Finite Group Databasebeta

← #20287586

Subgroups — #20287586

Order 6784.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#16811333C211yesmaximal, characteristic
2#16811022C212yesmaximal, complemented
3#16811019C211yesmaximal, characteristic, complemented
4#16810930C212yesmaximal, complemented
5#16810920C212yesmaximal
6#16810918C211yesmaximal, characteristic, complemented
7#16810900C211yesmaximal, characteristic
8Q8×#88322C211yesmaximal, characteristic, complemented
9C4×#11719924C211yesmaximal, characteristic
10C4×#11719924C212yesmaximal
11C53×#557C211yesmaximal, characteristic, complemented
12C2×#11719924C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133209C2212yes—
15#13133195—22no—
16#13133193—21no—
17C4×D212C2211yescharacteristic
18C4×D212C2212yes—
19C4⋊C4×D53—21no—
20#13133150C2212yes—
21#13133150C2212yes—
22#13133148C2212yes—
23#13133148C2212yes—
24#13133148C2211yescharacteristic
25C4⋊#88322C2211yescharacteristic
26C4⋊#88322C2212yes—
27#13133141C2212yes—
28#13133141C2211yescharacteristic
29#13133137C2212yes—
30#13133137C2211yescharacteristic
31C4×#88322C2211yescharacteristic
32C4×#88322C2212yes—
33C4⋊Q8×C53—21no—
34C4⋊Q8×C53C2211yescharacteristic
35#125×C53—22no—
36#125×C53C2212yes—
37C4×Q8×C53—21no—
38C4×Q8×C53C2211yescharacteristic
39C4×Q8×C53C2211yescharacteristic
40C4×Q8×C53C2212yes—
41C2×Q8×C53—21no—
42C2×Q8×C53D411yescharacteristic
43C2×Q8×C53C2311yescharacteristic
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×C4×D53—22no—
47C2×Dic106—21no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53D411yescharacteristic
54C4⋊C4×C53D412yes—
55C4⋊C4×C53C2311yescharacteristic
56C4⋊C4×C53C2312yes—
57C42×C53—22no—
58C42×C53—21no—
59C42×C53C2312yes—
60C42×C53C2311yescharacteristic
61#11719924Q814yescomplemented
62#11719921—22no—
63#11719921—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53C2312yes—
66C4⋊Dic53C2311yescharacteristic
67#11719919—22no—
68#11719919—22no—
69C2×#88322C2311yescharacteristic
70C2×#88322C2312yes—
71C2×#88322C2311yescharacteristic
72C22×D53—21no—
73Q8×C53—22no—
74Q8×C53—21no—
75Q8×C53—42no—
76Q8×C53C2×Q812yes—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×D412yes—
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C2×Dic53—22no—
86C2×Dic53—21no—
87D212—21no—
88D212C2×Q812yes—
89C4×D53—42no—
90C4×D53—82no—
91Dic106—81no—
92#88322—21no—
93#88322—22no—
94#88322C2×Q812yes—
95C22×C53C22×D411yescharacteristic
96C2×D53—22no—
97C2×D53—41no—
98C2×D53—41no—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53#13512yes—
106C4×C53C22×Q811yescharacteristic
107C4×C532−1+411yescharacteristic
108Dic53—41no—
109Dic53—42no—
110#3381—531nomaximal, Sylow
111C2×C53D4×Q811yescharacteristic
112C2×C53C2×#13511yescharacteristic
113C2×C53#57711yescharacteristic
114D53—42no—
115#557D5311yescharacteristic, complemented
116#554—531no—
117#499—532no—
118#498—531no—
119#479—532no—
120#478—532no—
121#477—531no—
122#474—531no—
123C8×Q8—531no—
124C4×SD16—531no—
125C4×SD16—532no—
126C53#338111yescharacteristic, complemented, Sylow
127C2×SD16—531no—
128C4⋊Q8—21no—
129C4⋊Q8—531no—
130C4⋊Q8C2×D5311yescharacteristic
131#125—22no—
132#125—532no—
133#125C2×D5312yes—
134C22⋊Q8—1061no—
135C22⋊Q8—1062no—
136C4×Q8—21no—
137C4×Q8C2×D5311yescharacteristic
138C4×Q8C2×D5312yes—
139C4×Q8C2×D5311yescharacteristic
140C4×D4—531no—
141C4×D4—532no—
142C2×C4⋊C4—1061no—
143#107—532no—
144#107—532no—
145#106—532no—
146#106—531no—
147#106—532no—
148C4⋊C8—531no—
149C4⋊C8—532no—
150#103—532no—
151#103—531no—
152#102—532no—
153#102—531no—
154C4×C8—531no—
155C4×C8—532no—
156C2×Q8—21no—
157C2×Q8—1061no—
158C2×Q8C22⋊D5311yescharacteristic
159C2×Q8C22×D5311yescharacteristic
160C2×D4—531no—
161C22×C4—1062no—
162C22×C4—1061no—
163SD16—534no—
164C2×C8—531no—
165C2×C8—531no—
166C2×C8—532no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—532no—
173C4⋊C4—531no—
174C4⋊C4—1062no—
175C4⋊C4—1061no—
176C4⋊C4—1062no—
177C4⋊C4C22⋊D5311yescharacteristic
178C4⋊C4C22⋊D5312yes—
179C4⋊C4C22×D5312yes—
180C4⋊C4C22×D5311yescharacteristic
181C22⋊C4—1061no—
182C22⋊C4—1062no—
183C42—22no—
184C42—21no—
185C42C22×D5312yes—
186C42C22×D5311yescharacteristic
187C23—1061no—
188Q8—22no—
189Q8—21no—
190Q8—42no—
191Q8—4241no—
192Q8Q8×D5312yes—
193D4—532no—
194D4—1061no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—1061no—
199C2×C4—1062no—
200C2×C4—4242no—
201C2×C4—2122no—
202C2×C4C2×C22⋊D5312yes—
203C2×C4C2×C22⋊D5312yes—
204C2×C4C2×C22⋊D5311yescharacteristic
205C2×C4C2×C22⋊D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—532no—
208C8—1061no—
209C8—1062no—
210C22—1062no—
211C22—2121no—
212C22—2121no—
213C22C22×C22⋊D5311yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—2121no—
221C4—2122no—
222C4#1313323712yes—
223C4C2×Q8×D5311yescharacteristic
224C4#1313326511yescharacteristic
225C2—2122no—
226C2Q8×C22⋊D5311yescharacteristic
227C2C2×#1313323711yescharacteristic
228C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#557C2
socleC22×C53C22×D4
hypercenter#557D53
hypocenterC53#3381
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #16811022C2semidirect
3. #16811019C2semidirect
4. #16810930C2semidirect
6. #16810918C2semidirect
8. Q8×#88322C2semidirect
11. C53×#557C2semidirect
61. #11719924Q8semidirect
115. #557D53semidirect
126. C53#3381semidirect