Finite Group Databasebeta

← #44560603

Subgroups — #44560603

Order 32128.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#30632161C211yesmaximal, characteristic, complemented
2#30632008C212yesmaximal, complemented
3#30631984C212yesmaximal, complemented
4#30631879C212yesmaximal, complemented
5#30631820C212yesmaximal, complemented
6#30631797C212yesmaximal, complemented
7C4×#16860795C212yesmaximal, complemented
8#30631748C211yesmaximal, characteristic, complemented
9#540×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C2×#16860795C2212yes—
12C2×#16860795C2212yes—
13C4×D1004C2212yes—
14C4×D1004C2212yes—
15#22548677—22no—
16#22548674C2211yescharacteristic
17C22⋊C4×D251—22no—
18#22548648C2212yescomplemented
19#22548646C2212yescomplemented
20C4⋊(C251⋊C8)C2212yescomplemented
21#22548636C2212yescomplemented
22#22548636C2212yescomplemented
23#22548635C2212yes—
24#22548635C2212yes—
25C22⋊(C251⋊C8)C2212yes—
26#22548626C2211yescharacteristic, complemented
27C4×C251⋊C8C2211yescharacteristic, complemented
28#127×C251C2211yescharacteristic, complemented
29#125×C251C2211yescharacteristic, complemented
30#123×C251—22no—
31#121×C251—22no—
32#121×C251C2212yes—
33C4×D4×C251C2212yescomplemented
34#117×C251C2211yescharacteristic
35C23×D251—21no—
36C2×D1004—24no—
37C2×D1004—21no—
38C2×D1004C2312yes—
39C2×C4×D251—22no—
40C2×C4×D251—22no—
41#16860795—24no—
42#16860795—24no—
43C2×C251⋊C8C2312yes—
44C2×C251⋊C8C2312yes—
45C4⋊Dic251C2312yes—
46C22⋊Dic251—21no—
47(C2×D251)⋊C4—22no—
48(C2×D251)⋊C4—22no—
49C4×Dic251—21no—
50C2×D4×C251—22no—
51C2×D4×C251—22no—
52C2×D4×C251C2312yes—
53C22×C4×C251—22no—
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—22no—
56C4⋊C4×C251D412yes—
57C4⋊C4×C251C2312yes—
58C22⋊C4×C251—22no—
59C22⋊C4×C251—22no—
60C22⋊C4×C251D412yes—
61C42×C251C2311yescharacteristic
62C42×C251C2311yescharacteristic
63C22×D251—22no—
64C22×D251—24no—
65C22×D251—82no—
66D1004—24no—
67D1004—22no—
68D1004#4114yes—
69C2×Dic251—22no—
70C4×D251—82no—
71C4×D251—82no—
72C251⋊C8—22no—
73C251⋊C8—22no—
74C23×C251—22no—
75C23×C251C2×D411yescharacteristic
76D4×C251—22no—
77D4×C251—22no—
78D4×C251—82no—
79D4×C251—82no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—82no—
83C2×C4×C251C2×D412yes—
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2411yescharacteristic
87C22×C251—21no—
88C22×C251—22no—
89C22×C251—42no—
90C22×C251—42no—
91C22×C251—42no—
92C22×C251C22×D411yescharacteristic
93C2×D251—24no—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—84no—
97C4×C251—21no—
98C4×C251—22no—
99C4×C251—22no—
100C4×C251—42no—
101C4×C251C2×#4112yes—
102Dic251—42no—
103C2×C251—41no—
104C2×C251—82no—
105C2×C251#52511yescharacteristic
106C2×C251#57612yes—
107D251—44no—
108C251#312211yescharacteristic, complemented, Sylow
109#3122—2511nomaximal, Sylow
110#540D25111yescharacteristic, complemented
111#518—2511no—
112#482—2512no—
113#478—2512no—
114#459—2512no—
115#447—2512no—
116#442—2512no—
117C4×D8—2512no—
118#433—2511no—
119C22×D4—2511no—
120C2×D8—2512no—
121C2×D8—2512no—
122#127C2×D25111yescharacteristic, complemented
123#125C2×D25111yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D25112yes—
127C4×D4—5022no—
128C4×D4—2512no—
129C4×D4—2512no—
130C4×D4C2×D25112yescomplemented
131#117—2511no—
132#117C2×D25111yescharacteristic
133C2×C22⋊C4—5022no—
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—5021no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—5024no—
148C2×D4—5021no—
149C2×D4—2512no—
150C2×D4C22×D25112yes—
151C22×C4—22no—
152C22×C4—5022no—
153C22×C4—5022no—
154C22×C4C22×D25111yescharacteristic
155D8—5024no—
156D8—5024no—
157C2×C8—2512no—
158C2×C8—2512no—
159C4⋊C4—22no—
160C4⋊C4—2512no—
161C4⋊C4C22⋊D25112yes—
162C4⋊C4C22×D25112yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—5021no—
166C22⋊C4—5022no—
167C22⋊C4—5022no—
168C22⋊C4C22⋊D25112yes—
169C42—5021no—
170C42C22×D25111yescharacteristic
171C42C22×D25111yescharacteristic
172C23—22no—
173C23—20082no—
174C23—5024no—
175C23—5022no—
176C23C2×C22⋊D25111yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—5024no—
182D4—5022no—
183D4—2514no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—20082no—
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—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—20084no—
202C22—10042no—
203C22—10042no—
204C22—5024no—
205C22C22×C22⋊D25111yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—10042no—
211C4#41×D25112yes—
212C2—41no—
213C2—82no—
214C2—10044no—
215C2#3063219411yescharacteristic
216C2#3063249012yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#540×C251C2
socleC22×C251C22×D4
hypercenter#540D251
hypocenterC251#3122
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #30632161C2semidirect
2. #30632008C2semidirect
3. #30631984C2semidirect
4. #30631879C2semidirect
5. #30631820C2semidirect
6. #30631797C2semidirect
7. C4×#16860795C2semidirect
8. #30631748C2semidirect
9. #540×C251C2semidirect
18. #22548648C22semidirect
19. #22548646C22semidirect
20. C4⋊(C251⋊C8)C22semidirect
21. #22548636C22semidirect
22. #22548636C22semidirect
26. #22548626C22semidirect
27. C4×C251⋊C8C22semidirect
28. #127×C251C22semidirect
29. #125×C251C22semidirect
33. C4×D4×C251C22semidirect
108. C251#3122semidirect
110. #540D251semidirect
122. #127C2×D251semidirect
123. #125C2×D251semidirect
130. C4×D4C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306321612
#306320082
#306319842
#306318792
#306318202
#306317972
C4×#168607952
#306317482
#540×C2512
#3122251