Finite Group Databasebeta

← #44558645

Subgroups — #44558645

Order 32128.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic251C211yesmaximal, characteristic, complemented
2C2×#22548687C213yesmaximal, complemented
3C2×#22548684C211yesmaximal, characteristic, complemented
4#30631567C213yesmaximal, complemented
5#30631564C213yesmaximal, complemented
6#30631545C213yesmaximal, complemented
7C2×#120×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251C2213yes—
9C22×C22⋊D251C2211yescharacteristic, complemented
10C23×Dic251—212no—
11C23×Dic251C2213yes—
12C23×Dic251C2213yes—
13#22548687—212no—
14#22548684—24no—
15C2×Dic251⋊C4C2213yescomplemented
16C2×C22⋊Dic251C2213yes—
17C2×C22⋊Dic251C2213yescomplemented
18C2×(C2×D251)⋊C4C2213yescomplemented
19C251⋊#95C2216yescomplemented
20C25×C251C2211yescharacteristic
21C22×D4×C251C2213yes—
22#120×C251—24no—
23C2×C22⋊C4×C251C2213yes—
24C23×D251C2311yescharacteristic
25C2×C22⋊D251—26no—
26C2×C22⋊D251—26no—
27C2×C22⋊D251—26no—
28C2×C22⋊D251—26no—
29C22×Dic251—232no—
30C22×Dic251—212no—
31C22×Dic251—212no—
32C22×Dic251—26no—
33C22×Dic251D4112yescomplemented
34C22×Dic251C2313yes—
35C22×Dic251C2313yes—
36C22×Dic251C2311yescharacteristic
37Dic251⋊C4—26no—
38C22⋊Dic251—26no—
39C22⋊Dic251—26no—
40(C2×D251)⋊C4—26no—
41C24×C251—26no—
42C24×C251—26no—
43C24×C251D414yescomplemented
44C24×C251C2311yescharacteristic
45C24×C251C2313yes—
46C2×D4×C251—26no—
47C2×D4×C251—26no—
48C2×D4×C251—26no—
49C22×C4×C251C2313yes—
50C22⋊C4×C251—26no—
51C22×D251—23no—
52C22×D251—21no—
53C22×D251—23no—
54C22⋊D251—412no—
55C22⋊D251—44no—
56C2×Dic251—224no—
57C2×Dic251—26no—
58C2×Dic251—224no—
59C2×Dic251—23no—
60C2×Dic251—23no—
61C2×Dic251—23no—
62C2×Dic251C2×D416yes—
63C2×Dic251C2×D416yes—
64C23×C251—224no—
65C23×C251—28no—
66C23×C251—224no—
67C23×C251—26no—
68C23×C251—26no—
69C23×C251—21no—
70C23×C251—23no—
71C23×C251—23no—
72C23×C251C2×D416yes—
73C23×C251C2×D416yes—
74C23×C251#4116yes—
75C23×C251C2411yescharacteristic
76D4×C251—412no—
77C2×C4×C251—23no—
78C2×C4×C251—23no—
79C2×C4×C251—23no—
80C22×C251—232no—
81C22×C251—26no—
82C22×C251—212no—
83C22×C251—26no—
84C22×C251—212no—
85C22×C251—41no—
86C22×C251—43no—
87C22×C251—43no—
88C22×C251#121112yescomplemented
89C22×C251C22×D413yes—
90C22×C251C22×D411yescharacteristic
91C22×C251C2×#4113yes—
92C2×D251—41no—
93C2×D251—43no—
94C2×D251—43no—
95C4×C251—43no—
96Dic251—26no—
97Dic251—43no—
98Dic251#12014yescomplemented
99C2×C251—212no—
100C2×C251—81no—
101C2×C251C2×#12011yescharacteristic
102C2×C251C2×#12113yes—
103C2×C251#52513yes—
104D251—81no—
105C251#284711yescharacteristic, complemented, Sylow
106#2847—2511nomaximal, Sylow
107C24×C4—2511no—
108C2×#121—2513no—
109C2×#120—2511no—
110C2×#120D25111yescharacteristic, complemented
111#386—2513no—
112#386—2513no—
113#380—2513no—
114C25C2×D25111yescharacteristic
115C22×D4—2513no—
116C22×D4—2511no—
117C22×D4C2×D25113yes—
118C23×C4—50212no—
119C23×C4—2513no—
120C23×C4—2513no—
121#121—50212no—
122#120—24no—
123#120—5024no—
124C2×C4⋊C4—2513no—
125C2×C22⋊C4—2513no—
126C2×C22⋊C4—2513no—
127C2×C22⋊C4—2513no—
128C2×C22⋊C4C2×D25113yes—
129#95—2516no—
130C24—26no—
131C24—26no—
132C24—2511no—
133C24C22⋊D25114yescomplemented
134C24C22×D25111yescharacteristic
135C24C22×D25113yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—5026no—
140C2×D4—5026no—
141C2×D4—5026no—
142C2×D4—5026no—
143C22×C4—50232no—
144C22×C4—5026no—
145C22×C4—50212no—
146C22×C4—50212no—
147C22×C4—2513no—
148C22×C4—25112no—
149C22×C4—2513no—
150C22×C4—2511no—
151C22×C4C22×D25113yes—
152C4⋊C4—5026no—
153C22⋊C4—26no—
154C22⋊C4—5026no—
155C22⋊C4—5026no—
156C22⋊C4—5026no—
157C23—28no—
158C23—224no—
159C23—26no—
160C23—26no—
161C23—224no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—5023no—
166C23—5021no—
167C23—5023no—
168C23D4×D25116yes—
169C23C2×C22⋊D25116yes—
170C23#1686080916yes—
171C23C23×D25111yescharacteristic
172D4—412no—
173D4—100412no—
174D4—10044no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—5023no—
179C2×C4—5023no—
180C2×C4—5023no—
181C2×C4—50224no—
182C2×C4—5026no—
183C2×C4—50224no—
184C2×C4—2516no—
185C2×C4—2516no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—43no—
192C22—41no—
193C22—43no—
194C22—10043no—
195C22—10041no—
196C22—10043no—
197C22#22548687112yescomplemented
198C22C2×D4×D25113yes—
199C22C22×C22⋊D25111yescharacteristic
200C22C2×#1686080913yes—
201C4—43no—
202C4—10043no—
203C4—5026no—
204C4—2514no—
205C2—212no—
206C2—81no—
207C2—20081no—
208C2#120×D25111yescharacteristic
209C2C2×#2254868713yes—
210C2#3063219313yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#120×C251C2
hypercenterC2×#120D251
hypocenterC251#2847
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic251C2semidirect
2. C2×#22548687C2semidirect
3. C2×#22548684C2semidirect
4. #30631567C2semidirect
5. #30631564C2semidirect
6. #30631545C2semidirect
7. C2×#120×C251C2semidirect
9. C22×C22⋊D251C22semidirect
15. C2×Dic251⋊C4C22semidirect
17. C2×C22⋊Dic251C22semidirect
18. C2×(C2×D251)⋊C4C22semidirect
19. C251⋊#95C22semidirect
33. C22×Dic251D4semidirect
43. C24×C251D4semidirect
88. C22×C251#121semidirect
98. Dic251#120semidirect
105. C251#2847semidirect
110. C2×#120D251semidirect
133. C24C22⋊D251semidirect
197. C22#22548687semidirect

Maximal subgroups

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

subgroupindex
C24×Dic2512
C2×#225486872
C2×#225486842
#306315672
#306315642
#306315452
C2×#120×C2512
#2847251