Finite Group Databasebeta

← #44560608

Subgroups — #44560608

Order 32128.

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

normal subgroups only

This group has 418 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#30632292C211yesmaximal, characteristic, complemented
2#30632011C212yesmaximal, complemented
3#30631987C212yesmaximal, complemented
4#30631882C212yesmaximal, complemented
5#30631821C212yesmaximal, complemented
6#30631797C212yesmaximal, complemented
7C4×#16860795C212yesmaximal, complemented
8#30631748C211yesmaximal, characteristic, complemented
9#518×C251C211yesmaximal, characteristic, complemented
10C2×#16860795C2212yes—
11C2×#16860795C2212yes—
12#22548712C2211yescharacteristic, complemented
13#22548707C2211yescharacteristic, complemented
14#22548700—22no—
15#22548689C2212yes—
16#22548688—22no—
17C4×D1004C2212yescomplemented
18#22548648C2212yescomplemented
19#22548646C2212yescomplemented
20C4⋊(C251⋊C8)C2212yescomplemented
21#22548636C2212yes—
22#22548636C2212yes—
23#22548635C2212yescomplemented
24#22548635C2212yescomplemented
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×D1004—22no—
36C2×D1004C2312yes—
37C2×C22⋊D251—22no—
38C2×C4×D251—22no—
39#16860795—24no—
40#16860795—24no—
41C2×C251⋊C8C2312yes—
42C2×C251⋊C8C2312yes—
43C4⋊Dic251C2312yes—
44Dic251⋊C4—22no—
45(C2×D251)⋊C4—22no—
46(C2×D251)⋊C4—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
63C22×D251—22no—
64D1004—22no—
65D1004—22no—
66D1004—82no—
67C22⋊D251—82no—
68C2×Dic251—22no—
69C4×D251—82no—
70C251⋊C8—22no—
71C251⋊C8—22no—
72C23×C251—24no—
73C23×C251—22no—
74C23×C251—82no—
75C23×C251C2×D411yescharacteristic
76D4×C251—24no—
77D4×C251—22no—
78D4×C251#4114yes—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—82no—
82C2×C4×C251—82no—
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2×D412yes—
86C2×C4×C251C2411yescharacteristic
87C22×C251—21no—
88C22×C251—24no—
89C22×C251—22no—
90C22×C251—42no—
91C22×C251—42no—
92C22×C251—84no—
93C22×C251C22×D411yescharacteristic
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—42no—
97C4×C251—21no—
98C4×C251—22no—
99C4×C251—22no—
100C4×C251—42no—
101C4×C251C2×#4112yes—
102Dic251—42no—
103C2×C251—41no—
104C2×C251—44no—
105C2×C251#52511yescharacteristic
106C2×C251#57612yes—
107D251—82no—
108C251#312211yescharacteristic, complemented, Sylow
109#3122—2511nomaximal, Sylow
110#540—2511no—
111#518D25111yescharacteristic, complemented
112#482—2512no—
113#478—2512no—
114#459—2512no—
115#447—2512no—
116#442—2512no—
117C4×D8—2512no—
118#433—2511no—
119C22×D4C2×D25111yescharacteristic
120C2×D8—2512no—
121C2×D8—2512no—
122#127—2511no—
123#125—2511no—
124#123—5022no—
125#121—5022no—
126#121—2512no—
127C4×D4—22no—
128C4×D4—2512no—
129C4×D4C2×D25112yes—
130C4×D4C2×D25112yes—
131#117C2×D25111yescharacteristic
132#117C2×D25111yescharacteristic
133C2×C22⋊C4—22no—
134#107—2512no—
135#106—2512no—
136C4⋊C8—2512no—
137#102—2512no—
138#102—2512no—
139#102—2512no—
140#102—2512no—
141C22⋊C8—2512no—
142#97—2511no—
143C4×C8—2511no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—5022no—
148C2×D4—5022no—
149C2×D4—2512no—
150C2×D4C22×D25112yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—5022no—
154C22×C4C22×D25111yescharacteristic
155D8—5024no—
156D8—5024no—
157C2×C8—2512no—
158C2×C8—2512no—
159C4⋊C4—5022no—
160C4⋊C4—2512no—
161C4⋊C4C22⋊D25112yes—
162C4⋊C4C22×D25112yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—5022no—
167C22⋊C4—5022no—
168C22⋊C4C22⋊D25112yes—
169C42—21no—
170C42C22×D25111yescharacteristic
171C42C22×D25111yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—5022no—
176C23C2×C22⋊D25111yescharacteristic
177D4—24no—
178D4—22no—
179D4—20082no—
180D4—20082no—
181D4—5022no—
182D4—5022no—
183D4#1686081014yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—20082no—
189C2×C4—5022no—
190C2×C4C2×C22⋊D25111yescharacteristic
191C2×C4C2×C22⋊D25112yes—
192C2×C4C2×C22⋊D25112yes—
193C2×C4C23×D25111yescharacteristic
194C8—5022no—
195C8—5022no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—10042no—
203C22—10042no—
204C22—10042no—
205C22C22×C22⋊D25111yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—10042no—
211C4C2×#1686081012yes—
212C2—41no—
213C2—44no—
214C2—20082no—
215C2#3063219811yescharacteristic
216C2#3063249012yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#518×C251C2
socleC22×C251C22×D4
hypercenter#518D251
hypocenterC251#3122
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. #30632292C2semidirect
2. #30632011C2semidirect
3. #30631987C2semidirect
4. #30631882C2semidirect
5. #30631821C2semidirect
6. #30631797C2semidirect
7. C4×#16860795C2semidirect
8. #30631748C2semidirect
9. #518×C251C2semidirect
12. #22548712C22semidirect
13. #22548707C22semidirect
17. C4×D1004C22semidirect
18. #22548648C22semidirect
19. #22548646C22semidirect
20. C4⋊(C251⋊C8)C22semidirect
23. #22548635C22semidirect
24. #22548635C22semidirect
26. #22548626C22semidirect
27. C4×C251⋊C8C22semidirect
108. C251#3122semidirect
111. #518D251semidirect

Maximal subgroups

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

subgroupindex
#306322922
#306320112
#306319872
#306318822
#306318212
#306317972
C4×#168607952
#306317482
#518×C2512
#3122251