Finite Group Databasebeta

← #44560686

Subgroups — #44560686

Order 32128.

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

normal subgroups only

This group has 400 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#30632243C211yesmaximal, characteristic, complemented
2#30632236C211yesmaximal, characteristic
3#30632044C211yesmaximal, characteristic, complemented
4#30632042C211yesmaximal, characteristic, complemented
5#30632041C211yesmaximal, characteristic, complemented
6#30632036C212yesmaximal, complemented
7#30632032C212yesmaximal, complemented
8#30631682C212yesmaximal, complemented
9#30631680C212yesmaximal, complemented
10C4×C8×D251C211yesmaximal, characteristic, complemented
11#488×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C2×#16860809C2211yescharacteristic
14C2×C8×D251C2212yes—
15#22548714C2212yes—
16#22548713C2211yescharacteristic
17#22548711C2211yescharacteristic
18#22548707C2211yescharacteristic
19#22548703C2211yescharacteristic
20#22548693—22no—
21#22548688—22no—
22#22548680—22no—
23(C22⋊D251)⋊C4—22no—
24#22548673C2212yes—
25C42×D251C2211yescharacteristic
26#22548640C2212yes—
27#22548639C2212yes—
28#22548638C2212yes—
29#22548637C2212yes—
30#22548636C2212yes—
31#22548635C2212yes—
32C4×C251⋊C8C2211yescharacteristic
33C8×Dic251C2212yes—
34#125×C251C2211yescharacteristic
35#124×C251C2211yescharacteristic
36#103×C251C2212yes—
37#102×C251C2212yes—
38C4×C8×C251C2211yescharacteristic
39#16860811—24no—
40#16860809—24no—
41C2×Dic502—22no—
42C2×Dic502C2311yescharacteristic
43C2×D1004—21no—
44C2×D1004C2311yescharacteristic
45C2×C22⋊D251—21no—
46C2×C22⋊D251—22no—
47C22×Dic251—21no—
48C2×C4×D251—21no—
49C2×C4×D251D412yes—
50C2×C4×D251C2311yescharacteristic
51C2×C251⋊C8C2312yes—
52C8×D251—24no—
53C4⋊Dic251C2312yes—
54Dic251⋊C4—22no—
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57(C2×D251)⋊C4—22no—
58(C2×D251)⋊C4—22no—
59C4×Dic251—22no—
60C4×Dic251D412yes—
61C4×Dic251C2311yescharacteristic
62C2×Q8×C251C2311yescharacteristic
63C2×D4×C251C2311yescharacteristic
64C2×C8×C251C2312yes—
65C4⋊C4×C251—22no—
66C4⋊C4×C251C2312yes—
67C22⋊C4×C251—22no—
68C42×C251C2311yescharacteristic
69C22×D251—21no—
70C22×D251C2×D411yescharacteristic
71Dic502—22no—
72Dic502—44no—
73D1004—22no—
74D1004—42no—
75C22⋊D251—42no—
76C22⋊D251—44no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—42no—
80C2×Dic251C2×D411yescharacteristic
81C2×Dic251C2×D412yes—
82C4×D251—24no—
83C4×D251—42no—
84C4×D251#4114yes—
85C251⋊C8—22no—
86C23×C251—21no—
87Q8×C251—22no—
88D4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—22no—
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2411yescharacteristic
93C8×C251—22no—
94C22×C251—41no—
95C22×C251—42no—
96C22×C251C22×D411yescharacteristic
97C2×D251—22no—
98C2×D251—41no—
99C2×D251—42no—
100C2×D251#12412yes—
101C4×C251—22no—
102C4×C251—41no—
103C4×C251—42no—
104C4×C251C2×#4112yes—
105Dic251—41no—
106Dic251—42no—
107Dic251#12412yes—
108Dic251#13514yes—
