Finite Group Databasebeta

← #20290701

Subgroups — #20290701

Order 6784.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×#13133296C211yesmaximal, characteristic
2C2×#13133280C212yesmaximal, complemented
3D53⋊(C22⋊C8)C211yesmaximal, characteristic
4#16811496C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C2×D53⋊C8C2212yes—
7#13133296—24no—
8C2×C4⋊#6157C2212yes—
9C2×C22⋊#6157—21no—
10C2×C4×#6157—21no—
11#13133280—24no—
12#13133274—21no—
13C22⋊#88324—21no—
14C2×Q8×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133197C411yescharacteristic, complemented
17#13133196C411yescharacteristic, complemented
18#13133195—21no—
19#13133194—21no—
20#13133193C411yescharacteristic, complemented
21C4⋊C4×D53—21no—
22C4⋊C4×D53C2211yescharacteristic
23C22⋊C4×D53—21no—
24C22⋊Q8×C53C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—22no—
27C2×Q8×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C22×Dic53D411yescharacteristic
30Q8×D53—22no—
31Q8×D53—22no—
32C2×C4×D53—22no—
33C2×C4×D53—22no—
34C2×C4×D53—21no—
35C2×C4×D53—21no—
36C2×C4×D53C2×C411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53C2311yescharacteristic
40C2×Dic106—21no—
41C2×Dic106C2×C411yescharacteristic
42C22⋊#6157—24no—
43C2×#88324—22no—
44C4⋊#6157—22no—
45C4⋊#6157D414yes—
46C4×#6157—24no—
47D53⋊C8—22no—
48D53⋊C8—22no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53C2×C411yescharacteristic
51C22⋊C4×C53—21no—
52C22⋊Dic53—21no—
53#11719921—21no—
54#11719921—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53C2×C411yescharacteristic
57#11719919—21no—
58#11719919—21no—
59#11719919—21no—
60C23×C53C22⋊C411yescharacteristic
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53C22⋊C411yescharacteristic
66C22×D53C2×D411yescharacteristic
67C2×#6157—24no—
68C2×#6157—42no—
69C2×#6157—42no—
70Q8×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53C22⋊C411yescharacteristic
75C2×C4×C53C22×C411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53C22⋊C411yescharacteristic
80C2×Dic53C2×D411yescharacteristic
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—41no—
86C4×D53—41no—
87C4×D53—42no—
88C4×D53C2×D412yes—
89C4×D53#4112yes—
90Dic106—22no—
91Dic106—42no—
92#88324—42no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53C2×C22⋊C411yescharacteristic
97C2×D53—21no—
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53#12011yescharacteristic
103C2×D53#12311yescharacteristic
104C2×D53#13512yes—
105C2×D53#13712yes—
106#6157—44no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53C4×D412yes—
111Dic53—21no—
112Dic53—41no—
113Dic53—41no—
114Dic53#12112yes—
115#1873—531nomaximal, Sylow
116C2×C53—41no—
117C2×C53#38611yescharacteristic
118C2×C53#41611yescharacteristic
119C2×C53#41911yescharacteristic
120D53—41no—
121D53#45212yes—
122D53#48512yes—
123C2×C22⋊Q8—531no—
124C2×#117—531no—
125C2×#103—532no—
126C2×C22⋊C8—531no—
127#340—532no—
128C53#187311yescharacteristic, complemented, Sylow
129C22×Q8—531no—
130C23×C4—531no—
131C22×C8—532no—
132C22⋊Q8—531no—
133C22⋊Q8—531no—
134C22⋊Q8—531no—
135C22⋊Q8—1061no—
136C22⋊Q8—1061no—
137C22⋊Q8#615711yescharacteristic, complemented
138#117—1064no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—1061no—
142C2×C22⋊C4—1061no—
143C2×C22⋊C4—1061no—
144C2×C42—1061no—
145#103—1064no—
146C22⋊C8—1061no—
147C22⋊C8—1061no—
148C24—531no—
149C2×Q8—531no—
150C2×Q8—1061no—
151C2×Q8—1062no—
152C2×Q8—1062no—
153C2×Q8C2×#615711yescharacteristic
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—1061no—
162C22×C4—1061no—
163C22×C4—1062no—
164C22×C4C2×#615711yescharacteristic
165C2×C8—1062no—
166C2×C8—1062no—
167C2×C8—1062no—
168C4⋊C4—21no—
169C4⋊C4—534no—
170C4⋊C4—531no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4—1062no—
175C4⋊C4—1061no—
176C4⋊C4C2×#615711yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1064no—
181C22⋊C4—1061no—
182C42—1064no—
183C23—531no—
184C23—531no—
185C23—1061no—
186C23—1061no—
187C23—1062no—
188C23—1062no—
189C23C22⋊#615711yescharacteristic
190Q8—22no—
191Q8—1062no—
192Q8—2122no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—531no—
197C2×C4—532no—
198C2×C4—532no—
199C2×C4—531no—
200C2×C4—1062no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1064no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—2122no—
209C2×C4—2121no—
210C2×C4—2121no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4C22⋊#615711yescharacteristic
214C2×C4C22×#615711yescharacteristic
215C8—2122no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—531no—
220C22—531no—
221C22—532no—
222C22—532no—
223C22—1061no—
224C22—1061no—
225C22—1061no—
226C22—2122no—
227C22—2122no—
228C22C2×C22⋊#615711yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—532no—
233C4—1061no—
234C4—2124no—
235C4—2121no—
236C4—2121no—
237C4D4×#615712yes—
238C2—41no—
239C2—532no—
240C2—532no—
241C2—2121no—
242C2#1681157111yescharacteristic
243C2#1681161311yescharacteristic
244C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1873
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13133280C2semidirect
4. #16811496C2semidirect
16. #13133197C4semidirect
17. #13133196C4semidirect
20. #13133193C4semidirect
24. C22⋊Q8×C53C4semidirect
128. C53#1873semidirect
137. C22⋊Q8#6157semidirect