Finite Group Databasebeta

← #24915513

Subgroups — #24915513

Order 10624.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14581780C211yesmaximal, characteristic
2#19127636C212yesmaximal, complemented
3#19127627C212yesmaximal, complemented
4#19127623C214yesmaximal, complemented
5#19127621C212yesmaximal, complemented
6#19127618C212yesmaximal
7#19127567C211yesmaximal, characteristic, complemented
8C2×#125×C83C211yesmaximal, characteristic, complemented
9C22×Dic166C2211yescharacteristic
10C22×D332C2211yescharacteristic
11#14581780—24no—
12C2×C4⋊Dic83C2212yes—
13C2×Dic83⋊C4C2214yes—
14C2×(C2×D83)⋊C4C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2214yes—
17C2×(C2×D83)⋊C4C2214yes—
18C83⋊#95C2212yes—
19C83⋊#95C2212yes—
20C83⋊#95C2214yes—
21#125×C83—24no—
22C2×C4⋊C4×C83C2212yes—
23C2×C4⋊C4×C83C2212yes—
24C2×C4⋊C4×C83C2211yescharacteristic
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C42×C83C2211yescharacteristic
27C23×D83C2312yes—
28C2×Dic166—22no—
29C2×Dic166—42no—
30C2×D332—22no—
31C2×D332—42no—
32C22×Dic83C2312yes—
33C22×Dic83C2314yes—
34C4⋊Dic83—42no—
35Dic83⋊C4—44no—
36(C2×D83)⋊C4—24no—
37(C2×D83)⋊C4—24no—
38(C2×D83)⋊C4—44no—
39(C2×D83)⋊C4—44no—
40C22×C4×C83C2312yes—
41C22×C4×C83C2312yes—
42C22×C4×C83C2311yescharacteristic
43C22×C4×C83C2311yescharacteristic
44C22×C4×C83C2311yescharacteristic
45C4⋊C4×C83—24no—
46C4⋊C4×C83—22no—
47C4⋊C4×C83—22no—
48C4⋊C4×C83—24no—
49C42×C83D414yes—
50C23×C83C2411yescharacteristic
51C22×D83—22no—
52C22×D83—22no—
53C22×D83—24no—
54C22×D83—22no—
55C22×D83—24no—
56C2×C4×C83—21no—
57C2×C4×C83—22no—
58C2×C4×C83—21no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—24no—
64C2×C4×C83—22no—
65C2×C4×C83C2×D412yes—
66C2×C4×C83C2×D412yes—
67C2×C4×C83C2×D412yes—
68C2×Dic83—22no—
69C2×Dic83—24no—
70C2×Dic83—24no—
71C2×Dic83—24no—
72C2×Dic83—24no—
73D332—44no—
74Dic166—44no—
75C22×C83C22×D411yescharacteristic
76C22×C83H(5,𝔽2)12yes—
77C22×C83H(5,𝔽2)11yescharacteristic
78C22×C832−1+411yescharacteristic
79C22×C832−1+412yes—
80C2×D83—42no—
81C2×D83—42no—
82C2×D83—42no—
83C2×D83—44no—
84C2×D83—44no—
85C4×C83—22no—
86C4×C83—22no—
87C4×C83—22no—
88C4×C83—42no—
89C4×C83—42no—
90Dic83—42no—
91Dic83—44no—
92C2×C83#53511yescharacteristic
93C2×C83#53611yescharacteristic
94C2×C83#53711yescharacteristic
95C2×C83#56212yes—
96C2×C83#56312yes—
97D83—82no—
98#2820—831nomaximal, Sylow
99C83#282011yescharacteristic, complemented, Sylow
100C2×#125D8311yescharacteristic, complemented
101C2×#124—831no—
102#399—832no—
103#397—832no—
104#396—834no—
105#395—832no—
106#394—832no—
107#382—831no—
108C22×Q8—831no—
109C22×D4—831no—
110#125—24no—
111#124—1664no—
112C2×C4⋊C4—832no—
113C2×C4⋊C4—834no—
114C2×C4⋊C4C2×D8312yes—
115C2×C4⋊C4C2×D8312yes—
116C2×C4⋊C4C2×D8311yescharacteristic
117C2×C4⋊C4C2×D8311yescharacteristic
118C2×C22⋊C4—834no—
119C2×C22⋊C4—834no—
120C2×C22⋊C4—832no—
121C2×C22⋊C4—832no—
122C2×C42C2×D8311yescharacteristic
123#95—832no—
124#95—832no—
125#95—834no—
126C24—832no—
127C2×Q8—3322no—
128C2×Q8—1662no—
129C2×D4—3322no—
130C2×D4—1662no—
131C22×C4—834no—
132C22×C4—832no—
133C22×C4C22×D8312yes—
134C22×C4C22×D8312yes—
135C22×C4C22×D8311yescharacteristic
136C22×C4C22×D8311yescharacteristic
137C22×C4C22×D8311yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—3324no—
143C4⋊C4—3322no—
144C22⋊C4—3324no—
145C22⋊C4—3324no—
146C22⋊C4—1664no—
147C22⋊C4—1664no—
148C42C22⋊D8314yes—
149C23—1662no—
150C23—1662no—
151C23—1664no—
152C23—1662no—
153C23—1664no—
154C23C23×D8311yescharacteristic
155Q8—3324no—
156D4—3324no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—1662no—
167C2×C4—1664no—
168C2×C4—1664no—
169C2×C4—1664no—
170C2×C4—1664no—
171C2×C4C2×C22⋊D8312yes—
172C2×C4C2×C22⋊D8312yes—
173C2×C4C2×C22⋊D8312yes—
174C22—3324no—
175C22—3324no—
176C22—3322no—
177C22—3322no—
178C22—3322no—
179C22C22×C22⋊D8311yescharacteristic
180C22#1458183811yescharacteristic
181C22#1458183812yes—
182C22#1458183911yescharacteristic
183C22#1458184012yes—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—3324no—
190C4—3322no—
191C2—6642no—
192C2#1912826511yescharacteristic
193C2#1912827511yescharacteristic
194C2#1912827911yescharacteristic
195C2#1912844912yes—
196C2#1912845512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#125×C83C2
prime radicalC22×D332C22
hypercenterC2×#125D83
hypocenterC83#2820
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #19127636C2semidirect
3. #19127627C2semidirect
4. #19127623C2semidirect
5. #19127621C2semidirect
7. #19127567C2semidirect
8. C2×#125×C83C2semidirect
99. C83#2820semidirect
100. C2×#125D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817802
#191276362
#191276272
#191276232
#191276212
#191276182
#191275672
C2×#125×C832
#282083