Finite Group Databasebeta

← #24919819

Subgroups — #24919819

Order 10624.

388 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128412C211yesmaximal, characteristic, complemented
2#19128354C211yesmaximal, characteristic
3#19128033C212yesmaximal, complemented
4#19128021C212yesmaximal, complemented
5#19127988C211yesmaximal, characteristic, complemented
6#19127983C211yesmaximal, characteristic, complemented
7#19127931C212yesmaximal, complemented
8C4×#13050579C211yesmaximal, characteristic, complemented
9C8×D332C211yesmaximal, characteristic, complemented
10#106×D83C212yesmaximal, complemented
11#498×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×#13050579C2211yescharacteristic
14C2×C8×D83C2212yes—
15#14581783—21no—
16#14581777—22no—
17#14581772C2212yes—
18#14581768—21no—
19#14581768—22no—
20#14581767C2212yes—
21C4×Dic166C2211yescharacteristic
22C4×D332C2211yescharacteristic
23C4×D332C2211yescharacteristic
24#14581754—21no—
25C4⋊C4×D83C2212yes—
26C4⋊C4×D83C2212yes—
27#14581724C2211yescharacteristic
28#14581723C2212yes—
29#14581723C2212yes—
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581717C2211yescharacteristic
32#14581715C2212yes—
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35(C2×D83)⋊C8C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#106×C83C2212yes—
39#106×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41#13050591—44no—
42C2×Dic166C2311yescharacteristic
43C2×D332—22no—
44C2×D332D411yescharacteristic
45C2×D332C2311yescharacteristic
46C2×C4×D83—22no—
47C2×C4×D83—21no—
48C2×C4×D83—22no—
49C2×C4×D83C2312yes—
50#13050579—22no—
51C2×C83⋊C8C2312yes—
52C8×D83—24no—
53C4⋊Dic83—22no—
54C4⋊Dic83D411yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—21no—
58Dic83⋊C4—22no—
59Dic83⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4D412yes—
64C4×Dic83—21no—
65C2×Q8×C83—21no—
66C2×Q8×C83—21no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83—21no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83C2312yes—
72C4⋊C4×C83C2312yes—
73C42×C83C2311yescharacteristic
74C22×D83—21no—
75C22×D83C2×D412yes—
76Q8×C83—42no—
77Q8×C83—81no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83C2×D412yes—
87D332—21no—
88D332—21no—
89D332—44no—
90D332SD1614yescomplemented
91C4×D83—22no—
92C4×D83—22no—
93C4×D83—44no—
94C4×D83—42no—
95C4×D83—82no—
96Dic166—21no—
97Dic166—21no—
98C8×C83—21no—
99C8×C83#4112yescomplemented
100C83⋊C8—22no—
101C22×C83C22×D411yescharacteristic
102C2×D83—22no—
103C2×D83—22no—
104C2×D83—41no—
105C2×D83—41no—
106C2×D83—41no—
107C2×D83C2×SD1614yes—
108C4×C83—21no—
109C4×C83—42no—
110C4×C83—42no—
111C4×C83C2×SD1612yes—
112C4×C83C2×#4111yescharacteristic
113C4×C832−1+411yescharacteristic
114Dic83—22no—
115Dic83—41no—
116Dic83—41no—
117Dic83—41no—
118C2×C83#54711yescharacteristic
119C2×C83C22×SD1611yescharacteristic
120C2×C83#57711yescharacteristic
121D83—24no—
122D83—81no—
123#3335—831nomaximal, Sylow
124C83#333511yescharacteristic, complemented, Sylow
125#554—831no—
126#547—831no—
127#498D8311yescharacteristic, complemented
128#483—832no—
129#481—832no—
130#476—831no—
131#475—831no—
132#465—832no—
133C4×SD16—831no—
134C8×D4—831no—
135C2×#106—832no—
136C2×#41—1661no—
137C2×SD16—831no—
138C22×C8—832no—
139C4⋊Q8C2×D8311yescharacteristic
140C4⋊Q8C2×D8311yescharacteristic
141#125—1661no—
142#123—1662no—
143C22⋊Q8—832no—
144C22⋊Q8—1662no—
145C22⋊Q8—1661no—
146#121—832no—
147C4×Q8—831no—
148C4×D4—831no—
149C4×D4—831no—
150C4×D4—1661no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—832no—
153#106—831no—
154#106—832no—
155#106—832no—
156#106C2×D8312yes—
157#106C2×D8312yes—
158C4⋊C8—831no—
159#103—831no—
160#103—832no—
161#102—831no—
162#102—832no—
163C22⋊C8—832no—
164C4×C8C2×D8311yescharacteristic
165#41—3324no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—831no—
169C2×D4—831no—
170C2×D4—831no—
171C2×D4—1662no—
172C22×C4—832no—
173C22×C4—1662no—
174C22×C4—1662no—
175C22×C4—1661no—
176SD16—1662no—
177C2×C8—832no—
178C2×C8—1664no—
179C2×C8C22×D8311yescharacteristic
180C2×C8C22×D8311yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—831no—
184C4⋊C4—831no—
185C4⋊C4—831no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1662no—
189C4⋊C4—1661no—
190C4⋊C4C22×D8312yes—
191C4⋊C4C22×D8312yes—
192C22⋊C4—832no—
193C22⋊C4—1662no—
194C22⋊C4—1662no—
195C22⋊C4—1661no—
196C42—1661no—
197C42C22×D8311yescharacteristic
198C23—832no—
199C23—1661no—
200Q8—42no—
201Q8—81no—
202Q8—1661no—
203Q8—1661no—
204D4—3324no—
205D4—834no—
206D4—1661no—
207D4—1661no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—3324no—
211C2×C4—3322no—
212C2×C4—832no—
213C2×C4—6642no—
214C2×C4—1661no—
215C2×C4—1661no—
216C2×C4—1661no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4D4×D8311yescharacteristic
221C2×C4C23×D8311yescharacteristic
222C8—21no—
223C8—1662no—
224C8#1305059112yescomplemented
225C22—3321no—
226C22—3321no—
227C22—3321no—
228C22—834no—
229C22—1662no—
230C22—1662no—
231C22C2×D4×D8311yescharacteristic
232C4—21no—
233C4—42no—
234C4—42no—
235C4—3321no—
236C4—3321no—
237C4—3321no—
238C4—1662no—
239C4SD16×D8312yes—
240C4C2×#1305059111yescharacteristic
241C4#1458184011yescharacteristic
242C2—6641no—
243C2—1664no—
244C2#1912835411yescharacteristic
245C2C2×SD16×D8311yescharacteristic
246C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#498×C83C2
socleC22×C83C22×D4
prime radical#19128354C2
hypercenter#498D83
hypocenterC83#3335
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #19128412C2semidirect
3. #19128033C2semidirect
4. #19128021C2semidirect
5. #19127988C2semidirect
6. #19127983C2semidirect
7. #19127931C2semidirect
8. C4×#13050579C2semidirect
9. C8×D332C2semidirect
10. #106×D83C2semidirect
11. #498×C83C2semidirect
90. D332SD16semidirect
99. C8×C83#41semidirect
124. C83#3335semidirect
127. #498D83semidirect
224. C8#13050591semidirect

Maximal subgroups

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

subgroupindex
#191284122
#191283542
#191280332
#191280212
#191279882
#191279832
#191279312
C4×#130505792
C8×D3322
#106×D832
#498×C832
#333583