Finite Group Databasebeta

← #24918421

Subgroups — #24918421

Order 10624.

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

normal subgroups only

This group has 394 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#19128258C211yesmaximal, characteristic, complemented
2#19128251C211yesmaximal, characteristic
3#19128081C211yesmaximal, characteristic, complemented
4#19128074C211yesmaximal, characteristic, complemented
5#19128052C211yesmaximal, characteristic, complemented
6#19128048C211yesmaximal, characteristic, complemented
7#19127924C214yesmaximal, complemented
8#19127865C214yesmaximal, complemented
9#431×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2212yes—
11C2×Dic332C2214yes—
12C2×#13050579C2214yes—
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581789C2211yescharacteristic
17#14581784—21no—
18#14581780C2212yes—
19#14581771—21no—
20#14581771C2212yes—
21#14581766—21no—
22#14581766—21no—
23C4×Dic166—21no—
24C4×Dic166C2212yes—
25C4×D332—21no—
26C4×D332—21no—
27#14581751—21no—
28#14581717C2214yes—
29#14581714C2214yes—
30C2×C42×C83C2211yescharacteristic
31C4⋊C8×C83C2212yes—
32C22⋊C8×C83C2212yes—
33C4×C8×C83C2211yescharacteristic
34C4×C8×C83C2211yescharacteristic
35#13050590—44no—
36C2×Dic166—21no—
37C2×Dic166C2312yes—
38C2×Dic166C2312yes—
39C2×D332—21no—
40C2×D332—21no—
41C2×D332C2312yes—
42C2×C22⋊D83—22no—
43C2×C4×D83—22no—
44Dic332—44no—
45#13050579—44no—
46C4⋊Dic83—21no—
47C4⋊Dic83—21no—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54C22⋊Dic83—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57C4×Dic83—22no—
58C22×C4×C83D411yescharacteristic
59C22×C4×C83D411yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C2×C8×C83C2312yes—
62C2×C8×C83C2312yes—
63C42×C83D411yescharacteristic
64C42×C83D411yescharacteristic
65C42×C83C2311yescharacteristic
66C42×C83C2311yescharacteristic
67C23×C83C2×D411yescharacteristic
68C22×D83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—44no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82D332—24no—
83D332—42no—
84D332—42no—
85C4×D83—44no—
86Dic166—24no—
87Dic166—24no—
88Dic166—42no—
89C8×C83—22no—
90C8×C83—22no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83C22×D411yescharacteristic
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83#13512yes—
102C4×C83#13712yes—
103C4×C83H(5,𝔽2)12yes—
104Dic83—42no—
105Dic83—42no—
106Dic83—42no—
107C2×C83—41no—
108C2×C83#53511yescharacteristic
109C2×C83C2×#13511yescharacteristic
110C2×C83C2×#13711yescharacteristic
111D83—82no—
112#3210—831nomaximal, Sylow
113C83#321011yescharacteristic, complemented, Sylow
114#533—831no—
115#532—831no—
116#494—831no—
117#492—831no—
118#487—831no—
119#486—831no—
120#464—834no—
121#452—834no—
122#431D8311yescharacteristic, complemented
123C2×#41—832no—
124C2×Q16—834no—
125C2×SD16—834no—
126C4⋊Q8—831no—
127C4⋊Q8—831no—
128C4⋊Q8—831no—
129#127—831no—
130#125—1661no—
131#124—832no—
132C22⋊Q8—832no—
133C22⋊Q8—1661no—
134#121—1661no—
135#121—1661no—
136C4×Q8—832no—
137C4×Q8—1661no—
138C4×D4—1661no—
139C4×D4—1661no—
140#117—1661no—
141C2×C42C2×D8311yescharacteristic
142C4⋊C8C2×D8312yes—
143#103—834no—
144#102—834no—
145C22⋊C8C2×D8312yes—
146C4×C8C2×D8311yescharacteristic
147C4×C8C2×D8311yescharacteristic
148#41—3324no—
149C2×Q8—832no—
150C2×Q8—832no—
151C2×Q8—1661no—
152C2×D4—832no—
153C2×D4—1662no—
154C2×D4—1661no—
155C2×D4—1661no—
156C22×C4—1662no—
157C22×C4D33211yescharacteristic
158C22×C4D33211yescharacteristic
159C22×C4C22×D8311yescharacteristic
160Q16—3324no—
161SD16—3324no—
162C2×C8C22×D8312yes—
163C2×C8C22×D8312yes—
164C4⋊C4—832no—
165C4⋊C4—1661no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C4⋊C4—1661no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1661no—
172C22⋊C4—1662no—
173C22⋊C4—1662no—
174C22⋊C4—1662no—
175C42—1662no—
176C42D33211yescharacteristic
177C42D33211yescharacteristic
178C42C22×D8311yescharacteristic
179C42C22×D8311yescharacteristic
180C23—1662no—
181C23C2×D33211yescharacteristic
182Q8—3322no—
183Q8—1664no—
184Q8—1664no—
185D4—3324no—
186D4—3322no—
187D4—3322no—
188D4—1664no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—3324no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4C2×D33211yescharacteristic
197C2×C4C2×D33211yescharacteristic
198C2×C4C2×D33211yescharacteristic
199C2×C4C2×D33211yescharacteristic
200C2×C4C2×D33211yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—22no—
203C8—22no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—3322no—
208C22—3322no—
209C22—3322no—
210C22C22×D33211yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—3322no—
215C4—3322no—
216C4—3322no—
217C4#1458181412yes—
218C4#1458182512yes—
219C4#1458183812yes—
220C2—41no—
221C2—6642no—
222C2#1912827011yescharacteristic
223C2C2×#1458181411yescharacteristic
224C2C2×#1458182511yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#431×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#431D83
hypocenterC83#3210
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
3. #19128081C2semidirect
4. #19128074C2semidirect
5. #19128052C2semidirect
6. #19128048C2semidirect
7. #19127924C2semidirect
8. #19127865C2semidirect
9. #431×C83C2semidirect
113. C83#3210semidirect
122. #431D83semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191282512
#191280812
#191280742
#191280522
#191280482
#191279242
#191278652
#431×C832
#321083