Finite Group Databasebeta

← #44560829

Subgroups — #44560829

Order 32128.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22548746C212yesmaximal, complemented
2C2×#22548741C212yesmaximal, complemented
3#30632237C211yesmaximal, characteristic, complemented
4#30632082C212yesmaximal, complemented
5#30632077C212yesmaximal, complemented
6#30632069C214yesmaximal, complemented
7C4×#16860794C211yesmaximal, characteristic, complemented
8C2×#124×C251C211yesmaximal, characteristic, complemented
9C2×#16860810C2212yes—
10#22548746—28no—
11#22548741—28no—
12C2×#16860800C2214yescomplemented
13C2×#16860798C2214yes—
14C2×#16860797C2214yescomplemented
15C2×#16860795C2214yes—
16C2×#16860794C2212yes—
17#22548715C2211yescharacteristic, complemented
18#22548712C2211yescharacteristic, complemented
19#22548703C2212yescomplemented
20#22548689—22no—
21C4×D1004—22no—
22#22548626C2212yescomplemented
23C4×C251⋊C8C2212yescomplemented
24C22×Q8×C251C2211yescharacteristic
25C22×D4×C251C2211yescharacteristic
26#124×C251—22no—
27#124×C251C2212yes—
28#124×C251C2212yes—
29C2×C22⋊C4×C251—22no—
30C2×C42×C251C2211yescharacteristic
31#16860810—24no—
32#16860810—24no—
33C2×Dic502C2312yes—
34C2×D1004—22no—
35C2×D1004C2312yes—
36C2×C22⋊D251—22no—
37C2×C4×D251—22no—
38#16860800—28no—
39#16860798—28no—
40#16860797—28no—
41#16860795—28no—
42#16860794D418yescomplemented
43C2×C251⋊C8C2314yes—
44C4⋊Dic251—22no—
45(C2×D251)⋊C4—24no—
46C24×C251—21no—
47C2×Q8×C251—21no—
48C2×Q8×C251—24no—
49C2×Q8×C251C2312yes—
50C2×D4×C251—24no—
51C2×D4×C251—21no—
52C2×D4×C251C2312yes—
53C22×C4×C251—21no—
54C22×C4×C251D412yes—
55C22×C4×C251C2311yescharacteristic
56C22⋊C4×C251—24no—
57C22⋊C4×C251—24no—
58C42×C251D412yes—
59C42×C251C2311yescharacteristic
60C42×C251C2311yescharacteristic
61C22×D251—22no—
62Dic502—22no—
63Dic502—22no—
64D1004—22no—
65D1004—22no—
66D1004—44no—
67C22⋊D251—44no—
68C2×Dic251—22no—
69C4×D251—44no—
70C251⋊C8C2×D418yes—
71C23×C251—22no—
72C23×C251—44no—
73C23×C251—42no—
74C23×C251C2×D411yescharacteristic
75Q8×C251—24no—
76Q8×C251—42no—
77D4×C251—24no—
78D4×C251—42no—
79C2×C4×C251—22no—
80C2×C4×C251—42no—
81C2×C4×C251—42no—
82C2×C4×C251C2×D414yes—
83C2×C4×C251C2×D412yes—
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2411yescharacteristic
87C22×C251—22no—
88C22×C251—42no—
89C22×C251—44no—
90C22×C251—84no—
91C22×C251#12712yes—
92C22×C251C22×D411yescharacteristic
93C2×D251—42no—
94C2×D251—44no—
95C4×C251—22no—
96C4×C251—22no—
97C4×C251—42no—
98C4×C251#12712yes—
99C4×C251C22×D412yes—
100Dic251—42no—
101C2×C251—22no—
102C2×C251—82no—
103C2×C251C2×#12711yescharacteristic
104C2×C251#57512yes—
105D251—82no—
106C251#315311yescharacteristic, complemented, Sylow
107#3153—2511nomaximal, Sylow
108C2×#137—2512no—
109C2×AGL(1,ℤ/8)—2512no—
110#532—2511no—
111C2×#124D25111yescharacteristic, complemented
112#497—2512no—
113#496—2512no—
114#495—2514no—
115C4×M16—2511no—
116C2×#41—2512no—
117C22×Q8C2×D25111yescharacteristic
118C22×D4C2×D25111yescharacteristic
119#137—5028no—
120AGL(1,ℤ/8)—5028no—
121C2×Q16—2514no—
122C2×SD16—2514no—
123C2×SD16—2514no—
124C2×D8—2514no—
125C2×M16—2512no—
126C4⋊Q8—2511no—
127#127—2511no—
128#124—22no—
129#124—2512no—
130#124C2×D25112yes—
131#124C2×D25112yes—
132#121—5022no—
133C4×D4—5022no—
134C2×C22⋊C4—22no—
135C2×C42C2×D25111yescharacteristic
136#97—2512no—
137C4×C8—2512no—
138C24—21no—
139#41—5024no—
140#41—5024no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—2512no—
144C2×Q8C22×D25112yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—5022no—
148C2×D4—5022no—
149C2×D4—2512no—
150C2×D4C22×D25112yes—
151C22×C4—21no—
152C22×C4—5022no—
153C22×C4C22⋊D25112yes—
154C22×C4C22×D25111yescharacteristic
155Q16—5028no—
156SD16—5028no—
157SD16—5028no—
158D8—5028no—
159M16—2518no—
160C2×C8—2514no—
161C4⋊C4—5022no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—5024no—
165C42C22⋊D25112yes—
166C42C22×D25111yescharacteristic
167C42C22×D25111yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—5022no—
172C23C2×C22⋊D25111yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—5022no—
176Q8—5022no—
177D4—24no—
178D4—42no—
179D4—10044no—
180D4—10044no—
181D4—5022no—
182D4—5022no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—10044no—
187C2×C4—5022no—
188C2×C4D4×D25114yes—
189C2×C4C2×C22⋊D25112yes—
190C2×C4C2×C22⋊D25112yes—
191C2×C4C2×C22⋊D25111yescharacteristic
192C2×C4C23×D25111yescharacteristic
193C8—2518no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—10042no—
199C22—10044no—
200C22#2254871112yes—
201C22C22×C22⋊D25111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—10042no—
206C4#2254871112yes—
207C4C2×D4×D25112yes—
208C2—22no—
209C2—82no—
210C2—20082no—
211C2C2×#2254871111yescharacteristic
212C2#3063248512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#124×C251C2
socleC22×C251C22×D4
hypercenterC2×#124D251
hypocenterC251#3153
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548741C2semidirect
3. #30632237C2semidirect
4. #30632082C2semidirect
5. #30632077C2semidirect
6. #30632069C2semidirect
7. C4×#16860794C2semidirect
8. C2×#124×C251C2semidirect
12. C2×#16860800C22semidirect
14. C2×#16860797C22semidirect
17. #22548715C22semidirect
18. #22548712C22semidirect
19. #22548703C22semidirect
22. #22548626C22semidirect
23. C4×C251⋊C8C22semidirect
42. #16860794D4semidirect
106. C251#3153semidirect
111. C2×#124D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487412
#306322372
#306320822
#306320772
#306320692
C4×#168607942
C2×#124×C2512
#3153251