Finite Group Databasebeta

← #44548710

Subgroups — #44548710

Order 32128.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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×#22548697C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic, complemented
3C22⋊C8×D251C211yesmaximal, characteristic, complemented
4#30631412C211yesmaximal, characteristic, complemented
5#30631412C211yesmaximal, characteristic, complemented
6#30631411C211yesmaximal, characteristic, complemented
7#323×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251C411yescharacteristic
9C22×C4×D251C2211yescharacteristic
10C2×#16860793—21no—
11C2×C8×D251—21no—
12#22548697—21no—
13#22548697—21no—
14#22548697—21no—
15#22548697—21no—
16C2×C4⋊Dic251C411yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×C22⋊Dic251C411yescharacteristic
19C2×(C2×D251)⋊C4C411yescharacteristic
20C22⋊(C251⋊C8)C2211yescharacteristic
21C22⋊(C251⋊C8)C2211yescharacteristic
22(C2×D251)⋊C8—21no—
23(C2×D251)⋊C8—21no—
24C2×C22⋊C4×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26C22⋊C8×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic
28C2×C22⋊D251—21no—
29C2×C22⋊D251—21no—
30C2×C22⋊D251—42no—
31C22×Dic251C2×C411yescharacteristic
32C22×Dic251C2×C411yescharacteristic
33C22×Dic251C2×C411yescharacteristic
34C2×C4×D251—42no—
35C2×C4×D251D411yescharacteristic
36C2×C4×D251D411yescharacteristic
37C2×C4×D251D411yescharacteristic
38C2×C4×D251D411yescharacteristic
39#16860793—42no—
40C2×C251⋊C8—21no—
41C2×C251⋊C8—21no—
42C8×D251—42no—
43C4⋊Dic251—21no—
44C4⋊Dic251—21no—
45Dic251⋊C4—21no—
46Dic251⋊C4—21no—
47C22⋊Dic251—21no—
48C22⋊Dic251—21no—
49(C2×D251)⋊C4—21no—
50(C2×D251)⋊C4—21no—
51C24×C251C2×C411yescharacteristic
52C22×C4×C251C2×C411yescharacteristic
53C22×C4×C251C2311yescharacteristic
54C2×C8×C251—21no—
55C2×C8×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—21no—
58C22×D251—21no—
59C22×D251—21no—
60C22×D251—22no—
61C22×D251—41no—
62C22×D251C22⋊C411yescharacteristic
63C22×D251C22⋊C411yescharacteristic
64C22⋊D251—42no—
65C22⋊D251—42no—
66C2×Dic251—21no—
67C2×Dic251—21no—
68C2×Dic251—21no—
69C2×Dic251—21no—
70C2×Dic251—41no—
71C2×Dic251—41no—
72C2×Dic251C22⋊C411yescharacteristic
73C2×Dic251C22⋊C411yescharacteristic
74C4×D251—22no—
75C4×D251—22no—
76C4×D251—41no—
77C4×D251—41no—
78C251⋊C8—41no—
79C251⋊C8—41no—
80C23×C251—21no—
81C23×C251—21no—
82C23×C251—21no—
83C23×C251—41no—
84C23×C251—41no—
85C23×C251C22×C411yescharacteristic
86C2×C4×C251—21no—
87C2×C4×C251—41no—
88C2×C4×C251—41no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C8×C251—41no—
92C8×C251—41no—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—81no—
99C22×C251—81no—
100C22×C251C2×C22⋊C411yescharacteristic
101C22×C251#13111yescharacteristic
102C22×C251#13111yescharacteristic
103C2×D251—21no—
104C2×D251—21no—
105C2×D251—21no—
106C2×D251—21no—
107C2×D251—21no—
108C2×D251—42no—
109C2×D251—42no—
110C2×D251#9912yes—
111C4×C251—21no—
112C4×C251—21no—
113C4×C251—41no—
114Dic251—21no—
115Dic251—21no—
116Dic251—41no—
117Dic251—41no—
