Finite Group Databasebeta

← #44558351

Subgroups — #44558351

Order 32128.

368 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 137 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#30631629C212yesmaximal
2#30631621C212yesmaximal
3#30631620C212yesmaximal
4#30631594C211yesmaximal, characteristic
5#30631594C211yesmaximal, characteristic
6#30631584C212yesmaximal
7#30631571C212yesmaximal
8#30631570C212yesmaximal
9#391×C251C211yesmaximal, characteristic
10C2×C4⋊Dic251C2212yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C4×Dic251C2211yescharacteristic
18C2×C4×Dic251C2211yescharacteristic
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2212yes—
31C22×Dic251C2312yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C4⋊Dic251—24no—
36C4⋊Dic251—41no—
37C4⋊Dic251—41no—
38Dic251⋊C4—24no—
39Dic251⋊C4—24no—
40Dic251⋊C4—42no—
41Dic251⋊C4—42no—
42C4×Dic251—41no—
43C4×Dic251—41no—
44C2×Q8×C251—41no—
45C2×Q8×C251—41no—
46C2×Q8×C251Q814yes—
47C22×C4×C251C2312yes—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2311yescharacteristic
51C22×C4×C251C2311yescharacteristic
52C4⋊C4×C251—41no—
53C42×C251—41no—
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—
63C2×Dic251—24no—
64C2×Dic251#4114yes—
65C23×C251C2411yescharacteristic
66Q8×C251—44no—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251C2×Q812yes—
80C2×C4×C251C2×Q812yes—
81C2×C4×C251C2×Q812yes—
82C22×C251C22×Q811yescharacteristic
83C22×C251C2×#4112yes—
84C22×C251H(5,𝔽2)11yescharacteristic
85C22×C251H(5,𝔽2)11yescharacteristic
86C22×C2512−1+412yes—
87C4×C251—22no—
88C4×C251—22no—
89C4×C251—22no—
90C4×C251—42no—
91C4×C251—42no—
92Dic251—24no—
93Dic251—42no—
94Dic251—42no—
95Dic251—42no—
96C2×C251#53912yes—
97C2×C251#54311yescharacteristic
98C2×C251#55511yescharacteristic
99C2×C251#55712yes—
100C2×C251#56311yescharacteristic
101C251#280911yescharacteristic, complemented, Sylow
102#2809—2511nomaximal, Sylow
103#400—2512no—
104#398—2512no—
105#398—2512no—
106#391—2511no—
107#391—2511no—
108#391D25111yescharacteristic
109#389—2512no—
110#387—2512no—
111#387—2512no—
112C22×Q8C2×D25111yescharacteristic
113C2×C4⋊C4—2512no—
114C2×C4⋊C4—2511no—
115C2×C4⋊C4—2511no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4—2512no—
118C2×C4⋊C4—2512no—
119C2×C4⋊C4—2512no—
120C2×C4⋊C4C2×D25111yescharacteristic
121C2×C42—2511no—
122C2×C42—2511no—
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#95C2×D25112yes—
132#95C2×D25112yes—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8Dic50214yes—
136C22×C4—2512no—
137C22×C4—2512no—
138C22×C4—2512no—
139C22×C4—2512no—
140C22×C4C22×D25112yes—
141C22×C4C22×D25112yes—
142C22×C4C22×D25111yescharacteristic
143C22×C4C22×D25111yescharacteristic
144C22×C4C22×D25111yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—10042no—
147C4⋊C4—10042no—
148C4⋊C4—10041no—
149C4⋊C4—10041no—
150C4⋊C4—5024no—
151C4⋊C4—5024no—
152C4⋊C4—5024no—
153C42—41no—
154C42—10041no—
155C42—10041no—
156C23C23×D25111yescharacteristic
157Q8—44no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—5024no—
178C2×C4—5022no—
179C2×C4—5022no—
180C2×C4—2514no—
181C2×C4C2×Dic50212yes—
182C2×C4C2×Dic50212yes—
183C2×C4C2×Dic50212yes—
184C22C22×Dic50211yescharacteristic
185C22#41×D25112yes—
186C22#2254876011yescharacteristic
187C22#2254876011yescharacteristic
188C22#2254876312yes—
189C4—22no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—10042no—
195C4—10042no—
196C4—10042no—
197C4—5024no—
198C2#3063228312yes—
199C2#3063231311yescharacteristic
200C2#3063239911yescharacteristic
201C2#3063241312yes—
202C2#3063243611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#391×C251C2
hypercenter#391D251
hypocenterC251#2809
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
101. C251#2809semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316212
#306316202
#306315942
#306315842
#306315712
#306315702
#391×C2512
#2809251