109C2×C251—81no—
110C2×C251C2×#12411yescharacteristic
111C2×C251C2×#13512yes—
112D251—22no—
113D251—81no—
114C251#313111yescharacteristic, complemented, Sylow
115#3131—2511nomaximal, Sylow
116#533—2511no—
117#532—2511no—
118#488—2511no—
119#488—2511no—
120#488—2511no—
121#488D25111yescharacteristic, complemented
122#487—2512no—
123#486—2512no—
124#416—2512no—
125#416—2512no—
126C2×C4×C8—2511no—
127C2×#41—2511no—
128C2×#41—2511no—
129C22×C8—2512no—
130C4⋊Q8—2511no—
131C4⋊Q8—2512no—
132#127—2511no—
133#125—2511no—
134#125C2×D25111yescharacteristic
135#124—2511no—
136#124C2×D25111yescharacteristic
137C22⋊Q8—5022no—
138#121—5022no—
139C4×Q8—5022no—
140C4×D4—5022no—
141#117—2512no—
142C2×C42—2511no—
143#103—2512no—
144#103—2512no—
145#103—2512no—
146#103C2×D25112yes—
147#102—2512no—
148#102—2512no—
149#102—2512no—
150#102C2×D25112yes—
151C4×C8—2512no—
152C4×C8—2511no—
153C4×C8C2×D25111yescharacteristic
154#41—5024no—
155#41—5024no—
156C2×Q8—5022no—
157C2×Q8—2511no—
158C2×Q8C22×D25111yescharacteristic
159C2×D4—5021no—
160C2×D4—5021no—
161C2×D4—5022no—
162C2×D4—2511no—
163C2×D4C22×D25111yescharacteristic
164C22×C4—5021no—
165C22×C4—5021no—
166C22×C4—2512no—
167C22×C4—2511no—
168C2×C8—5024no—
169C2×C8—2512no—
170C2×C8C22×D25112yes—
171C4⋊C4—22no—
172C4⋊C4—5022no—
173C4⋊C4—5022no—
174C4⋊C4—5022no—
175C4⋊C4—2512no—
176C4⋊C4C22×D25112yes—
177C22⋊C4—22no—
178C22⋊C4—5022no—
179C22⋊C4—5022no—
180C42—5022no—
181C42—2512no—
182C42—2511no—
183C42C22×D25111yescharacteristic
184C23—21no—
185C23—5021no—
186C23—2511no—
187Q8—22no—
188Q8—10044no—
189Q8—5022no—
190D4—22no—
191D4—10042no—
192D4—10042no—
193D4—10044no—
194D4—5022no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—10042no—
198C2×C4—10042no—
199C2×C4—5024no—
200C2×C4—5022no—
201C2×C4—5021no—
202C2×C4—2511no—
203C2×C4—2512no—
204C2×C4—2514no—
205C2×C4D4×D25112yes—
206C2×C4C23×D25111yescharacteristic
207C8—22no—
208C8—5022no—
209C22—41no—
210C22—42no—
211C22—10041no—
212C22—10042no—
213C22—5022no—
214C22—2512no—
215C22C2×D4×D25111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—10041no—
220C4—10042no—
221C4—2512no—
222C4—2514no—
223C4#41×D25112yes—
224C2—81no—
225C2—20081no—
226C2—5022no—
227C2#124×D25111yescharacteristic
228C2#135×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#488×C251C2
socleC22×C251C22×D4
prime radical#30632236C2
hypercenter#488D251
hypocenterC251#3131
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #30632243C2semidirect
3. #30632044C2semidirect
4. #30632042C2semidirect
5. #30632041C2semidirect
6. #30632036C2semidirect
7. #30632032C2semidirect
8. #30631682C2semidirect
9. #30631680C2semidirect
10. C4×C8×D251C2semidirect
11. #488×C251C2semidirect
114. C251#3131semidirect
121. #488D251semidirect

Maximal subgroups

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

subgroupindex
#306322432
#306322362
#306320442
#306320422
#306320412
#306320362
#306320322
#306316822
#306316802
C4×C8×D2512
#488×C2512
#3131251