Finite Group Databasebeta

← #24918585

Subgroups — #24918585

Order 10624.

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

normal subgroups only

This group has 404 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#19128252C211yesmaximal, characteristic
2C4⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128104C211yesmaximal, characteristic, complemented
4#19128049C211yesmaximal, characteristic, complemented
5#19128048C211yesmaximal, characteristic, complemented
6#19128016C214yesmaximal, complemented
7#19127980C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#500×C83C211yesmaximal, characteristic, complemented
10C2×#13050591—21no—
11C2×Q8×D83—21no—
12#14581792C2211yescharacteristic
13#14581791—21no—
14#14581791C2212yes—
15#14581790—21no—
16#14581789C2211yescharacteristic
17#14581781—21no—
18#14581767—21no—
19#14581767C2212yes—
20#14581754—21no—
21#14581754C2212yes—
22C4⋊C4×D83—21no—
23C4⋊C4×D83C2212yes—
24C42×D83C2211yescharacteristic
25#14581725C2214yes—
26Dic83⋊C8C2212yes—
27#14581714C2214yes—
28#14581712C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#107×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35#13050591—44no—
36C2×Dic166—21no—
37C2×Dic166—22no—
38Q8×D83—44no—
39C2×D332—22no—
40C2×D332—21no—
41C2×D332—21no—
42C2×D332C2312yes—
43C2×C4×D83—22no—
44C2×C4×D83—22no—
45C2×C4×D83D411yescharacteristic
46C2×C4×D83D411yescharacteristic
47C2×C4×D83C2311yescharacteristic
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—21no—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55C4×Dic83D411yescharacteristic
56C4×Dic83D411yescharacteristic
57C4×Dic83C2311yescharacteristic
58C2×Q8×C83—21no—
59C2×Q8×C83—21no—
60C2×C8×C83C2312yes—
61C4⋊C4×C83—21no—
62C4⋊C4×C83—21no—
63C4⋊C4×C83C2312yes—
64C4⋊C4×C83C2312yes—
65C42×C83C2311yescharacteristic
66C22×D83—22no—
67C22×D83C2×D411yescharacteristic
68Q8×C83—42no—
69Q8×C83—42no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2411yescharacteristic
75C2×Dic83—22no—
76C2×Dic83C2×D411yescharacteristic
77C2×Dic83C2×D411yescharacteristic
78C2×Dic83C2×D411yescharacteristic
79D332—24no—
80D332—44no—
81D332—42no—
82D332—42no—
83C4×D83—22no—
84C4×D83—22no—
85C4×D83—44no—
86C4×D83—44no—
87C4×D83D814yes—
88Dic166—42no—
89Dic166—44no—
90C8×C83—22no—
91C83⋊C8—22no—
92C22×C83C22×D411yescharacteristic
93C2×D83—21no—
94C2×D83—21no—
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C2×D83C2×D812yes—
99C4×C83—42no—
100C4×C83—42no—
101C4×C83C2×D812yes—
102C4×C83#13712yes—
103C4×C832−1+412yes—
104Dic83—21no—
105Dic83—21no—
106Dic83—42no—
107Dic83C2×D812yes—
108C2×C83#53611yescharacteristic
109C2×C83C22×D811yescharacteristic
110C2×C83C2×#13711yescharacteristic
111D83—22no—
112D83—82no—
113#3226—831nomaximal, Sylow
114C83#322611yescharacteristic, complemented, Sylow
115#532—831no—
116C2×C4⋊Q8—831no—
117#500—831no—
118#500D8311yescharacteristic, complemented
119#486—831no—
120#486—831no—
121#480—834no—
122#474—834no—
123#431—831no—
124C2×#41—1661no—
125C22×Q8—1661no—
126C4⋊Q8—832no—
127C4⋊Q8—831no—
128C4⋊Q8—1661no—
129C4⋊Q8—1661no—
130C4⋊Q8C2×D8311yescharacteristic
131C4⋊Q8C2×D8311yescharacteristic
132#127—831no—
133#124—1661no—
134#121—832no—
135#121—1661no—
136C4×D4—832no—
137C4×D4—1661no—
138C2×C4⋊C4—832no—
139C2×C4⋊C4—1661no—
140C2×C42—831no—
141#107—834no—
142#107C2×D8314yes—
143C4⋊C8—832no—
144#102—834no—
145#102—834no—
146C22⋊C8—832no—
147C4×C8—831no—
148C4×C8C2×D8311yescharacteristic
149#41—3324no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—3324no—
153C2×Q8—1662no—
154C2×Q8—1661no—
155C2×D4—832no—
156C2×D4—1662no—
157C2×D4—1661no—
158C2×D4—1661no—
159C22×C4—831no—
160C22×C4—831no—
161C22×C4—831no—
162C22×C4—1662no—
163C22×C4—1662no—
164C2×C8—832no—
165C2×C8C22×D8312yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—832no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1661no—
172C4⋊C4C22×D8312yes—
173C4⋊C4C22×D8312yes—
174C22⋊C4—1662no—
175C22⋊C4—1662no—
176C42—831no—
177C42—831no—
178C42—831no—
179C42C22×D8311yescharacteristic
180C23—831no—
181C23—1662no—
182Q8—42no—
183Q8—42no—
184Q8—3324no—
185Q8—3322no—
186D4—3324no—
187D4—3322no—
188D4—3322no—
189D4—1664no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—3324no—
193C2×C4—3324no—
194C2×C4—831no—
195C2×C4—834no—
196C2×C4—831no—
197C2×C4—831no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4D4×D8311yescharacteristic
202C2×C4D4×D8311yescharacteristic
203C2×C4C23×D8311yescharacteristic
204C8—22no—
205C8—1662no—
206C22—3322no—
207C22—3322no—
208C22—3322no—
209C22—832no—
210C22—1661no—
211C22—1661no—
212C22C2×D4×D8311yescharacteristic
213C4—42no—
214C4—42no—
215C4—3322no—
216C4—832no—
217C4—1661no—
218C4—1661no—
219C4D8×D8312yes—
220C4#1458182412yes—
221C4#1458184012yes—
222C2—6642no—
223C2—1662no—
224C2#1912827611yescharacteristic
225C2C2×D8×D8311yescharacteristic
226C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#500×C83C2
socleC22×C83C22×D4
prime radical#19128252C2
hypercenter#500D83
hypocenterC83#3226
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D83C2semidirect
3. #19128104C2semidirect
4. #19128049C2semidirect
5. #19128048C2semidirect
6. #19128016C2semidirect
7. #19127980C2semidirect
8. #19127752C2semidirect
9. #500×C83C2semidirect
114. C83#3226semidirect
118. #500D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
C4⋊Q8×D832
#191281042
#191280492
#191280482
#191280162
#191279802
#191277522
#500×C832
#322683