Finite Group Databasebeta

← #20290700

Subgroups — #20290700

Order 6784.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#6157C211yesmaximal, characteristic
2C2×#13133280C212yesmaximal, complemented
3D53⋊(C22⋊C8)C211yesmaximal, characteristic
4#16811496C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C23×#6157—21no—
7C2×D53⋊C8C2212yes—
8C2×C4⋊#6157—21no—
9C2×C4⋊#6157—24no—
10C2×C4⋊#6157C2212yes—
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—24no—
27C22×#6157—22no—
28C22×#6157—42no—
29C22×#6157—42no—
30C2×Q8×C53C2×C411yescharacteristic, complemented
31C22×C4×C53C2×C411yescharacteristic
32C22×Dic53D411yescharacteristic
33Q8×D53—22no—
34Q8×D53—22no—
35C2×C4×D53—22no—
36C2×C4×D53—22no—
37C2×C4×D53—21no—
38C2×C4×D53—21no—
39C2×C4×D53C2×C411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53C2311yescharacteristic
43C2×Dic106—21no—
44C2×Dic106C2×C411yescharacteristic, complemented
45C2×#88324—22no—
46C4⋊#6157—24no—
47C4⋊#6157—22no—
48C4⋊#6157D414yes—
49D53⋊C8—22no—
50D53⋊C8—22no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2×C411yescharacteristic, complemented
53C22⋊C4×C53—21no—
54C22⋊Dic53—21no—
55#11719921—21no—
56#11719921—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2×C411yescharacteristic, complemented
59#11719919—21no—
60#11719919—21no—
61#11719919—21no—
62C23×C53C22⋊C411yescharacteristic
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—21no—
67C22×D53C22⋊C411yescharacteristic
68C22×D53SD1612yes—
69C22×D53Q1612yes—
70C22×D53C2×D411yescharacteristic
71C2×#6157—24no—
72C2×#6157—48no—
73C2×#6157—42no—
74C2×#6157—42no—
75Q8×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53C22⋊C411yescharacteristic
80C2×C4×C53C22×C411yescharacteristic
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53C22⋊C411yescharacteristic
85C2×Dic53C2×D411yescharacteristic
86C4×D53—21no—
87C4×D53—21no—
88C4×D53—22no—
89C4×D53—22no—
90C4×D53—41no—
91C4×D53—41no—
92C4×D53—41no—
93C4×D53—41no—
94C4×D53C2×D412yes—
95C4×D53#4112yes—
96Dic106—22no—
97Dic106—42no—
98#88324—42no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53#10312yescomplemented
102C22×C53C2×C22⋊C411yescharacteristic
103C2×D53—21no—
104C2×D53—21no—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—22no—
109C2×D53#10312yescomplemented
110C2×D53#12011yescharacteristic
111C2×D53#12311yescharacteristic
112C2×D53C2×SD1611yescharacteristic
113C2×D53C2×Q1611yescharacteristic
114C2×D53AGL(1,ℤ/8)11yescharacteristic
115C2×D53#13711yescharacteristic
116#6157—44no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53C4×D412yes—
121Dic53—21no—
122Dic53—41no—
123Dic53—41no—
124Dic53#12112yes—
125#1872—531nomaximal, Sylow
126C2×C53—22no—
127C2×C53#38611yescharacteristic
128C2×C53C2×#10311yescharacteristic
129C2×C53#41711yescharacteristic
130D53—22no—
131D53#45011yescharacteristic
132D53#45111yescharacteristic
133D53#48311yescharacteristic
134D53#48411yescharacteristic
135C2×C22⋊Q8—531no—
136C22×C4⋊C4—531no—
137C2×#103—532no—
138C2×C22⋊C8—531no—
139#340—532no—
140C53#187211yescharacteristic, complemented, Sylow
141C22×Q8—531no—
142C23×C4—531no—
143C23×C4—1061no—
144C22×C8—532no—
145C22⋊Q8—531no—
146C22⋊Q8—531no—
147C22⋊Q8—531no—
148C22⋊Q8—1061no—
149C22⋊Q8—1061no—
150C22⋊Q8#615711yescharacteristic, complemented
151C2×C4⋊C4—532no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—1061no—
154C2×C4⋊C4—1064no—
155C2×C4⋊C4—1061no—
156C2×C22⋊C4—1061no—
157#103—1064no—
158C22⋊C8—1061no—
159C22⋊C8—1061no—
160C24—531no—
161C2×Q8—531no—
162C2×Q8—1061no—
163C2×Q8—1062no—
164C2×Q8—1062no—
165C2×Q8C2×#615711yescharacteristic, complemented
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—1061no—
174C22×C4—1061no—
175C22×C4—1062no—
176C22×C4—1064no—
177C22×C4—2122no—
178C22×C4—2122no—
179C22×C4C2×#615711yescharacteristic
180C2×C8—1062no—
181C2×C8—1062no—
182C2×C8—1062no—
183C4⋊C4—21no—
184C4⋊C4—534no—
185C4⋊C4—531no—
186C4⋊C4—1061no—
187C4⋊C4—1061no—
188C4⋊C4—1061no—
189C4⋊C4—1062no—
190C4⋊C4—1064no—
191C4⋊C4—1061no—
192C4⋊C4C2×#615711yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—1061no—
195C22⋊C4—1061no—
196C22⋊C4—1061no—
197C23—531no—
198C23—532no—
199C23—532no—
200C23—531no—
201C23—1061no—
202C23—1061no—
203C23—1061no—
204C23—1061no—
205C23C22⋊#615711yescharacteristic
206Q8—22no—
207Q8—1062no—
208Q8—2122no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—531no—
213C2×C4—532no—
214C2×C4—532no—
215C2×C4—531no—
216C2×C4—1062no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1064no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—2121no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2128no—
231C2×C4C22⋊#615711yescharacteristic
232C2×C4C22×#615711yescharacteristic
233C8—2122no—
234C22—21no—
235C22—21no—
236C22—531no—
237C22—531no—
238C22—531no—
239C22—531no—
240C22—531no—
241C22—531no—
242C22—532no—
243C22—1062no—
244C22—1062no—
245C22—1062no—
246C22—1062no—
247C22—1061no—
248C22—1061no—
249C22#1313328012yescomplemented
250C22C2×C22⋊#615711yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—532no—
255C4—1061no—
256C4—2124no—
257C4—2121no—
258C4—2121no—
259C4D4×#615712yes—
260C2—22no—
261C2—531no—
262C2—531no—
263C2—531no—
264C2—531no—
265C2—1062no—
266C2#1681157111yescharacteristic
267C2C2×#1313328011yescharacteristic
268C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1872
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
30. C2×Q8×C53C2×C4semidirect
44. C2×Dic106C2×C4semidirect
52. C4⋊C4×C53C2×C4semidirect
58. C4⋊Dic53C2×C4semidirect
101. C22×C53#103semidirect
109. C2×D53#103semidirect
140. C53#1872semidirect
150. C22⋊Q8#6157semidirect
165. C2×Q8C2×#6157semidirect
192. C4⋊C4C2×#6157semidirect
249. C22#13133280semidirect