Finite Group Databasebeta

← #44560689

Subgroups — #44560689

Order 32128.

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

normal subgroups only

This group has 400 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#30632242C211yesmaximal, characteristic, complemented
2#30632238C211yesmaximal, characteristic
3#30632043C214yesmaximal, complemented
4#30632037C211yesmaximal, characteristic, complemented
5#30632035C211yesmaximal, characteristic, complemented
6#30632034C211yesmaximal, characteristic, complemented
7#30631682C214yesmaximal, complemented
8C4×C8×D251C211yesmaximal, characteristic, complemented
9#487×C251C211yesmaximal, characteristic, complemented
10C2×#16860811C2212yes—
11C2×C8×D251C2212yes—
12#22548715C2211yescharacteristic
13#22548712C2211yescharacteristic
14#22548706C2212yes—
15#22548704C2212yes—
16#22548695—21no—
17#22548691—21no—
18#22548690—21no—
19#22548690—21no—
20Q8×Dic251—21no—
21#22548680—21no—
22#22548677—21no—
23#22548677—21no—
24#22548673C2212yes—
25C42×D251C2211yescharacteristic
26#22548639C2214yes—
27#22548637C2214yes—
28#22548635C2214yes—
29C4×C251⋊C8C2211yescharacteristic
30C8×Dic251C2212yes—
31C4⋊Q8×C251C2211yescharacteristic
32C4⋊Q8×C251C2211yescharacteristic
33#103×C251C2214yes—
34C4×C8×C251C2211yescharacteristic
35#16860811—28no—
36C2×Dic502—21no—
37C2×D1004—22no—
38C2×D1004—21no—
39C2×D1004—21no—
40C2×D1004C2312yes—
41C2×C4×D251—22no—
42C2×C4×D251D411yescharacteristic
43C2×C4×D251D411yescharacteristic
44C2×C4×D251C2311yescharacteristic
45C2×C251⋊C8C2312yes—
46C8×D251—24no—
47C4⋊Dic251—22no—
48C4⋊Dic251—21no—
49C4⋊Dic251C2312yes—
50Dic251⋊C4—22no—
51(C2×D251)⋊C4—22no—
52(C2×D251)⋊C4—22no—
53(C2×D251)⋊C4—22no—
54C4×Dic251—22no—
55C4×Dic251D411yescharacteristic
56C4×Dic251D411yescharacteristic
57C4×Dic251C2311yescharacteristic
58C2×Q8×C251—21no—
59C2×Q8×C251C2312yes—
60C2×C8×C251C2312yes—
61C4⋊C4×C251—21no—
62C4⋊C4×C251—21no—
63C4⋊C4×C251—21no—
64C4⋊C4×C251C2312yes—
65C42×C251C2311yescharacteristic
66C22×D251—22no—
67C22×D251C2×D411yescharacteristic
68Dic502—42no—
69D1004—24no—
70D1004—44no—
71D1004—42no—
72D1004—42no—
73C2×Dic251—22no—
74C2×Dic251C2×D411yescharacteristic
75C2×Dic251C2×D411yescharacteristic
76C2×Dic251C2×D411yescharacteristic
77C4×D251—22no—
78C4×D251—22no—
79C4×D251—44no—
80C4×D251#4114yes—
81C251⋊C8—22no—
82Q8×C251—24no—
83Q8×C251—42no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2411yescharacteristic
89C8×C251—22no—
90C22×C251C22×D411yescharacteristic
91C2×D251—21no—
92C2×D251—21no—
93C2×D251—42no—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251#12412yes—
97C4×C251—42no—
98C4×C251—42no—
99C4×C251#13512yes—
100C4×C251#13512yes—
101C4×C251C2×#4112yes—
102Dic251—21no—
103Dic251—21no—
104Dic251—42no—
105Dic251#12412yes—
106C2×C251C2×#12411yescharacteristic
107C2×C251C2×#13511yescharacteristic
108C2×C251C2×#13511yescharacteristic
109D251—22no—
110D251—82no—
111C251#313111yescharacteristic, complemented, Sylow
112#3131—2511nomaximal, Sylow
113#533—2511no—
114#532—2511no—
115#488—2514no—
116#487—2511no—
117#487D25111yescharacteristic, complemented
118#486—2511no—
119#486—2511no—
120#416—2514no—
121C2×C4×C8—2511no—
122C2×#41—2512no—
123C22×C8—2512no—
124C4⋊Q8—2511no—
125C4⋊Q8C2×D25111yescharacteristic
126C4⋊Q8C2×D25111yescharacteristic
127#127—2511no—
128#125—2512no—
129#124—2512no—
130C22⋊Q8—5021no—
131C22⋊Q8—5021no—
132#121—5021no—
133#121—5021no—
134C4×Q8—5021no—
135C4×Q8—5021no—
136C4×D4—5021no—
137C4×D4—5021no—
138#117—2512no—
139C2×C42—2511no—
140#103—2514no—
141#103C2×D25114yes—
142#102—2514no—
143#102—2514no—
144C4×C8—2512no—
145C4×C8—2511no—
146C4×C8C2×D25111yescharacteristic
147#41—5028no—
148C2×Q8—21no—
149C2×Q8—5021no—
150C2×Q8C22×D25112yes—
151C2×D4—5022no—
152C2×D4—5021no—
153C2×D4—5021no—
154C2×D4—2512no—
155C22×C4—5022no—
156C22×C4—2511no—
157C22×C4—2511no—
158C22×C4—2511no—
159C2×C8—5024no—
160C2×C8—2512no—
161C2×C8C22×D25112yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—5022no—
166C4⋊C4—5022no—
167C4⋊C4—5021no—
168C4⋊C4—2512no—
169C4⋊C4C22×D25112yes—
170C22⋊C4—5022no—
171C22⋊C4—5022no—
172C22⋊C4—5022no—
173C42—5022no—
174C42—2511no—
175C42—2511no—
176C42—2511no—
177C42C22×D25111yescharacteristic
178C23—5022no—
179C23—2511no—
180Q8—24no—
181Q8—42no—
182Q8—10042no—
183D4—10044no—
184D4—10042no—
185D4—10042no—
186D4—5024no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—10044no—
190C2×C4—5022no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—2511no—
194C2×C4—2511no—
195C2×C4—2511no—
196C2×C4—2514no—
197C2×C4D4×D25111yescharacteristic
198C2×C4D4×D25111yescharacteristic
199C2×C4C23×D25111yescharacteristic
200C8—22no—
201C8—5022no—
202C22—10042no—
203C22—10042no—
204C22—10042no—
205C22—5021no—
206C22—5021no—
207C22—2512no—
208C22C2×D4×D25111yescharacteristic
209C4—42no—
210C4—42no—
211C4—10042no—
212C4—5021no—
213C4—5021no—
214C4—2512no—
215C4#2254873612yes—
216C4#2254873812yes—
217C4#41×D25112yes—
218C2—20082no—
219C2—5022no—
220C2#124×D25111yescharacteristic
221C2C2×#2254873611yescharacteristic
222C2C2×#2254873811yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#487×C251C2
socleC22×C251C22×D4
prime radical#30632238C2
hypercenter#487D251
hypocenterC251#3131
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #30632242C2semidirect
3. #30632043C2semidirect
4. #30632037C2semidirect
5. #30632035C2semidirect
6. #30632034C2semidirect
7. #30631682C2semidirect
8. C4×C8×D251C2semidirect
9. #487×C251C2semidirect
111. C251#3131semidirect
117. #487D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
#306322382
#306320432
#306320372
#306320352
#306320342
#306316822
C4×C8×D2512
#487×C2512
#3131251