Finite Group Databasebeta

← #24918601

Subgroups — #24918601

Order 10624.

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

normal subgroups only

This group has 370 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#19128106C211yesmaximal, characteristic, complemented
4#19128099C211yesmaximal, characteristic, complemented
5#19128048C211yesmaximal, characteristic, complemented
6#19128048C211yesmaximal, characteristic, complemented
7#19128026C214yesmaximal, complemented
8#19128002C214yesmaximal, complemented
9#431×C83C211yesmaximal, characteristic, complemented
10C2×#13050590—21no—
11#14581792C2211yescharacteristic
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581784C2212yes—
16#14581780—21no—
17#14581771—21no—
18#14581771—21no—
19#14581766—21no—
20#14581766C2212yes—
21C4×Dic166—21no—
22C4×Dic166—21no—
23C4×D332—21no—
24C4×D332C2212yes—
25#14581751C2212yes—
26#14581726C2214yes—
27#14581724C2214yes—
28#14581714C2214yes—
29#14581714C2214yes—
30C2×C42×C83C2211yescharacteristic
31C4⋊C8×C83C2212yes—
32C22⋊C8×C83C2212yes—
33C4×C8×C83C2211yescharacteristic
34C4×C8×C83C2211yescharacteristic
35#13050590—44no—
36C2×Dic166—21no—
37C2×Dic166—21no—
38C2×Dic166—21no—
39C2×D332—21no—
40C2×D332—21no—
41C2×D332C2312yes—
42C2×C22⋊D83—22no—
43C2×C4×D83—22no—
44C4⋊Dic83—21no—
45C4⋊Dic83—21no—
46C4⋊Dic83—21no—
47C4⋊Dic83C2312yes—
48C4⋊Dic83C2312yes—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52C22⋊Dic83—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55C4×Dic83—22no—
56C22×C4×C83D411yescharacteristic
57C22×C4×C83D411yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C2×C8×C83C2312yes—
60C2×C8×C83C2312yes—
61C42×C83D411yescharacteristic
62C42×C83D411yescharacteristic
63C42×C83C2311yescharacteristic
64C42×C83C2311yescharacteristic
65C23×C83C2×D411yescharacteristic
66C22×D83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C22⋊D83—44no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80D332—24no—
81D332—42no—
82D332—42no—
83C4×D83—44no—
84Dic166—42no—
85Dic166—42no—
86Dic166—42no—
87C8×C83—22no—
88C8×C83—22no—
89C22×C83—21no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83C22×D411yescharacteristic
93C2×D83—42no—
94C2×D83—42no—
95C2×D83—42no—
96C4×C83—21no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83#13512yes—
100C4×C83AGL(1,ℤ/8)12yes—
101C4×C832−1+412yes—
102Dic83—42no—
103Dic83—42no—
104Dic83—42no—
105C2×C83—41no—
106C2×C83#53611yescharacteristic
107C2×C83C2×#13511yescharacteristic
108C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
109D83—82no—
110#3228—831nomaximal, Sylow
111C83#322811yescharacteristic, complemented, Sylow
112#533—831no—
113#532—831no—
114#500—831no—
115#498—831no—
116#486—831no—
117#486—831no—
118#482—834no—
119#478—834no—
120#431D8311yescharacteristic, complemented
121C2×#41—1661no—
122C4⋊Q8—831no—
123C4⋊Q8—831no—
124C4⋊Q8—831no—
125#127—831no—
126#125—832no—
127#124—1661no—
128C22⋊Q8—1661no—
129C22⋊Q8—1661no—
130#121—832no—
131#121—1661no—
132C4×Q8—1661no—
133C4×Q8—1661no—
134C4×D4—832no—
135C4×D4—1661no—
136#117—832no—
137C2×C42C2×D8311yescharacteristic
138#107—834no—
139#106—834no—
140C4⋊C8C2×D8312yes—
141#102—834no—
142#102—834no—
143C22⋊C8C2×D8312yes—
144C4×C8C2×D8311yescharacteristic
145C4×C8C2×D8311yescharacteristic
146#41—3324no—
147C2×Q8—1661no—
148C2×Q8—1661no—
149C2×Q8—1661no—
150C2×D4—832no—
151C2×D4—1662no—
152C2×D4—1661no—
153C2×D4—1661no—
154C22×C4—1662no—
155C22×C4D33211yescharacteristic
156C22×C4D33211yescharacteristic
157C22×C4C22×D8311yescharacteristic
158C2×C8C22×D8312yes—
159C2×C8C22×D8312yes—
160C4⋊C4—832no—
161C4⋊C4—832no—
162C4⋊C4—832no—
163C4⋊C4—1661no—
164C4⋊C4—1662no—
165C4⋊C4—1662no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C22⋊C4—1662no—
169C22⋊C4—1662no—
170C22⋊C4—1662no—
171C42—1662no—
172C42D33211yescharacteristic
173C42D33211yescharacteristic
174C42C22×D8311yescharacteristic
175C42C22×D8311yescharacteristic
176C23—1662no—
177C23C2×D33211yescharacteristic
178Q8—3322no—
179Q8—3322no—
180Q8—3322no—
181D4—3324no—
182D4—3322no—
183D4—3322no—
184D4—1664no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—3324no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4C2×D33211yescharacteristic
193C2×C4C2×D33211yescharacteristic
194C2×C4C2×D33211yescharacteristic
195C2×C4C2×D33211yescharacteristic
196C2×C4C2×D33211yescharacteristic
197C2×C4C23×D8311yescharacteristic
198C8—22no—
199C8—22no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—3322no—
204C22—3322no—
205C22—3322no—
206C22C22×D33211yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—3322no—
211C4—3322no—
212C4—3322no—
213C4#1458181412yes—
214C4#1458182012yes—
215C4#1458183912yes—
216C2—41no—
217C2—6642no—
218C2#1912827711yescharacteristic
219C2C2×#1458181411yescharacteristic
220C2C2×#1458182011yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#431×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#431D83
hypocenterC83#3228
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. #19128106C2semidirect
4. #19128099C2semidirect
5. #19128048C2semidirect
6. #19128048C2semidirect
7. #19128026C2semidirect
8. #19128002C2semidirect
9. #431×C83C2semidirect
111. C83#3228semidirect
120. #431D83semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191282512
#191281062
#191280992
#191280482
#191280262
#191280022
#431×C832
#322883