Finite Group Databasebeta

← #24917297

Subgroups — #24917297

Order 10624.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128329C211yesmaximal, characteristic, complemented
2#19128175C211yesmaximal, characteristic
3#19128004C212yesmaximal, complemented
4#19128000C212yesmaximal, complemented
5#19127990C212yesmaximal, complemented
6#19127980C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127727C211yesmaximal, characteristic, complemented
9#19127705C211yesmaximal, characteristic
10C2×#14581714C211yesmaximal, characteristic
11#427×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14#14581792—21no—
15#14581791C2212yescomplemented
16#14581784—21no—
17#14581783C2212yescomplemented
18#14581769—22no—
19C4×D332—21no—
20C4×D332—21no—
21#14581754C2212yes—
22#14581754C2212yes—
23#14581751C2211yescharacteristic
24C2×C4⋊Dic83C2211yescharacteristic
25C22⋊C4×D83—22no—
26#14581725C2212yescomplemented
27#14581723C2212yescomplemented
28Dic83⋊C8C2212yescomplemented
29Dic83⋊C8C2212yescomplemented
30#14581714C2212yes—
31#14581714C2212yes—
32#14581712C2212yes—
33#14581712C2212yes—
34C22×C8×C83C2211yescharacteristic
35#117×C83C2211yescharacteristic
36#117×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic, complemented
38#107×C83C2211yescharacteristic, complemented
39#106×C83C2211yescharacteristic, complemented
40#106×C83C2211yescharacteristic, complemented
41C23×D83—21no—
42C2×Dic166—22no—
43C2×D332—21no—
44C2×D332—22no—
45C2×D332Q814yescomplemented
46C2×D332C2312yes—
47C22×Dic83—21no—
48C2×C4×D83—22no—
49C2×C4×D83—22no—
50#13050574—22no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83D411yescharacteristic
55C4⋊Dic83D411yescharacteristic
56C4⋊Dic83C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59C22⋊Dic83D412yes—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62C4×Dic83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C2×C8×C83—22no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83C2311yescharacteristic
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83—21no—
74C42×C83—21no—
75C23×C83C2×D411yescharacteristic
76C22×D83—22no—
77C22×D83—24no—
78C22×D83—81no—
79C22×D83—81no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×Q812yes—
86C2×C4×C83#4112yes—
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—22no—
89C2×Dic83—81no—
90C2×Dic83C2×D412yes—
91C2×Dic83C2×D412yes—
92D332—24no—
93D332—22no—
94D332C2×Q814yes—
95C4×D83—82no—
96C4×D83—82no—
97Dic166—82no—
98C8×C83—21no—
99C8×C83—21no—
100C83⋊C8—22no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83C22⋊Q812yes—
104C22×C83C22×D411yescharacteristic
105C2×D83—24no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—84no—
109C4×C83—41no—
110C4×C83—41no—
111C4×C83—41no—
112C4×C83—41no—
113C4×C83C22⋊Q812yes—
114C4×C83C22×Q811yescharacteristic
115C4×C83C2×#4111yescharacteristic
116Dic83—22no—
117Dic83—22no—
118Dic83—42no—
119C2×C83—22no—
120C2×C83C2×C22⋊Q811yescharacteristic
121C2×C83#57611yescharacteristic
122C2×C83#57711yescharacteristic
123D83—44no—
124#3081—831nomaximal, Sylow
125C83#308111yescharacteristic, complemented, Sylow
126#544—831no—
127#518—831no—
128#478—832no—
129#478—832no—
130#476—832no—
131#474—832no—
132#428—831no—
133#427D8311yescharacteristic, complemented
