Finite Group Databasebeta

← #24920450

Subgroups — #24920450

Order 10624.

360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128425C211yesmaximal, characteristic, complemented
2#19128421C211yesmaximal, characteristic, complemented
3#19128082C212yesmaximal
4#19128080C211yesmaximal, characteristic, complemented
5#19127926C212yesmaximal, complemented
6#19127921C212yesmaximal, complemented
7#19127907C211yesmaximal, characteristic, complemented
8#19127825C211yesmaximal, characteristic, complemented
9#19127793C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050580C2212yes—
14C2×#13050579C2212yes—
15C2×#13050575C2211yescharacteristic
16#14581792C2211yescharacteristic
17#14581787—22no—
18#14581785—22no—
19#14581781C2212yes—
20#14581779C2212yes—
21#14581772—21no—
22#14581771—21no—
23Q8×Dic83C2212yes—
24C4×Dic166C2211yescharacteristic
25#14581757C2212yes—
26C4×D332C2211yescharacteristic
27#14581751—21no—
28#14581750—21no—
29#14581725C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C2×SD16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166D411yescharacteristic
42C2×Dic166C2312yes—
43C2×D332C2311yescharacteristic
44C2×C4×D83—21no—
45#13050580—24no—
46#13050580—42no—
47#13050579—42no—
48#13050575—22no—
49C2×C83⋊C8C2311yescharacteristic
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83D411yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58C22⋊Dic83—22no—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63C4×Dic83—22no—
64C4×Dic83—22no—
65C4×Dic83C2312yes—
66C2×Q8×C83C2312yes—
67C2×D4×C83C2311yescharacteristic
68C22×C4×C83—21no—
69SD16×C83—42no—
70C2×C8×C83C2312yes—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2311yescharacteristic
74C22⋊C4×C83—21no—
75C42×C83C2311yescharacteristic
76C23×C83—21no—
77C22×D83—21no—
78Q8×C83—22no—
79Q8×C83—22no—
80D4×C83—21no—
81D4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—42no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83C2×D412yes—
91C2×Dic83C2×D412yes—
92D332—21no—
93D332—21no—
94C4×D83—42no—
95Dic166—22no—
96Dic166—22no—
97Dic166—22no—
98C8×C83—22no—
99C83⋊C8—21no—
100C83⋊C8#4112yes—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83C22×D411yescharacteristic
105C2×D83—41no—
106C2×D83—41no—
107C2×D83—41no—
108C4×C83—21no—
109C4×C83—42no—
110C4×C83—41no—
111C4×C83#13512yes—
112C4×C83C2×#4111yescharacteristic
113C4×C83H(5,𝔽2)11yescharacteristic
114Dic83—22no—
115Dic83—22no—
116Dic83—42no—
117Dic83—41no—
118C2×C83—81no—
119C2×C83#55011yescharacteristic
120C2×C83C2×#13511yescharacteristic
121C2×C83#57811yescharacteristic
122D83—81no—
123#3393—831nomaximal, Sylow
124C83#339311yescharacteristic, complemented, Sylow
125#556—831no—
126#556—831no—
127#495—832no—
128#494—831no—
129#464—832no—
130#464—832no—
131#461—831no—
132#461D8311yescharacteristic, complemented
133C8×Q8—831no—
134C4×Q16—832no—
135C4×D8—831no—
136C2×Q16—832no—
137C2×Q16—832no—
138C2×SD16—832no—
139C2×SD16C2×D8312yes—
140C2×D8—831no—
141C4⋊Q8—831no—
142C4⋊Q8C2×D8311yescharacteristic
143#126—1662no—
144#126—1662no—
145#124—832no—
146#124—832no—
147C22⋊Q8—1661no—
148C22⋊Q8—1661no—
149C4×Q8—831no—
150C4×Q8—832no—
151C4×Q8—832no—
152C4×D4—831no—
153C4×D4C2×D8311yescharacteristic
154#117—1661no—
155#117—1661no—
156#107—831no—
157C4⋊C8—832no—
158C4⋊C8C2×D8311yescharacteristic
159#103—832no—
160#103C2×D8312yes—
161#102—831no—
162#102—831no—
163C4×C8—832no—
164C4×C8—831no—
165C2×Q8—831no—
166C2×Q8—832no—
167C2×Q8C22×D8312yes—
168C2×D4—831no—
169C2×D4C22×D8311yescharacteristic
170C22×C4—21no—
171C22×C4—1661no—
172Q16—3322no—
173Q16—1664no—
174SD16—42no—
175SD16—3322no—
176D8—1662no—
177C2×C8—831no—
178C2×C8—831no—
179C2×C8C22×D8312yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—831no—
183C4⋊C4—832no—
184C4⋊C4—831no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1661no—
188C4⋊C4—1661no—
189C4⋊C4C22×D8311yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C22⋊C4—1662no—
194C22⋊C4—1662no—
195C22⋊C4—1661no—
196C42—832no—
197C42—1662no—
198C42—1662no—
199C42C22×D8311yescharacteristic
200C23—21no—
201C23—1661no—
202Q8—22no—
203Q8—22no—
204Q8—1662no—
205Q8—1662no—
206Q8—1662no—
207D4—21no—
208D4—21no—
209D4—1661no—
210D4—1661no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—3322no—
215C2×C4—832no—
216C2×C4—832no—
217C2×C4—1661no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4D4×D8311yescharacteristic
221C2×C4C23×D8311yescharacteristic
222C8—22no—
223C8—832no—
224C8—1661no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—3321no—
229C22—3321no—
230C22—3321no—
231C22C2×D4×D8311yescharacteristic
232C4—21no—
233C4—42no—
234C4—41no—
235C4—3322no—
236C4—3321no—
237C4—1662no—
238C4—1662no—
239C4#1458181312yes—
240C4#41×D8311yescharacteristic
241C4#1458183711yescharacteristic
242C2—81no—
243C2—6641no—
244C2#1912836911yescharacteristic
245C2C2×#1458181311yescharacteristic
246C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#461D83
hypocenterC83#3393
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128425C2semidirect
2. #19128421C2semidirect
4. #19128080C2semidirect
5. #19127926C2semidirect
6. #19127921C2semidirect
7. #19127907C2semidirect
8. #19127825C2semidirect
9. #19127793C2semidirect
11. #461×C83C2semidirect
124. C83#3393semidirect
132. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191284252
#191284212
#191280822
#191280802
#191279262
#191279212
#191279072
#191278252
#191277932
C4×#130505752
#461×C832
#339383