Finite Group Databasebeta

← #20290880

Subgroups — #20290880

Order 6784.

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

normal subgroups only

This group has 336 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×#13133282C212yesmaximal, complemented
2C2×#13133281C212yesmaximal, complemented
3C4×#11719936C211yesmaximal, characteristic, complemented
4C4×C4⋊#6157C211yesmaximal, characteristic, complemented
5#16811172C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7C2×C4⋊#6157—21no—
8C2×C4×#6157C2212yes—
9#13133282—24no—
10#13133281—24no—
11#13133272—21no—
12C4×#88324—21no—
13(C2×#6157)⋊C4—21no—
14C2×#11719949C2211yescharacteristic
15C2×#11719947C2211yescharacteristic
16#13133215C411yescharacteristic, complemented
17#13133213C411yescharacteristic, complemented
18#13133205C411yescharacteristic, complemented
19#13133190—21no—
20#13133190—21no—
21#13133177—21no—
22#13133177—21no—
23C42×D53C2211yescharacteristic
24#124×C53C411yescharacteristic, complemented
25C22×#6157—22no—
26C2×Q8×C53C2×C411yescharacteristic
27C2×D4×C53C2×C411yescharacteristic
28C2×C22⋊D53—21no—
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32#11719949—22no—
33#11719949—22no—
34#11719947—22no—
35#11719947—22no—
36C2×D212—21no—
37C2×D212C2×C411yescharacteristic
38C2×C4×D53—21no—
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×Dic106C2×C411yescharacteristic
43C2×#88324—22no—
44C4⋊#6157—24no—
45C4×#6157—22no—
46C4×#6157—22no—
47#11719936—22no—
48#11719936D414yescomplemented
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C42×C53C2×C411yescharacteristic
52#11719921—21no—
53#11719921—21no—
54#11719919—21no—
55#11719919—21no—
56C4×Dic53C2×C411yescharacteristic
57C4×Dic53D411yescharacteristic
58C4×Dic53D411yescharacteristic
59C23×C53—21no—
60C22×D53—21no—
61C22×D53C2×D411yescharacteristic
62C2×#6157—24no—
63C2×#6157—42no—
64C2×#6157—42no—
65Q8×C53—22no—
66D4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C22×C411yescharacteristic
71C22⋊D53—42no—
72C22⋊D53—41no—
73C22⋊D53—41no—
74C22⋊D53—42no—
75C2×Dic53—21no—
76C2×Dic53—42no—
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C22⋊C411yescharacteristic
79C2×Dic53C2×D411yescharacteristic
80D212—22no—
81D212—41no—
82D212—41no—
83C4×D53—22no—
84C4×D53—41no—
85C4×D53—42no—
86C4×D53C2×D412yes—
87C4×D53#4112yes—
88Dic106—22no—
89#88324—24no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53C2×C22⋊C411yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—41no—
99C2×D53#12411yescharacteristic
100C2×D53#12711yescharacteristic
101#6157—44no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53C4×D412yes—
106Dic53—21no—
107Dic53—41no—
108Dic53#10412yescomplemented
109Dic53#12112yes—
110#1980—531nomaximal, Sylow
111C2×C53—81no—
112C2×C53#39011yescharacteristic
113C2×C53C2×#10411yescharacteristic
114C2×C53#42111yescharacteristic
115D53—21no—
116D53—21no—
117D53—81no—
118#532—531no—
119C2×#104—532no—
120C2×#104—532no—
121C4×M16—531no—
122C4×C4⋊C4—531no—
123C53#198011yescharacteristic, complemented, Sylow
124C2×#41—531no—
125C2×#41—531no—
126C2×M16—532no—
127C4⋊Q8—531no—
128#127—531no—
129#124—531no—
130#124#615711yescharacteristic, complemented
131#121—1061no—
132#121—1061no—
133C4×D4—1061no—
134C4×D4—1061no—
135C2×C4⋊C4—1061no—
136C2×C42—531no—
137C2×C42—532no—
138#104—1064no—
139#104—1064no—
140#97—1061no—
141C4×C8—1061no—
142#95—1061no—
143#41—1062no—
144#41—1062no—
145#41—1062no—
146#41—1062no—
147C2×Q8—531no—
148C2×Q8C2×#615711yescharacteristic
149C2×D4—531no—
150C2×D4—1061no—
151C2×D4—1061no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4C2×#615711yescharacteristic
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—1061no—
159C22×C4—1061no—
160C22×C4—1062no—
161M16—534no—
162M16—1062no—
163C2×C8—1062no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1064no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C42—531no—
172C42—531no—
173C42—531no—
174C42—1062no—
175C42—1062no—
176C42C2×#615711yescharacteristic
177C23—21no—
178C23—531no—
179C23—1061no—
180Q8—22no—
181Q8—1062no—
182D4—22no—
183D4—1062no—
184D4—2121no—
185D4—2121no—
186D4—2122no—
187D4—2122no—
188D4—2121no—
189D4—2121no—
190C2×C4—21no—
191C2×C4—531no—
192C2×C4—531no—
193C2×C4—532no—
194C2×C4—532no—
195C2×C4—531no—
196C2×C4—1062no—
197C2×C4—1064no—
198C2×C4—1061no—
199C2×C4—2121no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4C22⋊#615711yescharacteristic
205C2×C4C22⋊#615711yescharacteristic
206C2×C4C22×#615711yescharacteristic
207C8—1064no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—531no—
212C22—531no—
213C22—1061no—
214C22—1061no—
215C22—2121no—
216C22—2121no—
217C22—2121no—
218C22C2×C22⋊#615711yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—532no—
223C4—532no—
224C4—1061no—
225C4—2124no—
226C4—2121no—
227C4D4×#615712yes—
228C2—81no—
229C2—1061no—
230C2—1061no—
231C2—4241no—
232C2#1681157511yescharacteristic
233C2D53⋊#10411yescharacteristic
234C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#124#6157
hypocenterC53#1980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#13133282C2semidirect
2. C2×#13133281C2semidirect
3. C4×#11719936C2semidirect
4. C4×C4⋊#6157C2semidirect
16. #13133215C4semidirect
17. #13133213C4semidirect
18. #13133205C4semidirect
24. #124×C53C4semidirect
48. #11719936D4semidirect
108. Dic53#104semidirect
123. C53#1980semidirect
130. #124#6157semidirect