134#424—831no—
135#418—831no—
136C2×#102—831no—
137C22×D4—831no—
138C2×M16—831no—
139C22×C8C2×D8311yescharacteristic
140C4⋊Q8—832no—
141C4⋊Q8—1661no—
142#125—832no—
143#125—1661no—
144C22⋊Q8—1662no—
145C4×D4—832no—
146C4×D4—832no—
147C4×D4—1661no—
148C4×D4—1661no—
149#117—831no—
150#117C2×D8311yescharacteristic
151#117C2×D8311yescharacteristic
152C2×C4⋊C4—831no—
153C2×C22⋊C4—1662no—
154#107—832no—
155#107C2×D8311yescharacteristic, complemented
156#107C2×D8311yescharacteristic, complemented
157#106—832no—
158#106C2×D8311yescharacteristic, complemented
159#106C2×D8311yescharacteristic, complemented
160C4⋊C8—832no—
161C4⋊C8—832no—
162#102—832no—
163#102—832no—
164#102—832no—
165#102—832no—
166C24—1661no—
167C2×Q8—1662no—
168C2×D4—834no—
169C2×D4—832no—
170C2×D4—1661no—
171C2×D4—1662no—
172C22×C4—1662no—
173C22×C4—1662no—
174C22×C4—1661no—
175C22×C4C22×D8311yescharacteristic
176M16—1662no—
177C2×C8—22no—
178C2×C8—832no—
179C2×C8C22×D8311yescharacteristic
180C2×C8C22×D8311yescharacteristic
181C4⋊C4—831no—
182C4⋊C4—831no—
183C4⋊C4—832no—
184C4⋊C4—1662no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1661no—
188C4⋊C4C22×D8311yescharacteristic
189C4⋊C4C22×D8311yescharacteristic
190C4⋊C4C22×D8311yescharacteristic
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—832no—
195C22⋊C4—1662no—
196C22⋊C4—1662no—
197C42—21no—
198C42—21no—
199C42—832no—
200C23—6641no—
201C23—6641no—
202C23—1662no—
203C23—1664no—
204C23D4×D8311yescharacteristic
205Q8—6642no—
206D4—834no—
207D4—1664no—
208D4—1662no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—832no—
214C2×C4—832no—
215C2×C4—6642no—
216C2×C4—6642no—
217C2×C4—6641no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4Q8×D8312yes—
221C2×C4#1305059012yes—
222C2×C4C23×D8311yescharacteristic
223C8—21no—
224C8—21no—
225C8—1662no—
226C22—21no—
227C22—21no—
228C22—3322no—
229C22—3322no—
230C22—6644no—
231C22—1664no—
232C22#1458177012yes—
233C22C2×D4×D8311yescharacteristic
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—3322no—
239C4—1662no—
240C4—1662no—
241C4#1458177012yes—
242C4C2×Q8×D8311yescharacteristic
243C4C2×#1305059011yescharacteristic
244C2—22no—
245C2—3324no—
246C2C2×#1458177011yescharacteristic
247C2#1912850311yescharacteristic
248C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#427×C83C2
socleC22×C83C22×D4
prime radicalC22×D332C22
hypercenter#427D83
hypocenterC83#3081
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #19128329C2semidirect
3. #19128004C2semidirect
4. #19128000C2semidirect
5. #19127990C2semidirect
6. #19127980C2semidirect
7. #19127744C2semidirect
8. #19127727C2semidirect
11. #427×C83C2semidirect
15. #14581791C22semidirect
17. #14581783C22semidirect
26. #14581725C22semidirect
27. #14581723C22semidirect
28. Dic83⋊C8C22semidirect
29. Dic83⋊C8C22semidirect
37. #107×C83C22semidirect
38. #107×C83C22semidirect
39. #106×C83C22semidirect
40. #106×C83C22semidirect
45. C2×D332Q8semidirect
125. C83#3081semidirect
133. #427D83semidirect
155. #107C2×D83semidirect
156. #107C2×D83semidirect
158. #106C2×D83semidirect
159. #106C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283292
#191281752
#191280042
#191280002
#191279902
#191279802
#191277442
#191277272
#191277052
C2×#145817142
#427×C832
#308183