Finite Group Databasebeta

← #44558432

Subgroups — #44558432

Order 32128.

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

normal subgroups only

This group has 380 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
1C2×#22548707C211yesmaximal, characteristic
2#30631626C212yesmaximal
3#30631625C212yesmaximal
4#30631620C214yesmaximal
5#30631618C212yesmaximal
6#30631617C212yesmaximal
7#30631553C211yesmaximal, characteristic
8C2×#125×C251C211yesmaximal, characteristic
9#22548707—24no—
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2214yes—
16C251⋊#95C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2214yes—
19C251⋊#95C2214yes—
20C251⋊#95C2214yes—
21#125×C251—24no—
22C2×C4⋊C4×C251C2212yes—
23C2×C4⋊C4×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2212yes—
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×C42×C251C2211yescharacteristic
27C22×Dic251C2314yes—
28C22×Dic251C2312yes—
29C22×Dic251C2312yes—
30C4⋊Dic251—22no—
31C4⋊Dic251—22no—
32C4⋊Dic251—42no—
33Dic251⋊C4—24no—
34Dic251⋊C4—24no—
35Dic251⋊C4—44no—
36C22×C4×C251C2312yes—
37C22×C4×C251C2312yes—
38C22×C4×C251C2311yescharacteristic
39C22×C4×C251C2311yescharacteristic
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—22no—
42C4⋊C4×C251—24no—
43C4⋊C4×C251—22no—
44C4⋊C4×C251—24no—
45C42×C251D414yes—
46C2×Dic251—22no—
47C2×Dic251—22no—
48C2×Dic251—22no—
49C2×Dic251—24no—
50C2×Dic251—22no—
51C2×Dic251—24no—
52C2×Dic251—24no—
53C2×Dic251—24no—
54C23×C251C2411yescharacteristic
55C2×C4×C251—21no—
56C2×C4×C251—22no—
57C2×C4×C251—21no—
58C2×C4×C251—22no—
59C2×C4×C251—22no—
60C2×C4×C251—22no—
61C2×C4×C251—22no—
62C2×C4×C251—24no—
63C2×C4×C251—22no—
64C2×C4×C251C2×D412yes—
65C2×C4×C251C2×D412yes—
66C2×C4×C251C2×D412yes—
67C22×C251C22×D411yescharacteristic
68C22×C251H(5,𝔽2)11yescharacteristic
69C22×C2512−1+412yes—
70C22×C2512−1+411yescharacteristic
71C22×C2512−1+412yes—
72C4×C251—22no—
73C4×C251—22no—
74C4×C251—22no—
75C4×C251—42no—
76C4×C251—42no—
77Dic251—42no—
78Dic251—42no—
79Dic251—44no—
80C2×C251#53611yescharacteristic
81C2×C251#53611yescharacteristic
82C2×C251#53711yescharacteristic
83C2×C251#56312yes—
84C2×C251#56412yes—
85C251#282211yescharacteristic, complemented, Sylow
86#2822—2511nomaximal, Sylow
87C2×#125—2511no—
88C2×#125D25111yescharacteristic
89#400—2512no—
90#400—2512no—
91#398—2512no—
92#398—2512no—
93#398—2514no—
94#382—2511no—
95#125—24no—
96#125—5024no—
97C2×C4⋊C4—2512no—
98C2×C4⋊C4—2514no—
99C2×C4⋊C4—2512no—
100C2×C4⋊C4—2511no—
101C2×C4⋊C4—2512no—
102C2×C4⋊C4—2511no—
103C2×C4⋊C4C2×D25112yes—
104C2×C4⋊C4C2×D25111yescharacteristic
105C2×C4⋊C4C2×D25112yes—
106C2×C4⋊C4C2×D25111yescharacteristic
107C2×C42C2×D25111yescharacteristic
108#95—2512no—
109#95—2512no—
110#95—2514no—
111#95—2514no—
112#95—2514no—
113C22×C4—2514no—
114C22×C4—2512no—
115C22×C4—2512no—
116C22×C4C22×D25112yes—
117C22×C4C22×D25112yes—
118C22×C4C22×D25111yescharacteristic
119C22×C4C22×D25111yescharacteristic
120C22×C4C22×D25111yescharacteristic
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—10044no—
126C4⋊C4—10042no—
127C4⋊C4—5024no—
128C4⋊C4—5024no—
129C4⋊C4—5022no—
130C4⋊C4—5022no—
131C42C22⋊D25114yes—
132C23C23×D25111yescharacteristic
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—5022no—
143C2×C4—5022no—
144C2×C4—5022no—
145C2×C4—5024no—
146C2×C4—5022no—
147C2×C4—5024no—
148C2×C4—5024no—
149C2×C4—5024no—
150C2×C4C2×C22⋊D25112yes—
151C2×C4C2×C22⋊D25112yes—
152C2×C4C2×C22⋊D25112yes—
153C22C22×C22⋊D25111yescharacteristic
154C22#2254876111yescharacteristic
155C22#2254876212yes—
156C22#2254876211yescharacteristic
157C22#2254876312yes—
158C4—22no—
159C4—22no—
160C4—22no—
161C4—42no—
162C4—42no—
163C4—10042no—
164C4—10042no—
165C4—10044no—
166C2#3063225811yescharacteristic
167C2#3063226111yescharacteristic
168C2#3063226511yescharacteristic
169C2#3063244112yes—
170C2C251⋊#56412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
FittingC2×#125×C251C2
hypercenterC2×#125D251
hypocenterC251#2822
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
85. C251#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#225487072
#306316262
#306316252
#306316202
#306316182
#306316172
#306315532
C2×#125×C2512
#2822251