Finite Group Databasebeta

← #24918699

Subgroups — #24918699

Order 10624.

328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128258C211yesmaximal, characteristic, complemented
2#19128111C211yesmaximal, characteristic, complemented
3#19128111C211yesmaximal, characteristic, complemented
4#19128060C211yesmaximal, characteristic, complemented
5#19128060C211yesmaximal, characteristic, complemented
6#19128039C212yesmaximal, complemented
7#19128022C212yesmaximal
8#19128008C212yesmaximal, complemented
9#19128001C212yesmaximal, complemented
10#19127758C211yesmaximal, characteristic, complemented
11#529×C83C211yesmaximal, characteristic
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581784C2212yes—
15#14581771—21no—
16#14581771—21no—
17C4×Dic166—21no—
18C4×Dic166—21no—
19#14581751C2212yes—
20#14581725C2212yes—
21#14581725C2212yes—
22#14581723C2212yes—
23#14581723C2212yes—
24C4⋊(C83⋊C8)C2212yes—
25#14581716C2212yes—
26#14581716C2212yes—
27#14581713C2212yes—
28#14581713C2212yes—
29C22⋊(C83⋊C8)C2212yes—
30#14581703C2211yescharacteristic
31#14581703C2211yescharacteristic
32#126×C83—21no—
33#125×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#123×C83—21no—
36C22⋊Q8×C83C2211yescharacteristic
37#121×C83C2211yescharacteristic
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83—21no—
40C4×D4×C83C2211yescharacteristic
41#117×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—21no—
45C2×C83⋊C8C2312yes—
46C2×C83⋊C8C2312yes—
47C4⋊Dic83—21no—
48C4⋊Dic83—21no—
49C4⋊Dic83C2312yes—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53C22⋊Dic83—22no—
54C4×Dic83—22no—
55C2×Q8×C83C2311yescharacteristic
56C2×D4×C83—21no—
57C2×D4×C83C2311yescharacteristic
58C22×C4×C83—21no—
59C22×C4×C83D411yescharacteristic
60C22×C4×C83D411yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—21no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83—21no—
66C4⋊C4×C83C2311yescharacteristic
67C4⋊C4×C83C2311yescharacteristic
68C22⋊C4×C83—21no—
69C22⋊C4×C83—21no—
70C22⋊C4×C83—21no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83—21no—
74C42×C83D411yescharacteristic
75C42×C83D411yescharacteristic
76C42×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C23×C83—21no—
79C23×C83C2×D411yescharacteristic
80Q8×C83—22no—
81D4×C83—22no—
82D4×C83—42no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—42no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2411yescharacteristic
96C2×Dic83—22no—
97C2×Dic83—22no—
98Dic166—42no—
99Dic166—42no—
100C83⋊C8—22no—
101C83⋊C8—22no—
102C22×C83—21no—
103C22×C83—21no—
104C22×C83—21no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83C22×D411yescharacteristic
109C4×C83—21no—
110C4×C83—21no—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—41no—
115C4×C83—41no—
116C4×C83—41no—
117C4×C832−1+412yes—
118Dic83—42no—
119Dic83—42no—
120C2×C83—41no—
121C2×C83—81no—
122C2×C83#53611yescharacteristic
123C2×C83#57511yescharacteristic
124C2×C83#57511yescharacteristic
125#3240—831nomaximal, Sylow
126C83#324011yescharacteristic, complemented, Sylow
127#533—831no—
128#529D8311yescharacteristic
129#501—831no—
130#501—831no—
131#489—831no—
132#489—831no—
133#485—832no—
134#482—832no—
135#479—832no—
136#478—832no—
137#432—831no—
138C4⋊Q8—831no—
139C4⋊Q8—831no—
140#126—21no—
141#125—832no—
142#125C2×D8311yescharacteristic
143#124C2×D8311yescharacteristic
144#123—21no—
145C22⋊Q8—1661no—
146C22⋊Q8—1661no—
147C22⋊Q8C2×D8311yescharacteristic
148#121C2×D8311yescharacteristic
149C4×Q8—1661no—
150C4×Q8—1661no—
151C4×Q8C2×D8311yescharacteristic
152C4×D4—21no—
153C4×D4C2×D8311yescharacteristic
154#117—21no—
155#117—832no—
156C2×C42C2×D8311yescharacteristic
157#107—832no—
158#107—832no—
159#106—832no—
160#106—832no—
161C4⋊C8—832no—
162#103—832no—
163#103—832no—
164#102—832no—
165#102—832no—
166C22⋊C8—832no—
167#97—831no—
168#97—831no—
169C2×Q8—1661no—
170C2×Q8—1661no—
171C2×Q8C22×D8311yescharacteristic
172C2×D4—21no—
173C2×D4C22×D8311yescharacteristic
174C22×C4—21no—
175C22×C4C22⋊D8311yescharacteristic
176C22×C4C22⋊D8311yescharacteristic
177C22×C4C22×D8311yescharacteristic
178C2×C8—832no—
179C2×C8—832no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—832no—
185C4⋊C4—832no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4C22×D8311yescharacteristic
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—1662no—
198C42—21no—
199C42—1662no—
200C42C22⋊D8311yescharacteristic
201C42C22⋊D8311yescharacteristic
202C42C22×D8311yescharacteristic
203C42C22×D8311yescharacteristic
204C23—21no—
205C23C2×C22⋊D8311yescharacteristic
206Q8—22no—
207Q8—3322no—
208Q8—3322no—
209D4—22no—
210D4—42no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4C2×C22⋊D8311yescharacteristic
221C2×C4C2×C22⋊D8311yescharacteristic
222C2×C4C2×C22⋊D8311yescharacteristic
223C2×C4C2×C22⋊D8311yescharacteristic
224C2×C4C2×C22⋊D8311yescharacteristic
225C2×C4C23×D8311yescharacteristic
226C8—1662no—
227C8—1662no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22C22×C22⋊D8311yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—3322no—
244C4—3322no—
245C4#1458183912yes—
246C2—41no—
247C2—81no—
248C2#1912827211yescharacteristic
249C2#1912849911yescharacteristic
250C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#529×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#529D83
hypocenterC83#3240
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
2. #19128111C2semidirect
3. #19128111C2semidirect
4. #19128060C2semidirect
5. #19128060C2semidirect
6. #19128039C2semidirect
8. #19128008C2semidirect
9. #19128001C2semidirect
10. #19127758C2semidirect
126. C83#3240semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191282582
#191281112
#191280602
#191280392
#191280222
#191280082
#191280012
#191277582
#529×C832
#324083