Finite Group Databasebeta

← #44560298

Subgroups — #44560298

Order 32128.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#30632198C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3#30632029C214yesmaximal, complemented
4#30632012C214yesmaximal, complemented
5#30631729C212yesmaximal, complemented
6#30631689C212yesmaximal, complemented
7#407×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9#22548698—21no—
10#22548694C2212yes—
11#22548689C2212yes—
12#22548684—21no—
13C4×C22⋊D251—22no—
14#22548674—22no—
15#22548674C2211yescharacteristic
16#22548674C2214yes—
17C2×C4⋊Dic251C2212yes—
18C2×C22⋊Dic251—21no—
19C2×C4×Dic251—21no—
20#22548649C2214yescomplemented
21#22548647C2214yescomplemented
22#22548640C2214yescomplemented
23#22548637C2214yescomplemented
24C23×C4×C251C2211yescharacteristic
25C2×M16×C251C2212yes—
26C22⋊C8×C251C2214yes—
27#16860810—24no—
28C2×Dic502C2311yescharacteristic
29C2×D1004C2311yescharacteristic
30C2×C22⋊D251—21no—
31C2×C22⋊D251—22no—
32C22×Dic251—22no—
33C2×C4×D251—21no—
34C4⋊Dic251—22no—
35C4⋊Dic251C2314yes—
36C4⋊Dic251C2312yes—
37Dic251⋊C4—22no—
38C22⋊Dic251—24no—
39C22⋊Dic251—21no—
40C22⋊Dic251—22no—
41(C2×D251)⋊C4—22no—
42C4×Dic251—24no—
43C4×Dic251—21no—
44C24×C251D411yescharacteristic
45C22×C4×C251—24no—
46C22×C4×C251D411yescharacteristic
47C22×C4×C251D412yes—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50M16×C251—24no—
51C2×C8×C251C2314yes—
52C22×D251—21no—
53Dic502—22no—
54D1004—22no—
55C22⋊D251—44no—
56C22⋊D251—42no—
57C2×Dic251—24no—
58C2×Dic251—22no—
59C2×Dic251—21no—
60C2×Dic251—44no—
61C4×D251—42no—
62C23×C251—22no—
63C23×C251—42no—
64C23×C251C2×D412yes—
65C23×C251C2×D411yescharacteristic
66C2×C4×C251—22no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251C2×D412yes—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251#4114yes—
72C2×C4×C251C2411yescharacteristic
73C8×C251—24no—
74C22×C251—22no—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—44no—
79C22×C251#12312yes—
80C22×C251C22×D411yescharacteristic
81C2×D251—41no—
82C2×D251—42no—
83C4×C251—22no—
84C4×C251#12312yes—
85C4×C251C2×#4112yes—
86Dic251—41no—
87Dic251—44no—
88Dic251—42no—
89C2×C251—22no—
90C2×C251—42no—
91C2×C251C2×#12311yescharacteristic
92C2×C251#57512yes—
93D251—81no—
94C251#309311yescharacteristic, complemented, Sylow
95#3093—2511nomaximal, Sylow
96#525—2511no—
97C2×#117—2511no—
98#485—2514no—
99#482—2514no—
100#428—2512no—
101#417—2512no—
102#407D25111yescharacteristic, complemented
103C2×#41—2511no—
104C23×C4C2×D25111yescharacteristic
105C2×M16C2×D25112yes—
106#123—5021no—
107C22⋊Q8—2512no—
108#121—2512no—
109#120—5021no—
110C4×D4—5022no—
111#117—5022no—
112#117—2514no—
113#117—2511no—
114C2×C4⋊C4—2512no—
115C2×C22⋊C4—5021no—
116C2×C42—5021no—
117#107—2514no—
118#106—2514no—
119#103—2514no—
120#102—2514no—
121C22⋊C8C2×D25114yes—
122C24D100411yescharacteristic
123#41—5024no—
124C2×Q8—2511no—
125C2×D4—5022no—
126C2×D4—5021no—
127C2×D4—2511no—
128C22×C4—24no—
129C22×C4—5022no—
130C22×C4—5021no—
131C22×C4D100411yescharacteristic
132C22×C4D100412yes—
133C22×C4C22×D25112yes—
134C22×C4C22×D25111yescharacteristic
135M16—24no—
136C2×C8C22×D25114yes—
137C4⋊C4—5022no—
138C4⋊C4—5022no—
139C4⋊C4—2514no—
140C4⋊C4—2512no—
141C22⋊C4—5024no—
142C22⋊C4—5021no—
143C22⋊C4—5022no—
144C22⋊C4—5022no—
145C42—5024no—
146C42—5021no—
147C23—22no—
148C23—42no—
149C23—5021no—
150C23C2×D100412yes—
151C23C2×D100411yescharacteristic
152Q8—5022no—
153D4—10044no—
154D4—10042no—
155D4—5022no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—10044no—
160C2×C4—10042no—
161C2×C4—5024no—
162C2×C4—5022no—
163C2×C4—5021no—
164C2×C4C2×D100412yes—
165C2×C4C2×D100411yescharacteristic
166C2×C4#1686080914yes—
167C2×C4C23×D25111yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—10042no—
175C22—10041no—
176C22#2254869912yes—
177C22C22×D100411yescharacteristic
178C4—22no—
179C4—10044no—
180C4—10042no—
181C4—10041no—
182C4#2254869912yes—
183C4C2×#1686080912yes—
184C2—22no—
185C2—42no—
186C2—20081no—
187C2C2×#2254869911yescharacteristic
188C2#3063248712yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenter#407D251
hypocenterC251#3093
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
3. #30632029C2semidirect
4. #30632012C2semidirect
5. #30631729C2semidirect
6. #30631689C2semidirect
7. #407×C251C2semidirect
20. #22548649C22semidirect
21. #22548647C22semidirect
22. #22548640C22semidirect
23. #22548637C22semidirect
94. C251#3093semidirect
102. #407D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×#225486742
#306320292
#306320122
#306317292
#306316892
#407×C2512
#3093251