Finite Group Databasebeta

← #20290698

Subgroups — #20290698

Order 6784.

376 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 74 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×#11719936C211yesmaximal, characteristic, complemented
2C2×#13133281C212yesmaximal, complemented
3C4×C22⋊#6157C211yesmaximal, characteristic, complemented
4#16811493C212yesmaximal, complemented
5#16811133C211yesmaximal, characteristic
6C2×#11719936—21no—
7C2×#11719936—24no—
8C2×#11719936C2212yes—
9C22×#88324—21no—
10C2×C22⋊#6157—21no—
11C2×C4×#6157C2212yes—
12#13133281—24no—
13(C2×#6157)⋊C4—21no—
14C2×#11719947C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133199—21no—
17#13133197C411yescharacteristic, complemented
18#13133196C411yescharacteristic, complemented
19#13133188C411yescharacteristic, complemented
20#13133185—21no—
21D4×Dic53—21no—
22#13133177—21no—
23#13133175C2211yescharacteristic
24#121×C53C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—22no—
27C2×D4×C53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34#11719947—22no—
35#11719947—22no—
36C2×C4×D53—22no—
37C2×C4×D53—22no—
38C2×C4×D53C2×C411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×Dic106C2×C411yescharacteristic
43C22⋊#6157—24no—
44C2×#88324—24no—
45C2×#88324—22no—
46C4×#6157—22no—
47C4×#6157—22no—
48#11719936—24no—
49#11719936—22no—
50#11719936D414yescomplemented
51C4⋊C4×C53C2×C411yescharacteristic
52C22⋊C4×C53—21no—
53C22⋊Dic53—21no—
54C22⋊Dic53—21no—
55#11719921—21no—
56#11719921—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2×C411yescharacteristic
59#11719919—21no—
60C4×Dic53—21no—
61C23×C53—21no—
62C23×C53C22⋊C411yescharacteristic
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—41no—
66C22×D53C22⋊C411yescharacteristic
67C22×D53C2×D411yescharacteristic
68C2×#6157—24no—
69C2×#6157—42no—
70C2×#6157—42no—
71D4×C53—22no—
72D4×C53—42no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22×C411yescharacteristic
77C22⋊D53—42no—
78C22⋊D53—42no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53C22⋊C411yescharacteristic
86C2×Dic53C2×D411yescharacteristic
87C4×D53—22no—
88C4×D53—22no—
89C4×D53—21no—
90C4×D53—21no—
91C4×D53C2×D412yes—
92C4×D53#4112yes—
93Dic106—22no—
94#88324—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53#10412yescomplemented
101C22×C53C2×C22⋊C411yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—21no—
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—44no—
108C2×D53#12112yes—
109#6157—44no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53C4×D412yes—
113Dic53—21no—
114Dic53—41no—
115Dic53—41no—
116Dic53#12011yescharacteristic
117Dic53#12311yescharacteristic
118#1871—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53—81no—
121C2×C53#38611yescharacteristic
122C2×C53C2×#10411yescharacteristic
123C2×C53#42111yescharacteristic
124D53—22no—
125D53—41no—
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—531no—
141#121—531no—
142#121#615711yescharacteristic, complemented
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×Q8—531no—
155C2×D4—21no—
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4C2×#615711yescharacteristic
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—531no—
175C4⋊C4—1061no—
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—21no—
188C23—531no—
189C23—531no—
190C23—1062no—
191C23—1062no—
192C23—2121no—
193C23C22⋊#615711yescharacteristic
194Q8—1062no—
195D4—22no—
196D4—42no—
197D4—2122no—
198D4—2122no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—531no—
202C2×C4—532no—
203C2×C4—532no—
204C2×C4—531no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1062no—
211C2×C4—1064no—
212C2×C4—1061no—
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—41no—
224C22—41no—
225C22—41no—
226C22—532no—
227C22—1061no—
228C22—1061no—
229C22—1061no—
230C22—1061no—
231C22—1061no—
232C22—2124no—
233C22#1313328112yescomplemented
234C22C2×C22⋊#615711yescharacteristic
235C4—21no—
236C4—41no—
237C4—531no—
238C4—531no—
239C4—1061no—
240C4—2124no—
241C4—2121no—
242C4—2121no—
243C4D4×#615712yes—
244C2—22no—
245C2—81no—
246C2—1062no—
247C2—2121no—
248C2#1681157111yescharacteristic
249C2C2×#1313328111yescharacteristic
250C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1871
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×#11719936C2semidirect
2. C2×#13133281C2semidirect
3. C4×C22⋊#6157C2semidirect
4. #16811493C2semidirect
17. #13133197C4semidirect
18. #13133196C4semidirect
19. #13133188C4semidirect
24. #121×C53C4semidirect
50. #11719936D4semidirect
100. C22×C53#104semidirect
131. C53#1871semidirect
142. #121#6157semidirect
233. C22#13133281semidirect