Finite Group Databasebeta

← #44560617

Subgroups — #44560617

Order 32128.

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#30632299C211yesmaximal, characteristic, complemented
2#30632028C212yesmaximal, complemented
3#30631987C212yesmaximal, complemented
4#30631892C212yesmaximal, complemented
5#30631838C212yesmaximal
6#30631787C212yesmaximal, complemented
7C4×#16860798C212yesmaximal, complemented
8#30631748C211yesmaximal, characteristic, complemented
9#518×C251C211yesmaximal, characteristic
10C2×#16860798C2212yes—
11C2×#16860798C2212yes—
12#22548715C2211yescharacteristic, complemented
13#22548708—22no—
14#22548707C2211yescharacteristic, complemented
15#22548705—22no—
16#22548694C2212yes—
17C4×Dic502C2212yescomplemented
18#22548648C2212yescomplemented
19#22548646C2212yescomplemented
20C4⋊(C251⋊C8)C2212yescomplemented
21#22548638C2212yescomplemented
22#22548638C2212yescomplemented
23#22548636C2212yes—
24#22548636C2212yes—
25C22⋊(C251⋊C8)C2212yes—
26#22548626C2211yescharacteristic, complemented
27C4×C251⋊C8C2211yescharacteristic, complemented
28C22×D4×C251C2211yescharacteristic
29C4×D4×C251—22no—
30C4×D4×C251C2212yes—
31C4×D4×C251C2212yes—
32#117×C251C2211yescharacteristic
33#117×C251C2211yescharacteristic
34C2×C22⋊C4×C251—22no—
35C2×Dic502C2312yes—
36#16860798—24no—
37#16860798—24no—
38C2×C251⋊C8C2312yes—
39C2×C251⋊C8C2312yes—
40C4⋊Dic251—22no—
41C4⋊Dic251C2312yes—
42Dic251⋊C4—22no—
43Dic251⋊C4—22no—
44Dic251⋊C4—22no—
45C22⋊Dic251—22no—
46C4×Dic251—22no—
47C24×C251—21no—
48C2×D4×C251—24no—
49C2×D4×C251—21no—
50C2×D4×C251C2312yes—
51C22×C4×C251—22no—
52C22×C4×C251—22no—
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251D412yes—
55C4⋊C4×C251C2312yes—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—22no—
58C22⋊C4×C251—22no—
59C22⋊C4×C251D412yes—
60C42×C251—21no—
61C42×C251C2311yescharacteristic
62C42×C251C2311yescharacteristic
63Dic502—22no—
64Dic502—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C251⋊C8—22no—
68C251⋊C8—22no—
69C23×C251—24no—
70C23×C251—22no—
71C23×C251—82no—
72C23×C251C2×D411yescharacteristic
73D4×C251—24no—
74D4×C251—22no—
75D4×C251#4114yes—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—82no—
79C2×C4×C251—82no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D412yes—
82C2×C4×C251C2×D412yes—
83C2×C4×C251C2411yescharacteristic
84C22×C251—21no—
85C22×C251—24no—
86C22×C251—22no—
87C22×C251—42no—
88C22×C251—42no—
89C22×C251—84no—
90C22×C251C22×D411yescharacteristic
91C4×C251—21no—
92C4×C251—22no—
93C4×C251—22no—
94C4×C251—42no—
95C4×C251C2×#4112yes—
96Dic251—42no—
97Dic251—42no—
98C2×C251—41no—
99C2×C251—44no—
100C2×C251#52511yescharacteristic
101C2×C251#57712yes—
102C251#312311yescharacteristic, complemented, Sylow
103#3123—2511nomaximal, Sylow
104#541—2511no—
105#518D25111yescharacteristic
106#485—2512no—
107#478—2512no—
108#461—2512no—
109#450—2512no—
110#440—2512no—
111C4×SD16—2512no—
112#433—2511no—
113C22×D4C2×D25111yescharacteristic
114C2×SD16—2512no—
115C2×SD16—2512no—
116C4⋊Q8—2511no—
117#126—5022no—
118#125—5022no—
119#125—2511no—
120C22⋊Q8—2512no—
121C4×Q8—2512no—
122C4×D4—22no—
123C4×D4C2×D25112yes—
124C4×D4C2×D25112yes—
125#117C2×D25111yescharacteristic
126#117C2×D25111yescharacteristic
127C2×C22⋊C4—22no—
128#107—2512no—
129#106—2512no—
130C4⋊C8—2512no—
131#103—2512no—
132#103—2512no—
133#102—2512no—
134#102—2512no—
135C22⋊C8—2512no—
136#97—2511no—
137C4×C8—2511no—
138C24—21no—
139C2×Q8—2512no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D25112yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D25111yescharacteristic
146SD16—5024no—
147SD16—5024no—
148C2×C8—2512no—
149C2×C8—2512no—
150C4⋊C4—5022no—
151C4⋊C4—5022no—
152C4⋊C4—5022no—
153C4⋊C4—5022no—
154C4⋊C4—2512no—
155C4⋊C4C22⋊D25112yes—
156C4⋊C4C22×D25112yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—5022no—
161C22⋊C4C22⋊D25112yes—
162C42—21no—
163C42—5022no—
164C42C22×D25111yescharacteristic
165C42C22×D25111yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D25111yescharacteristic
170Q8—5022no—
171Q8—5022no—
172D4—24no—
173D4—22no—
174D4#1686081014yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—5022no—
180C2×C4—5022no—
181C2×C4C2×C22⋊D25111yescharacteristic
182C2×C4C2×C22⋊D25112yes—
183C2×C4C2×C22⋊D25112yes—
184C2×C4C23×D25111yescharacteristic
185C8—5022no—
186C8—5022no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D25111yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—10042no—
199C4—10042no—
200C4C2×#1686081012yes—
201C2—41no—
202C2—44no—
203C2#3063219811yescharacteristic
204C2#3063249812yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#518×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenter#518D251
hypocenterC251#3123
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #30632299C2semidirect
2. #30632028C2semidirect
3. #30631987C2semidirect
4. #30631892C2semidirect
6. #30631787C2semidirect
7. C4×#16860798C2semidirect
8. #30631748C2semidirect
12. #22548715C22semidirect
14. #22548707C22semidirect
17. C4×Dic502C22semidirect
18. #22548648C22semidirect
19. #22548646C22semidirect
20. C4⋊(C251⋊C8)C22semidirect
21. #22548638C22semidirect
22. #22548638C22semidirect
26. #22548626C22semidirect
27. C4×C251⋊C8C22semidirect
102. C251#3123semidirect

Maximal subgroups

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

subgroupindex
#306322992
#306320282
#306319872
#306318922
#306318382
#306317872
C4×#168607982
#306317482
#518×C2512
#3123251