118C2×C251—21no—
119C2×C251—21no—
120C2×C251—81no—
121C2×C251#40811yescharacteristic
122C2×C251C2×#9911yescharacteristic
123C2×C251#41311yescharacteristic
124D251—22no—
125D251—41no—
126C251#146811yescharacteristic, complemented, Sylow
127#1468—2511nomaximal, Sylow
128C2×#123—2511no—
129#407—2511no—
130C2×C22⋊C8—2511no—
131#324—2511no—
132#324—2511no—
133#323—2511no—
134#323D25111yescharacteristic, complemented
135C22×D4—2511no—
136C23×C4—2511no—
137C2×M16—5021no—
138C22×C8—5021no—
139#123—5021no—
140#123—5021no—
141#123—5021no—
142#123—5021no—
143C2×C4⋊C4—2511no—
144C2×C4⋊C4—2511no—
145C2×C22⋊C4—2511no—
146C2×C22⋊C4—2511no—
147C2×C22⋊C4C2×D25111yescharacteristic
148C22⋊C8—5021no—
149C22⋊C8—5021no—
150C22⋊C8—2511no—
151C22⋊C8—2511no—
152C22⋊C8C2×D25111yescharacteristic
153C22⋊C8C2×D25111yescharacteristic
154C24—2511no—
155C24C4×D25111yescharacteristic
156C2×D4—10042no—
157C2×D4—5021no—
158C2×D4—5021no—
159C22×C4—10042no—
160C22×C4—2511no—
161C22×C4—2511no—
162C22×C4—2511no—
163C22×C4—2511no—
164C22×C4—2511no—
165C22×C4—2511no—
166C22×C4—2511no—
167C22×C4C4×D25111yescharacteristic
168C22×C4C22×D25111yescharacteristic
169M16—10042no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—10042no—
173C2×C8—5021no—
174C2×C8—5021no—
175C4⋊C4—5021no—
176C4⋊C4—5021no—
177C4⋊C4—5021no—
178C4⋊C4—5021no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—5021no—
182C22⋊C4—5021no—
183C22⋊C4—5021no—
184C22⋊C4—5021no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23—10041no—
191C23—5022no—
192C23—5021no—
193C23—5021no—
194C23—2511no—
195C23—2511no—
196C23C2×C4×D25111yescharacteristic
197D4—10042no—
198D4—10042no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—10041no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—10041no—
206C2×C4—5022no—
207C2×C4—5022no—
208C2×C4—5021no—
209C2×C4—5021no—
210C2×C4—5021no—
211C2×C4—5021no—
212C2×C4—2511no—
213C2×C4—2511no—
214C2×C4D4×D25111yescharacteristic
215C2×C4D4×D25111yescharacteristic
216C8—41no—
217C8—41no—
218C8—10041no—
219C8—10041no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22—10042no—
228C22—10042no—
229C22—5021no—
230C22—5021no—
231C22—5021no—
232C22—5021no—
233C22—5021no—
234C22—2512no—
235C22C22⋊C4×D25111yescharacteristic
236C22#2254872111yescharacteristic
237C22#2254872311yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—10041no—
242C4—10041no—
243C4—5021no—
244C4—5021no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—10041no—
249C2—5022no—
250C2#3063165211yescharacteristic
251C2#99×D25111yescharacteristic
252C2#3063166811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C4
cosocleC22×C4×C251C23
hypercenter#323D251
hypocenterC251#1468
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30631649C2semidirect
3. C22⋊C8×D251C2semidirect
4. #30631412C2semidirect
5. #30631412C2semidirect
6. #30631411C2semidirect
7. #323×C251C2semidirect
126. C251#1468semidirect
134. #323D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486972
#306316492
C22⋊C8×D2512
#306314122
#306314112
#323×C2512
#1468251