Finite Group Databasebeta

← #20276382

Subgroups — #20276382

Order 6784.

286 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#16811190C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3#16810417C212yesmaximal, complemented
4#16810415C212yesmaximal, complemented
5C2×#99×C53C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7#13133213C411yescharacteristic, complemented
8#13133213C2211yescharacteristic
9#13133204C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133188—21no—
12#13133188—21no—
13#13133186—21no—
14#13133186—21no—
15#13133176C411yescharacteristic, complemented
16C2×C22⋊Dic53—21no—
17#13133132—21no—
18#13133132C2212yes—
19C22×D4×C53C2211yescharacteristic
20C2×C22⋊C4×C53—21no—
21#99×C53—21no—
22#99×C53C2212yes—
23C24×C53D411yescharacteristic
24C24×C53D411yescharacteristic
25C2×D4×C53—41no—
26C2×D4×C53—41no—
27C2×D4×C53C2×C411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C2×D4×C53D411yescharacteristic
30C2×D4×C53C2311yescharacteristic
31C22×C4×C53—41no—
32C22×C4×C53C2×C411yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C2×C22⋊D53—21no—
38C22×Dic53—41no—
39#11719945—41no—
40#11719945—41no—
41C2×D212C2×C411yescharacteristic, complemented
42C2×C4×D53—21no—
43C2×Dic106C2×C411yescharacteristic, complemented
44C22⋊C4×C53—22no—
45C22⋊C4×C53—41no—
46C22⋊Dic53—21no—
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53—22no—
52C22⋊Dic53—41no—
53C4⋊Dic53—21no—
54C4×Dic53C2×C411yescharacteristic, complemented
55C4×Dic53C2×C411yescharacteristic, complemented
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—81no—
61C23×C53—81no—
62C23×C53C22⋊C411yescharacteristic
63C23×C53C22⋊C411yescharacteristic
64C23×C53C22⋊C411yescharacteristic
65C23×C53C2×D411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68D4×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71D4×C53—41no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—42no—
75C2×C4×C53—82no—
76C2×C4×C53C22⋊C411yescharacteristic
77C2×C4×C53C22×C411yescharacteristic
78C22⋊D53—41no—
79C22⋊D53—41no—
80C22⋊D53—81no—
81C22⋊D53—81no—
82C22⋊D53—81no—
83C22⋊D53—81no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—42no—
88C2×Dic53—82no—
89D212—41no—
90C4×D53—42no—
91Dic106—41no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—81no—
104C22×C53—81no—
105C22×C53#9911yescharacteristic
106C22×C53#9911yescharacteristic
107C22×C53C2×C22⋊C411yescharacteristic
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—41no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—82no—
114Dic53—41no—
115Dic53—41no—
116Dic53—41no—
117Dic53—82no—
118#2127—531nomaximal, Sylow
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—41no—
123C2×C53—41no—
124C2×C53—41no—
125C2×C53—41no—
126C2×C53C2×#9911yescharacteristic
127D53—81no—
128#535—531no—
129C2×#99—531no—
130C2×#99D5311yescharacteristic, complemented
131#354—532no—
132#353—532no—
133C53#212711yescharacteristic, complemented, Sylow
134C2×#41—531no—
135C22×D4C2×D5311yescharacteristic
136#127—531no—
137#127—531no—
138#124—531no—
139#124—531no—
140#121—1061no—
141#121—1061no—
142#120—1061no—
143#120—1061no—
144#117—531no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—1061no—
147#99—21no—
148#99—532no—
149#99—1061no—
150#99C2×D5312yes—
151C24D21211yescharacteristic
152C24C22⋊D5311yescharacteristic
153#41—2121no—
154#41—2121no—
155C2×Q8—531no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—531no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—1061no—
162C2×D4—1061no—
163C2×D4—1061no—
164C2×D4C4×D5311yescharacteristic
165C2×D4D21211yescharacteristic
166C2×D4C22⋊D5311yescharacteristic
167C2×D4C22×D5311yescharacteristic
168C22×C4—41no—
169C22×C4—1061no—
170C22×C4—2121no—
171C22×C4C4×D5311yescharacteristic
172C4⋊C4—1061no—
173C22⋊C4—22no—
174C22⋊C4—41no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1062no—
181C22⋊C4—2121no—
182C42—531no—
183C42—531no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—81no—
189C23—81no—
190C23—1061no—
191C23#1171992111yescharacteristic
192C23#1171992111yescharacteristic
193C23#1171992111yescharacteristic
194C23C2×D21211yescharacteristic
195C23C2×C22⋊D5311yescharacteristic
196Q8—2121no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—4241no—
202D4—4241no—
203D4—4241no—
204D4—4241no—
205D4—2121no—
206D4—2121no—
207D4—2121no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—4242no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4#1171992111yescharacteristic
219C2×C4C2×C4×D5311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—2121no—
234C22—2121no—
235C22—2121no—
236C22#1313313111yescharacteristic
237C22#1313313111yescharacteristic
238C22C2×#1171992111yescharacteristic
239C4—21no—
240C4—21no—
241C4—82no—
242C4—4242no—
243C4—2121no—
244C4—2121no—
245C4—2121no—
246C2—21no—
247C2—21no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—41no—
252C2—41no—
253C2—4241no—
254C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC2×#99×C53C2
socleC2×C53C2×#99
hypercenterC2×#99D53
hypocenterC53#2127
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. #16810417C2semidirect
4. #16810415C2semidirect
5. C2×#99×C53C2semidirect
6. C2×#11719945C4semidirect
7. #13133213C4semidirect
9. #13133204C4semidirect
15. #13133176C4semidirect
41. C2×D212C2×C4semidirect
43. C2×Dic106C2×C4semidirect
54. C4×Dic53C2×C4semidirect
55. C4×Dic53C2×C4semidirect
130. C2×#99D53semidirect
133. C53#2127semidirect