Finite Group Databasebeta

← #24918194

Subgroups — #24918194

Order 10624.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128236C212yesmaximal, complemented
2#19128085C211yesmaximal, characteristic, complemented
3#19128058C212yesmaximal, complemented
4#19127960C211yesmaximal, characteristic, complemented
5#19127955C211yesmaximal, characteristic
6#19127948C212yesmaximal, complemented
7#19127812C211yesmaximal, characteristic, complemented
8#19127807C211yesmaximal, characteristic, complemented
9#19127800C212yesmaximal, complemented
10C4×#13050573C211yesmaximal, characteristic, complemented
11#499×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2211yescharacteristic
13C2×#13050579C2212yes—
14C2×D664C2211yescharacteristic
15C2×#13050573C2211yescharacteristic
16C2×#13050573C2211yescharacteristic
17#14581787—22no—
18#14581780C2212yes—
19#14581777—22no—
20#14581772C2212yes—
21#14581767C2212yes—
22#14581757C2212yes—
23#14581754—22no—
24#14581754C2212yes—
25#14581750—22no—
26C42×D83C2211yescharacteristic
27#14581715C2212yes—
28#14581715C2212yes—
29#14581712C2212yes—
30#14581712C2212yes—
31#14581702C2211yescharacteristic
32#14581702C2211yescharacteristic
33C4×C83⋊C8C2211yescharacteristic
34#125×C83C2212yes—
35#107×C83C2212yes—
36#106×C83C2211yescharacteristic
37#106×C83C2211yescharacteristic
38C4×C8×C83C2211yescharacteristic
39C2×Dic166C2312yes—
40C2×D332—22no—
41C2×D332C2312yes—
42C2×C4×D83—22no—
43C2×C4×D83D412yes—
44C2×C4×D83C2311yescharacteristic
45Dic332—22no—
46#13050579—24no—
47D664—22no—
48#13050573—22no—
49#13050573—22no—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C4⋊Dic83—22no—
53Dic83⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59C4×Dic83—22no—
60C4×Dic83D412yes—
61C4×Dic83C2311yescharacteristic
62C2×C8×C83C2311yescharacteristic
63C2×C8×C83C2311yescharacteristic
64C4⋊C4×C83—22no—
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C4⋊C4×C83C2312yes—
68C42×C83C2311yescharacteristic
69C22×D83—22no—
70C22×D83C2×D411yescharacteristic
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2411yescharacteristic
75C2×Dic83—22no—
76C2×Dic83C2×D412yes—
77C2×Dic83C2×D411yescharacteristic
78D332—22no—
79D332—22no—
80D332—44no—
81C4×D83—21no—
82C4×D83—21no—
83C4×D83—42no—
84C4×D83—44no—
85Dic166—22no—
86Dic166—22no—
87C8×C83#4112yes—
88C8×C83#4112yes—
89C83⋊C8—21no—
90C83⋊C8—21no—
91C22×C83C22×D411yescharacteristic
92C2×D83—21no—
93C2×D83—22no—
94C2×D83—42no—
95C2×D83—42no—
96C2×D83—42no—
97C4×C83—22no—
98C4×C83—42no—
99C4×C83—42no—
100C4×C83C2×#4111yescharacteristic
101C4×C83C2×#4111yescharacteristic
102Dic83—21no—
103Dic83—22no—
104Dic83—42no—
105C2×C83#53211yescharacteristic
106C2×C83#57512yes—
107D83—41no—
108D83—82no—
109#3183—831nomaximal, Sylow
110C83#318311yescharacteristic, complemented, Sylow
111#529—832no—
112#499D8311yescharacteristic, complemented
113#495—831no—
114#488—832no—
115#470—831no—
116#469—831no—
117#468—832no—
118#442—831no—
119#441—831no—
120#440—832no—
121C4×M16—831no—
122C2×Q16—831no—
123C2×SD16—832no—
124C2×D8—831no—
125C2×M16—831no—
126C2×M16—831no—
127#126—1662no—
128#125C2×D8312yes—
129#124—832no—
130#123—1662no—
131C22⋊Q8—832no—
132#121—832no—
133C4×Q8—832no—
134C4×D4—832no—
135C4×D4—1662no—
136#117—1662no—
137C2×C42—831no—
138#107C2×D8312yes—
139#106C2×D8311yescharacteristic
140#106C2×D8311yescharacteristic
141#103—832no—
142#103—832no—
143#102—832no—
144#102—832no—
145#97—831no—
146#97—831no—
147C4×C8—831no—
148C4×C8C2×D8311yescharacteristic
149C2×Q8—832no—
150C2×D4—832no—
151C2×D4—1662no—
152C22×C4—832no—
153C22×C4—831no—
154C22×C4—1662no—
155Q16—1662no—
156SD16—1664no—
157D8—1662no—
158M16—1662no—
159M16—1662no—
160C2×C8—831no—
161C2×C8—831no—
162C2×C8C22×D8311yescharacteristic
163C2×C8C22×D8311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—1662no—
167C4⋊C4—1662no—
168C4⋊C4C22×D8312yes—
169C4⋊C4C22×D8312yes—
170C22⋊C4—1662no—
171C22⋊C4—1662no—
172C22⋊C4—1662no—
173C22⋊C4—1662no—
174C22⋊C4—1662no—
175C42—832no—
176C42—831no—
177C42—1662no—
178C42C22×D8311yescharacteristic
179C23—831no—
180C23—1662no—
181Q8—1662no—
182Q8—1662no—
183D4—3324no—
184D4—1662no—
185D4—1662no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—3322no—
189C2×C4—3324no—
190C2×C4—832no—
191C2×C4—831no—
192C2×C4—1662no—
193C2×C4—1661no—
194C2×C4—1661no—
195C2×C4D4×D8312yes—
196C2×C4C23×D8311yescharacteristic
197C8—1661no—
198C8—1661no—
199C8#1305059112yes—
200C8#1305059112yes—
201C22—3322no—
202C22—3322no—
203C22—3322no—
204C22—1662no—
205C22—1661no—
206C22C2×D4×D8311yescharacteristic
207C4—22no—
208C4—42no—
209C4—42no—
210C4—3322no—
211C4—1662no—
212C4—1661no—
213C4C2×#1305059111yescharacteristic
214C4C2×#1305059111yescharacteristic
215C2—3321no—
216C2—6642no—
217C2#1912825211yescharacteristic
218C2#1912849812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#499×C83C2
socleC22×C83C22×D4
prime radical#19127955C2
hypercenter#499D83
hypocenterC83#3183
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #19128236C2semidirect
2. #19128085C2semidirect
3. #19128058C2semidirect
4. #19127960C2semidirect
6. #19127948C2semidirect
7. #19127812C2semidirect
8. #19127807C2semidirect
9. #19127800C2semidirect
10. C4×#13050573C2semidirect
11. #499×C83C2semidirect
110. C83#3183semidirect
112. #499D83semidirect

Maximal subgroups

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

subgroupindex
#191282362
#191280852
#191280582
#191279602
#191279552
#191279482
#191278122
#191278072
#191278002
C4×#130505732
#499×C832
#318383