—41no—
153SD16—24no—
154SD16—2122no—
155D8—41no—
156D8—1062no—
157D8—2122no—
158C2×C8—532no—
159C2×C8C22×D5311yescharacteristic
160C2×C8C22×D5311yescharacteristic
161C4⋊C4—532no—
162C4⋊C4—531no—
163C4⋊C4—532no—
164C4⋊C4—1062no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—1062no—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—532no—
172C42—1062no—
173C42C22×D5311yescharacteristic
174C23—22no—
175C23—1062no—
176Q8—22no—
177Q8—22no—
178Q8—2121no—
179D4—22no—
180D4—22no—
181D4—1062no—
182D4—1062no—
183D4—4242no—
184D4—2124no—
185C2×C4—22no—
186C2×C4—532no—
187C2×C4—532no—
188C2×C4—1062no—
189C2×C4—4242no—
190C2×C4—4242no—
191C2×C4D4×D5311yescharacteristic
192C2×C4D4×D5311yescharacteristic
193C2×C4C23×D5311yescharacteristic
194C8—21no—
195C8—1062no—
196C8#1171994712yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—2122no—
201C22—2122no—
202C22—2122no—
203C22C2×D4×D5311yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—1062no—
208C4—1062no—
209C4—2122no—
210C4C2×#1171994711yescharacteristic
211C4#1313326211yescharacteristic
212C2—82no—
213C2—4242no—
214C2#1681129311yescharacteristic
215C2#1681142011yescharacteristic
216C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#495C2
socleC22×C53C22×D4
hypercenter#495D53
hypocenterC53#3403
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #16811355C2semidirect
2. #16811009C2semidirect
3. #16810847C2semidirect
4. #16810835C2semidirect
5. #16810816C2semidirect
6. #16810753C2semidirect
7. #16810724C2semidirect
8. C4×D424C2semidirect
9. C53×#495C2semidirect
111. #495D53semidirect
118. C53#3403semidirect