Finite Group Databasebeta

← #24918052

Subgroups — #24918052

Order 10624.

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

normal subgroups only

This group has 408 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#19128235C211yesmaximal, characteristic, complemented
2#19128107C211yesmaximal, characteristic, complemented
3#19128088C211yesmaximal, characteristic, complemented
4#19128065C211yesmaximal, characteristic, complemented
5#19128063C211yesmaximal, characteristic, complemented
6#19127952C212yesmaximal, complemented
7#19127945C212yesmaximal, complemented
8C4×#13050578C212yesmaximal, complemented
9C4×#13050575C212yesmaximal, complemented
10C4×#13050574C211yesmaximal, characteristic, complemented
11#532×C83C211yesmaximal, characteristic, complemented
12C2×#13050578C2212yes—
13C2×#13050575C2212yes—
14C2×#13050574C2212yes—
15#14581786—21no—
16#14581784C2211yescharacteristic
17#14581780C2211yescharacteristic
18#14581775—21no—
19#14581771C2211yescharacteristic
20#14581766C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22C4×D332C2211yescharacteristic
23C4×C22⋊D83—21no—
24#14581749—21no—
25#14581725C2212yes—
26#14581723C2212yes—
27#14581715C2212yes—
28#14581713C2212yes—
29#14581713C2212yes—
30#14581712C2212yes—
31#14581703C2211yescharacteristic
32#14581703C2211yescharacteristic
33C4×C83⋊C8C2212yes—
34C2×#41×C83—21no—
35C4⋊Q8×C83C2211yescharacteristic
36#127×C83C2211yescharacteristic
37#124×C83—21no—
38#121×C83—21no—
39#121×C83C2212yes—
40C4×D4×C83—21no—
41C4×D4×C83C2212yes—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050578—24no—
48#13050575—24no—
49#13050574—24no—
50C2×C83⋊C8C2312yes—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58C22⋊Dic83—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63C4×Dic83—21no—
64#41×C83—44no—
65C2×Q8×C83—21no—
66C2×D4×C83—22no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C2×D4×C83C2312yes—
70C22×C4×C83—22no—
71C22×C4×C83D411yescharacteristic
72C22×C4×C83D411yescharacteristic
73C22×C4×C83C2311yescharacteristic
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C22⋊C4×C83—22no—
77C22⋊C4×C83—22no—
78C42×C83D411yescharacteristic
79C42×C83D411yescharacteristic
80C42×C83C2311yescharacteristic
81C42×C83C2311yescharacteristic
82C23×C83—22no—
83C23×C83C2×D411yescharacteristic
84C22×D83—21no—
85Q8×C83—42no—
86D4×C83—22no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—42no—
90D4×C83—44no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—41no—
95C2×C4×C83—44no—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2411yescharacteristic
102C22⋊D83—42no—
103C2×Dic83—21no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106D332—22no—
107C4×D83—42no—
108Dic166—22no—
109C83⋊C8—22no—
110C83⋊C8#4114yescomplemented
111C22×C83—21no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83C22×D411yescharacteristic
118C2×D83—41no—
119C2×D83—41no—
120C2×D83—41no—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—21no—
124C4×C83—21no—
125C4×C83—42no—
126C4×C83AGL(1,ℤ/8)12yes—
127C4×C83C2×#4112yes—
128Dic83—41no—
129Dic83—41no—
130Dic83—41no—
131C2×C83—41no—
132C2×C83—82no—
133C2×C83#53211yescharacteristic
134C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C83#57511yescharacteristic
136D83—81no—
137#3171—831nomaximal, Sylow
138C83#317111yescharacteristic, complemented, Sylow
139#532D8311yescharacteristic, complemented
140#529—831no—
141#501—831no—
142#496—831no—
143#490—831no—
144#489—831no—
145#469—832no—
146#468—832no—
147C4×SD16—832no—
148C4×D8—832no—
149C4×M16—831no—
150C2×#41—21no—
151C2×SD16—832no—
152C2×D8—832no—
153C2×M16—832no—
154C4⋊Q8C2×D8311yescharacteristic
155#127C2×D8311yescharacteristic
156#126—1661no—
157#125—831no—
158#124—21no—
159#124—831no—
160#123—1661no—
161C22⋊Q8—831no—
162#121—21no—
163#121—831no—
164#121C2×D8312yes—
165C4×Q8—831no—
166C4×D4—21no—
167C4×D4—831no—
168C4×D4—1661no—
169C4×D4C2×D8312yes—
170#117—1661no—
171C2×C42C2×D8311yescharacteristic
172#107—832no—
173#106—832no—
174#103—832no—
175#102—832no—
176#102—832no—
177#102—832no—
178#97—831no—
179#97—831no—
180C4×C8—832no—
181#41—44no—
182C2×Q8—21no—
183C2×Q8—831no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—831no—
188C2×D4—1661no—
189C2×D4C22×D8312yes—
190C22×C4—22no—
191C22×C4—1661no—
192C22×C4C22⋊D8311yescharacteristic
193C22×C4C22⋊D8311yescharacteristic
194C22×C4C22×D8311yescharacteristic
195SD16—1664no—
196D8—1664no—
197M16—1664no—
198C2×C8—832no—
199C2×C8—832no—
200C4⋊C4—21no—
201C4⋊C4—831no—
202C4⋊C4—831no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4C22×D8312yes—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—1661no—
211C22⋊C4—1661no—
212C22⋊C4—1661no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C42—1661no—
216C42C22⋊D8311yescharacteristic
217C42C22⋊D8311yescharacteristic
218C42C22×D8311yescharacteristic
219C42C22×D8311yescharacteristic
220C23—22no—
221C23—1661no—
222C23C2×C22⋊D8311yescharacteristic
223Q8—42no—
224Q8—1662no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—44no—
230D4—3322no—
231D4—1662no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—44no—
237C2×C4—3322no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4C2×C22⋊D8311yescharacteristic
242C2×C4C2×C22⋊D8311yescharacteristic
243C2×C4C2×C22⋊D8311yescharacteristic
244C2×C4C2×C22⋊D8311yescharacteristic
245C2×C4C2×C22⋊D8311yescharacteristic
246C2×C4C23×D8311yescharacteristic
247C8—834no—
248C8—1662no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—3321no—
256C22—3321no—
257C22—3321no—
258C22C22×C22⋊D8311yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—3321no—
265C4—3321no—
266C4—3321no—
267C4#1458181812yes—
268C4#41×D8312yes—
269C2—41no—
270C2—82no—
271C2—6641no—
272C2#1912824711yescharacteristic
273C2C2×#1458181811yescharacteristic
274C2#1912849911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#532×C83C2
socleC22×C83C22×D4
hypercenter#532D83
hypocenterC83#3171
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128107C2semidirect
3. #19128088C2semidirect
4. #19128065C2semidirect
5. #19128063C2semidirect
6. #19127952C2semidirect
7. #19127945C2semidirect
8. C4×#13050578C2semidirect
9. C4×#13050575C2semidirect
10. C4×#13050574C2semidirect
11. #532×C83C2semidirect
110. C83⋊C8#41semidirect
138. C83#3171semidirect
139. #532D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191281072
#191280882
#191280652
#191280632
#191279522
#191279452
C4×#130505782
C4×#130505752
C4×#130505742
#532×C832
#317183