Finite Group Databasebeta

← #24918898

Subgroups — #24918898

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#19128306C211yesmaximal, characteristic, complemented
2#19128001C212yesmaximal, complemented
3#19127998C212yesmaximal, complemented
4#19127952C212yesmaximal, complemented
5#19127935C212yesmaximal, complemented
6#19127896C212yesmaximal, complemented
7#19127893C212yesmaximal, complemented
8#19127728C211yesmaximal, characteristic, complemented
9#540×C83C211yesmaximal, characteristic, complemented
10C2×#13050575C2212yes—
11C2×#13050575C2212yes—
12C2×#13050574C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581784C2211yescharacteristic
16#14581777—22no—
17#14581766C2212yes—
18#14581765—22no—
19C4×D332C2212yes—
20#14581725C2212yes—
21#14581723C2212yes—
22C4⋊(C83⋊C8)C2212yes—
23C4⋊(C83⋊C8)C2212yes—
24#14581713C2212yes—
25#14581713C2212yes—
26#14581712C2212yes—
27#14581712C2212yes—
28#127×C83C2211yescharacteristic
29#125×C83C2211yescharacteristic
30#123×C83—22no—
31#121×C83—22no—
32#121×C83C2212yes—
33C4×D4×C83C2212yes—
34#117×C83C2211yescharacteristic
35C2×D332—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050575—42no—
40#13050575—42no—
41#13050574—41no—
42C2×C83⋊C8—41no—
43C2×C83⋊C8C2312yes—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—22no—
47(C2×D83)⋊C4—22no—
48(C2×D83)⋊C4—22no—
49C2×D4×C83—22no—
50C2×D4×C83—22no—
51C2×D4×C83C2312yes—
52C22×C4×C83—22no—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—22no—
55C4⋊C4×C83D412yes—
56C4⋊C4×C83C2312yes—
57C22⋊C4×C83—22no—
58C22⋊C4×C83—22no—
59C22⋊C4×C83D412yes—
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83—22no—
63C23×C83C2×D411yescharacteristic
64C22×D83—22no—
65D4×C83—22no—
66D4×C83—22no—
67D4×C83—82no—
68D4×C83—82no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—82no—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2411yescharacteristic
77C22⋊D83—82no—
78C2×Dic83—22no—
79D332—22no—
80D332—22no—
81D332—82no—
82C4×D83—82no—
83C83⋊C8—22no—
84C83⋊C8—22no—
85C22×C83—21no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83C22×D411yescharacteristic
91C2×D83—42no—
92C2×D83—42no—
93C2×D83—42no—
94C4×C83—21no—
95C4×C83—22no—
96C4×C83—22no—
97C4×C83—42no—
98C4×C83H(5,𝔽2)11yescharacteristic
99C4×C832−1+411yescharacteristic
100Dic83—42no—
101C2×C83—41no—
102C2×C83—82no—
103C2×C83#53711yescharacteristic
104C2×C83#57612yes—
105D83—82no—
106#3259—831nomaximal, Sylow
107C83#325911yescharacteristic, complemented, Sylow
108#540—831no—
109#540D8311yescharacteristic, complemented
110#478—832no—
111#478—832no—
112#469—832no—
113#466—832no—
114#459—832no—
115#459—832no—
116#424—831no—
117C2×D8—832no—
118C2×D8—832no—
119C2×M16—831no—
120C22×C8—831no—
121#127—831no—
122#127C2×D8311yescharacteristic
123#125—831no—
124#125C2×D8311yescharacteristic
125#123—22no—
126#123—1662no—
127#121—22no—
128#121—832no—
129#121—1662no—
130#121C2×D8312yes—
131C4×D4—832no—
132C4×D4C2×D8312yes—
133#117C2×D8311yescharacteristic
134#107—832no—
135#106—832no—
136C4⋊C8—832no—
137C4⋊C8—832no—
138#102—832no—
139#102—832no—
140#102—832no—
141#102—832no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—832no—
145C2×D4—1662no—
146C2×D4—1662no—
147C2×D4C22×D8312yes—
148C22×C4—22no—
149C22×C4—1662no—
150C22×C4C22×D8311yescharacteristic
151D8—3322no—
152D8—3322no—
153M16—3321no—
154C2×C8—3321no—
155C2×C8—832no—
156C2×C8—832no—
157C4⋊C4—22no—
158C4⋊C4—832no—
159C4⋊C4—1662no—
160C4⋊C4C22⋊D8312yes—
161C4⋊C4C22×D8312yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—1662no—
165C22⋊C4—1662no—
166C22⋊C4C22⋊D8312yes—
167C42C22×D8311yescharacteristic
168C42C22×D8311yescharacteristic
169C23—22no—
170C23—1662no—
171C23C2×C22⋊D8311yescharacteristic
172D4—22no—
173D4—22no—
174D4—82no—
175D4—82no—
176D4—6642no—
177D4—6642no—
178D4—1662no—
179D4—1662no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—6642no—
185C2×C4—1662no—
186C2×C4C2×C22⋊D8312yes—
187C2×C4C2×C22⋊D8312yes—
188C2×C4C2×C22⋊D8311yescharacteristic
189C2×C4C23×D8311yescharacteristic
190C8—1662no—
191C8—1662no—
192C22—22no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—3322no—
198C22—3322no—
199C22—3322no—
200C22C22×C22⋊D8311yescharacteristic
201C4—22no—
202C4—22no—
203C4—21no—
204C4—42no—
205C4—3322no—
206C4#1458183811yescharacteristic
207C4#1458183911yescharacteristic
208C2—41no—
209C2—82no—
210C2—6642no—
211C2#1912827911yescharacteristic
212C2#1912850412yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#540×C83C2
socleC22×C83C22×D4
hypercenter#540D83
hypocenterC83#3259
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #19128306C2semidirect
2. #19128001C2semidirect
3. #19127998C2semidirect
4. #19127952C2semidirect
5. #19127935C2semidirect
6. #19127896C2semidirect
7. #19127893C2semidirect
8. #19127728C2semidirect
9. #540×C83C2semidirect
107. C83#3259semidirect
109. #540D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191280012
#191279982
#191279522
#191279352
#191278962
#191278932
#191277282
#540×C832
#325983