Finite Group Databasebeta

← #20290697

Subgroups — #20290697

Order 6784.

376 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 72 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×D53⋊M16C211yesmaximal, characteristic, complemented
2C2×#13133282C212yesmaximal, complemented
3C4×C22⋊#6157C211yesmaximal, characteristic, complemented
4#16811493C212yesmaximal, complemented
5#16811136C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7D53⋊M16—24no—
8C2×#11719940—21no—
9C2×D53⋊C8—21no—
10C2×C22⋊#6157—21no—
11C2×C4×#6157C2212yes—
12#13133282—24no—
13(C2×#6157)⋊C4—21no—
14C2×#11719949C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133202—21no—
17#13133193C411yescharacteristic, complemented
18#13133192C411yescharacteristic, complemented
19#13133191C411yescharacteristic, complemented
20#13133187—21no—
21#13133179—21no—
22#13133177—21no—
23#13133175C2211yescharacteristic
24C22⋊Q8×C53C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—22no—
27C2×Q8×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C2×C22⋊D53—21no—
30C22×Dic53D411yescharacteristic
31#11719949—22no—
32#11719949—22no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212C2×C411yescharacteristic
36C2×C4×D53—22no—
37C2×C4×D53—22no—
38C2×C4×D53—21no—
39C2×C4×D53C2×C411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53C2311yescharacteristic
43C22⋊#6157—24no—
44#11719940—24no—
45C2×#88324—22no—
46C4×#6157—22no—
47C4×#6157—22no—
48#11719936—22no—
49#11719936D414yescomplemented
50D53⋊C8—24no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2×C411yescharacteristic
53C22⋊C4×C53—21no—
54#11719921—21no—
55#11719921—21no—
56#11719921—21no—
57#11719921—21no—
58C4⋊Dic53C2×C411yescharacteristic
59#11719919—21no—
60C4×Dic53—21no—
61C23×C53C22⋊C411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—21no—
65C22×D53—41no—
66C22×D53C22⋊C411yescharacteristic
67C22×D53C2×D411yescharacteristic
68C2×#6157—24no—
69C2×#6157—42no—
70C2×#6157—42no—
71Q8×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22×C411yescharacteristic
77C22⋊D53—42no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C2×D411yescharacteristic
83D212—22no—
84D212—42no—
85D212—42no—
86C4×D53—22no—
87C4×D53—22no—
88C4×D53—21no—
89C4×D53—21no—
90C4×D53—41no—
91C4×D53—41no—
92C4×D53C2×D412yes—
93C4×D53#4112yes—
94#88324—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53C2×C22⋊C411yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—44no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—41no—
107C2×D53#10412yescomplemented
108C2×D53#12112yes—
109#6157—44no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53C4×D412yes—
114Dic53—21no—
115Dic53—41no—
116Dic53#12011yescharacteristic
117Dic53#12311yescharacteristic
118#1871—531nomaximal, Sylow
119C2×C53—41no—
120C2×C53#38611yescharacteristic
121C2×C53C2×#10411yescharacteristic
122C2×C53#42111yescharacteristic
123D53—22no—
124D53—22no—
125D53—81no—
126C22×M16—531no—
127#525—531no—
128C2×#104—532no—
129C4×C22⋊C4—531no—
130#339—532no—
131C53#187111yescharacteristic, complemented, Sylow
132C2×#41—531no—
133C23×C4—531no—
134C2×M16—532no—
135C2×M16—1064no—
136C2×M16—1061no—
137C22×C8—1061no—
138#123—1061no—
139C22⋊Q8—531no—
140C22⋊Q8#615711yescharacteristic, complemented
141#121—531no—
142#121—531no—
143#120—1061no—
144C4×D4—1061no—
145C4×D4—1061no—
146#117—531no—
147C2×C22⋊C4—1061no—
148C2×C42—532no—
149#104—1064no—
150#95—1061no—
151C24—531no—
152#41—1062no—
153#41—1062no—
154C2×Q8C2×#615711yescharacteristic
155C2×D4—531no—
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4—1061no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—1062no—
165C22×C4—1062no—
166C22×C4—1061no—
167C22×C4—1062no—
168C22×C4C2×#615711yescharacteristic
169M16—534no—
170M16—1062no—
171M16—1064no—
172C2×C8—1062no—
173C2×C8—1064no—
174C4⋊C4—21no—
175C4⋊C4—531no—
176C4⋊C4—1061no—
177C4⋊C4C2×#615711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1064no—
183C22⋊C4—1061no—
184C42—1061no—
185C42—1062no—
186C42—1062no—
187C23—531no—
188C23—531no—
189C23—1062no—
190C23—1062no—
191C23—1061no—
192C23—2121no—
193C23C22⋊#615711yescharacteristic
194Q8—22no—
195D4—1062no—
196D4—2122no—
197D4—2122no—
198D4—2122no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—531no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—531no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1062no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—1064no—
213C2×C4—1061no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4C22⋊#615711yescharacteristic
219C2×C4C22×#615711yescharacteristic
220C8—1064no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—532no—
225C22—532no—
226C22—1061no—
227C22—1061no—
228C22—1061no—
229C22—1061no—
230C22—2124no—
231C22—2121no—
232C22—2121no—
233C22—2121no—
234C22C2×C22⋊#615711yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—531no—
239C4—531no—
240C4—1061no—
241C4—2124no—
242C4—2121no—
243C4D4×#615712yes—
244C2—41no—
245C2—1062no—
246C2—1062no—
247C2—4241no—
248C2#1681157111yescharacteristic
249C2D53⋊#10411yescharacteristic
250C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1871
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
1. C2×D53⋊M16C2semidirect
2. C2×#13133282C2semidirect
3. C4×C22⋊#6157C2semidirect
4. #16811493C2semidirect
17. #13133193C4semidirect
18. #13133192C4semidirect
19. #13133191C4semidirect
24. C22⋊Q8×C53C4semidirect
49. #11719936D4semidirect
107. C2×D53#104semidirect
131. C53#1871semidirect
140. C22⋊Q8#6157semidirect