Finite Group Databasebeta

← #44548714

Subgroups — #44548714

Order 32128.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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
2C2×C22⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
3#30631412C212yesmaximal, complemented
4#30631410C212yesmaximal, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C411yescharacteristic
7C22×C251⋊C8—21no—
8#22548698—21no—
9#22548698—21no—
10#22548698—21no—
11#22548698—21no—
12C2×Dic251⋊C4C411yescharacteristic
13C2×Dic251⋊C4C2211yescharacteristic
14C2×C22⋊Dic251C411yescharacteristic
15C2×(C2×D251)⋊C4C411yescharacteristic
16C2×(C2×D251)⋊C4C2211yescharacteristic
17C22⋊(C251⋊C8)—21no—
18C22⋊(C251⋊C8)C2212yes—
19C23×C4×C251C2211yescharacteristic
20C2×M16×C251—21no—
21C22⋊C8×C251—21no—
22C22⋊C8×C251C2212yes—
23C23×D251C2×C411yescharacteristic
24C2×C22⋊D251—21no—
25C2×C22⋊D251—21no—
26C2×C22⋊D251—42no—
27C22×Dic251C2×C411yescharacteristic
28C22×Dic251C2×C411yescharacteristic
29C22×Dic251C2×C411yescharacteristic
30C2×C251⋊C8—22no—
31C2×C251⋊C8—42no—
32Dic251⋊C4—21no—
33Dic251⋊C4—21no—
34Dic251⋊C4—21no—
35Dic251⋊C4—21no—
36C22⋊Dic251—21no—
37C22⋊Dic251—21no—
38(C2×D251)⋊C4—21no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42C24×C251C2×C411yescharacteristic
43C22×C4×C251—42no—
44C22×C4×C251C2×C411yescharacteristic
45C22×C4×C251D411yescharacteristic
46C22×C4×C251D411yescharacteristic
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251C2311yescharacteristic
50M16×C251—42no—
51C2×C8×C251—22no—
52C22×D251—21no—
53C22×D251—22no—
54C22×D251—41no—
55C22×D251—41no—
56C22⋊D251—42no—
57C22⋊D251—42no—
58C2×Dic251—21no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C2×Dic251—22no—
62C2×Dic251—41no—
63C2×Dic251—41no—
64C251⋊C8—42no—
65C23×C251—22no—
66C23×C251—22no—
67C23×C251—41no—
68C23×C251C22⋊C411yescharacteristic
69C23×C251C22⋊C411yescharacteristic
70C23×C251C22×C411yescharacteristic
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—41no—
74C2×C4×C251—41no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251C22⋊C411yescharacteristic
78C2×C4×C251C22⋊C411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C8×C251—42no—
82C22×C251—21no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—44no—
90C22×C251#9912yes—
91C22×C251C2×C22⋊C411yescharacteristic
92C22×C251#13112yes—
93C2×D251—41no—
94C2×D251—41no—
95C2×D251—41no—
96C2×D251—82no—
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—21no—
101Dic251—41no—
102Dic251—41no—
103Dic251—41no—
104C2×C251—22no—
105C2×C251—22no—
106C2×C251—41no—
107C2×C251#40811yescharacteristic
108C2×C251C2×#9911yescharacteristic
109C2×C251#41311yescharacteristic
110D251—81no—
111C251#146811yescharacteristic, complemented, Sylow
112#1468—2511nomaximal, Sylow
113C2×#123—2511no—
114#407D25111yescharacteristic, complemented
115C2×C22⋊C8—2511no—
116#324—2512no—
117#323—2512no—
118C22×D4—2511no—
119C23×C4C2×D25111yescharacteristic
120C2×M16—21no—
121C22×C8—5021no—
122#123—5021no—
123#123—5021no—
124#123—5021no—
125#123—5021no—
126C2×C4⋊C4—2511no—
127C2×C4⋊C4—2511no—
128C2×C22⋊C4—2511no—
129C2×C22⋊C4—2511no—
130C2×C22⋊C4—2511no—
131C22⋊C8—21no—
132C22⋊C8—5021no—
133C22⋊C8—2512no—
134C22⋊C8C2×D25112yes—
135C24—2511no—
136C24C4×D25111yescharacteristic
137C2×D4—10042no—
138C2×D4—5021no—
139C2×D4—5021no—
140C22×C4—42no—
141C22×C4—2511no—
142C22×C4—2511no—
143C22×C4—2511no—
144C22×C4C4×D25111yescharacteristic
145C22×C4C22⋊D25111yescharacteristic
146C22×C4C22⋊D25111yescharacteristic
147C22×C4D100411yescharacteristic
148C22×C4D100411yescharacteristic
149C22×C4C22×D25111yescharacteristic
150M16—42no—
151C2×C8—22no—
152C2×C8—10042no—
153C2×C8—5022no—
154C4⋊C4—5021no—
155C4⋊C4—5021no—
156C4⋊C4—5021no—
157C4⋊C4—5021no—
158C22⋊C4—5021no—
159C22⋊C4—5021no—
160C22⋊C4—5021no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C22⋊C4—5021no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—10041no—
168C23—10041no—
169C23—5022no—
170C23—5021no—
171C23(C2×D251)⋊C411yescharacteristic
172C23(C2×D251)⋊C411yescharacteristic
173C23C2×C4×D25111yescharacteristic
174D4—10042no—
175D4—10042no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—10041no—
183C2×C4—10041no—
184C2×C4—5021no—
185C2×C4—5021no—
186C2×C4—5021no—
187C2×C4—5022no—
188C2×C4(C2×D251)⋊C411yescharacteristic
189C2×C4(C2×D251)⋊C411yescharacteristic
190C2×C4C2×C22⋊D25111yescharacteristic
191C2×C4C2×D100411yescharacteristic
192C8—42no—
193C8—10042no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—20082no—
203C22—10041no—
204C22—10041no—
205C22—10041no—
206C22#2254862912yes—
207C22C2×(C2×D251)⋊C411yescharacteristic
208C22#2254872312yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—10041no—
214C4—10041no—
215C4—10041no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—20081no—
220C2#3063165511yescharacteristic
221C2C2×#2254862911yescharacteristic
222C2#3063166911yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C4
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1468
LCSG ▷ C23×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×C22⋊(C251⋊C8)C2semidirect
3. #30631412C2semidirect
4. #30631410C2semidirect
5. #407×C251C2semidirect
111. C251#1468semidirect
114. #407D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
C2×C22⋊(C251⋊C8)2
#306314122
#306314102
#407×C2512
#1468251