Finite Group Databasebeta

← #44558688

Subgroups — #44558688

Order 32128.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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×#22548698C211yesmaximal, characteristic
2C22×C4×Dic251C211yesmaximal, characteristic, complemented
3#30631579C214yesmaximal, complemented
4#30631572C214yesmaximal, complemented
5#30631564C212yesmaximal
6#30631562C212yesmaximal, complemented
7#381×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251C2211yescharacteristic
9C23×Dic251C2212yes—
10#22548698—24no—
11C2×C4⋊Dic251C2214yes—
12C2×Dic251⋊C4C2212yes—
13C2×C22⋊Dic251C2211yescharacteristic
14C2×(C2×D251)⋊C4C2214yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×C4×Dic251—24no—
17C2×C4×Dic251C2214yes—
18C251⋊#95C2214yes—
19C251⋊#95C2214yes—
20C23×C4×C251C2211yescharacteristic
21C2×C22⋊C4×C251C2212yes—
22#95×C251C2214yes—
23C23×D251C2311yescharacteristic
24C2×C22⋊D251—22no—
25C2×C22⋊D251—24no—
26C22×Dic251—28no—
27C22×Dic251—24no—
28C22×Dic251C2311yescharacteristic
29C22×Dic251C2312yes—
30C22×Dic251C2314yes—
31C4⋊Dic251—28no—
32Dic251⋊C4—24no—
33C22⋊Dic251—22no—
34(C2×D251)⋊C4—28no—
35(C2×D251)⋊C4—24no—
36C4×Dic251—28no—
37C24×C251C2311yescharacteristic
38C22×C4×C251—24no—
39C22×C4×C251D414yes—
40C22×C4×C251C2314yes—
41C22×C4×C251C2312yes—
42C22⋊C4×C251—24no—
43C22×D251—21no—
44C22×D251—24no—
45C22×D251—22no—
46C22⋊D251—44no—
47C2×Dic251—216no—
48C2×Dic251—24no—
49C2×Dic251—24no—
50C2×Dic251—22no—
51C2×Dic251—21no—
52C2×Dic251—24no—
53C2×Dic251—22no—
54C2×Dic251#4118yes—
55C23×C251—24no—
56C23×C251C2×D412yes—
57C23×C251#4114yes—
58C23×C251C2411yescharacteristic
59C2×C4×C251—28no—
60C2×C4×C251—24no—
61C2×C4×C251—24no—
62C2×C4×C251—24no—
63C2×C4×C251—24no—
64C2×C4×C251C2×D414yes—
65C22×C251—24no—
66C22×C251—28no—
67C22×C251#12314yes—
68C22×C251C22×D411yescharacteristic
69C22×C251C2×#4114yes—
70C22×C251C2×#4112yes—
71C2×D251—41no—
72C2×D251—42no—
73C2×D251—44no—
74C4×C251—24no—
75C4×C251—44no—
76Dic251—28no—
77Dic251—41no—
78Dic251—42no—
79C2×C251—24no—
80C2×C251C2×#12311yescharacteristic
81C2×C251#52512yes—
82C2×C251#52914yes—
83D251—81no—
84C251#285311yescharacteristic, complemented, Sylow
85#2853—2511nomaximal, Sylow
86C2×#123—2511no—
87C22×C42—2511no—
88#388—2514no—
89#387—2514no—
90#386—2512no—
91#385—2512no—
92#381D25111yescharacteristic, complemented
93C22×D4—2511no—
94C23×C4—2512no—
95C23×C4C2×D25111yescharacteristic
96#123—5024no—
97C2×C4⋊C4—2512no—
98C2×C4⋊C4—2514no—
99C2×C22⋊C4—2511no—
100C2×C22⋊C4—2512no—
101C2×C22⋊C4—2514no—
102C2×C22⋊C4C2×D25112yes—
103C2×C42—5024no—
104C2×C42—2514no—
105#95—2514no—
106#95—2514no—
107#95C2×D25114yes—
108C24—2511no—
109C24C22×D25111yescharacteristic
110C2×D4—5022no—
111C2×D4—5024no—
112C22×C4—24no—
113C22×C4—5024no—
114C22×C4—5028no—
115C22×C4—2511no—
116C22×C4—2512no—
117C22×C4—2514no—
118C22×C4D100414yes—
119C22×C4C22×D25114yes—
120C22×C4C22×D25112yes—
121C4⋊C4—5028no—
122C4⋊C4—5024no—
123C22⋊C4—24no—
124C22⋊C4—5028no—
125C22⋊C4—5022no—
126C22⋊C4—5024no—
127C42—5028no—
128C23—24no—
129C23—5021no—
130C23—5024no—
131C23—5022no—
132C23C2×D100412yes—
133C23#1686080914yes—
134C23C23×D25111yescharacteristic
135D4—10044no—
136C2×C4—24no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—5022no—
142C2×C4—5021no—
143C2×C4—5024no—
144C2×C4—5022no—
145C2×C4—50216no—
146C2×C4—5024no—
147C2×C4—5024no—
148C2×C4—2518no—
149C2×C4C2×D100414yes—
150C22—24no—
151C22—28no—
152C22—10042no—
153C22—10044no—
154C22—10041no—
155C22#2254869914yes—
156C22C22×D100411yescharacteristic
157C22#41×D25114yes—
158C22C2×#1686080912yes—
159C4—24no—
160C4—44no—
161C4—10042no—
162C4—10041no—
163C4—5028no—
164C2—24no—
165C2—20081no—
166C2C2×#2254869911yescharacteristic
167C2#3063219912yes—
168C2#3063221414yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#381×C251C2
prime radicalC22×C22⋊D251C22
hypercenter#381D251
hypocenterC251#2853
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic251C2semidirect
3. #30631579C2semidirect
4. #30631572C2semidirect
6. #30631562C2semidirect
7. #381×C251C2semidirect
84. C251#2853semidirect
92. #381D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
C22×C4×Dic2512
#306315792
#306315722
#306315642
#306315622
#381×C2512
#2853251