Finite Group Databasebeta

← #44558356

Subgroups — #44558356

Order 32128.

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

normal subgroups only

This group has 344 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#30631627C212yesmaximal
2#30631626C212yesmaximal
3#30631625C212yesmaximal
4#30631585C211yesmaximal, characteristic
5#30631585C211yesmaximal, characteristic
6#30631574C212yesmaximal
7#30631573C212yesmaximal
8#30631568C212yesmaximal
9#389×C251C211yesmaximal, characteristic
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×C4×Dic251C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C42×C251C2211yescharacteristic
30#95×C251C2212yes—
31C22×Dic251C2312yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C4⋊Dic251—22no—
36C4⋊Dic251—22no—
37C4⋊Dic251—41no—
38C4⋊Dic251—41no—
39Dic251⋊C4—24no—
40Dic251⋊C4—24no—
41C4×Dic251—42no—
42C22×C4×C251C2312yes—
43C22×C4×C251C2312yes—
44C22×C4×C251C2311yescharacteristic
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C4⋊C4×C251—24no—
48C4⋊C4×C251—22no—
49C4⋊C4×C251—41no—
50C42×C251—41no—
51C2×Dic251—22no—
52C2×Dic251—22no—
53C2×Dic251—22no—
54C2×Dic251—22no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C23×C251C2411yescharacteristic
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251#4112yes—
77C2×C4×C251#4112yes—
78C2×C4×C251#4112yes—
79C22×C251C2×#4111yescharacteristic
80C22×C251C2×#4111yescharacteristic
81C22×C251C2×#4111yescharacteristic
82C22×C251H(5,𝔽2)11yescharacteristic
83C22×C2512−1+411yescharacteristic
84C22×C2512−1+411yescharacteristic
85C22×C2512−1+411yescharacteristic
86C4×C251—22no—
87C4×C251—22no—
88C4×C251—22no—
89C4×C251—42no—
90C4×C251—42no—
91Dic251—42no—
92Dic251—42no—
93Dic251—42no—
94Dic251—42no—
95C2×C251#53911yescharacteristic
96C2×C251#53911yescharacteristic
97C2×C251#53911yescharacteristic
98C2×C251#55611yescharacteristic
99C2×C251#55611yescharacteristic
100C2×C251#55611yescharacteristic
101C2×C251#56411yescharacteristic
102C251#281011yescharacteristic, complemented, Sylow
103#2810—2511nomaximal, Sylow
104#400—2512no—
105#400—2512no—
106#400—2512no—
107#389—2511no—
108#389—2511no—
109#389D25111yescharacteristic
110#387—2512no—
111#387—2512no—
112#387—2512no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4—2511no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4—2511no—
118C2×C4⋊C4—2511no—
119C2×C4⋊C4C2×D25112yes—
120C2×C4⋊C4C2×D25111yescharacteristic
121C2×C4⋊C4C2×D25111yescharacteristic
122C2×C42—2512no—
123C2×C42C2×D25111yescharacteristic
124#95—2512no—
125#95—2512no—
126#95—2512no—
127#95—2512no—
128#95—2512no—
129#95—2512no—
130#95—2512no—
131#95—2512no—
132#95—2512no—
133#95C2×D25112yes—
134C22×C4—2512no—
135C22×C4—2512no—
136C22×C4—2512no—
137C22×C4—2512no—
138C22×C4C22×D25112yes—
139C22×C4C22×D25112yes—
140C22×C4C22×D25111yescharacteristic
141C22×C4C22×D25111yescharacteristic
142C22×C4C22×D25111yescharacteristic
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—41no—
146C4⋊C4—10041no—
147C4⋊C4—10041no—
148C4⋊C4—5022no—
149C4⋊C4—5024no—
150C4⋊C4—5024no—
151C4⋊C4—5022no—
152C42—41no—
153C42—10042no—
154C23C23×D25111yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—5022no—
178C2×C4—5022no—
179C2×C4#1686080912yes—
180C2×C4#1686081012yes—
181C2×C4#1686081112yes—
182C22C2×#1686080911yescharacteristic
183C22C2×#1686081011yescharacteristic
184C22C2×#1686081111yescharacteristic
185C22#2254876011yescharacteristic
186C22#2254876211yescharacteristic
187C22#2254876211yescharacteristic
188C22#2254876311yescharacteristic
189C4—22no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—10042no—
195C4—10042no—
196C4—10042no—
197C4—10042no—
198C2#3063228211yescharacteristic
199C2#3063228411yescharacteristic
200C2#3063228611yescharacteristic
201C2#3063240711yescharacteristic
202C2#3063241011yescharacteristic
203C2#3063241111yescharacteristic
204C2C251⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#389×C251C2
hypercenter#389D251
hypocenterC251#2810
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
102. C251#2810semidirect

Maximal subgroups

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

subgroupindex
#306316272
#306316262
#306316252
#306315852
#306315742
#306315732
#306315682
#389×C2512
#2810251