Finite Group Databasebeta

← #44548907

Subgroups — #44548907

Order 32128.

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

normal subgroups only

This group has 286 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
1#30632200C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic, complemented
3#30631546C211yesmaximal, characteristic, complemented
4#30631424C211yesmaximal, characteristic, complemented
5#30631422C211yesmaximal, characteristic, complemented
6#30631420C211yesmaximal, characteristic, complemented
7#328×C251C211yesmaximal, characteristic, complemented
8C2×#16860811C411yescharacteristic, complemented
9C22×C4×D251C2211yescharacteristic
10C2×#16860793—21no—
11#22548700—21no—
12#22548691C411yescharacteristic, complemented
13#22548690C411yescharacteristic, complemented
14#22548689C2211yescharacteristic
15#22548685—21no—
16#22548677—21no—
17(C22⋊D251)⋊C4—21no—
18#22548673C411yescharacteristic, complemented
19C2×Dic251⋊C4—21no—
20C2×(C2×D251)⋊C4—21no—
21C22⋊(C251⋊C8)C2211yescharacteristic
22(C2×D251)⋊C8—21no—
23C251⋊#95C2211yescharacteristic
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28#16860811—42no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004C2×C411yescharacteristic
32C2×C22⋊D251—21no—
33C22×Dic251—21no—
34C22×Dic251D411yescharacteristic
35C2×C4×D251—21no—
36C2×C4×D251—42no—
37C2×C4×D251C2×C411yescharacteristic
38C2×C4×D251C2×C411yescharacteristic
39C2×C4×D251D411yescharacteristic
40C2×C4×D251D411yescharacteristic
41#16860793—42no—
42C2×C251⋊C8—21no—
43C4⋊Dic251C2×C411yescharacteristic
44Dic251⋊C4—21no—
45Dic251⋊C4—42no—
46(C2×D251)⋊C4—21no—
47(C2×D251)⋊C4—21no—
48(C2×D251)⋊C4—21no—
49(C2×D251)⋊C4—21no—
50(C2×D251)⋊C4—42no—
51C4×Dic251—21no—
52C2×Q8×C251C2×C411yescharacteristic
53C22×C4×C251—21no—
54C22×C4×C251C2311yescharacteristic
55C2×C8×C251—21no—
56C4⋊C4×C251—21no—
57C4⋊C4×C251C2×C411yescharacteristic
58C22⋊C4×C251—21no—
59C22×D251—21no—
60C22×D251—21no—
61C22×D251—21no—
62C22×D251—41no—
63C22×D251—41no—
64C22×D251C22⋊C411yescharacteristic
65C22×D251C22⋊C411yescharacteristic
66D1004—41no—
67D1004—42no—
68D1004—42no—
69C22⋊D251—41no—
70C22⋊D251—41no—
71C2×Dic251—22no—
72C2×Dic251—21no—
73C2×Dic251—41no—
74C2×Dic251—41no—
75C2×Dic251—41no—
76C2×Dic251C22⋊C411yescharacteristic
77C2×Dic251C22⋊C411yescharacteristic
78C4×D251—22no—
79C4×D251—22no—
80C4×D251—41no—
81C4×D251—41no—
82C4×D251—42no—
83C251⋊C8—41no—
84C23×C251C2×D411yescharacteristic
85Q8×C251—41no—
86C2×C4×C251—21no—
87C2×C4×C251—21no—
88C2×C4×C251—41no—
89C2×C4×C251—41no—
90C2×C4×C251—41no—
91C2×C4×C251—41no—
92C2×C4×C251C22×C411yescharacteristic
93C2×C4×C251C2×D411yescharacteristic
94C8×C251—41no—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251C2×C22⋊C411yescharacteristic
98C22×C251AGL(1,ℤ/8)11yescharacteristic
99C22×C251#13711yescharacteristic
100C2×D251—21no—
101C2×D251—21no—
102C2×D251—21no—
103C2×D251—21no—
104C2×D251—21no—
105C2×D251—21no—
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—41no—
109C2×D251—41no—
110C2×D251—41no—
111C4×C251—21no—
112C4×C251—21no—
113C4×C251—41no—
114C4×C251—41no—
115C4×C251—81no—
116Dic251—21no—
117Dic251—41no—
118Dic251—81no—
119Dic251#9912yes—
120C2×C251—21no—
121C2×C251—21no—
122C2×C251C2×#9911yescharacteristic
123C2×C251#41711yescharacteristic
124C2×C251#42111yescharacteristic
125D251—41no—
126D251—41no—
127D251—81no—
128C251#151311yescharacteristic, complemented, Sylow
129#1513—2511nomaximal, Sylow
130#525—2511no—
131#407—2511no—
132#380—2511no—
133#328—2511no—
134#328D25111yescharacteristic, complemented
135#327—2511no—
136#327—2511no—
137C2×#41—2511no—
138C23×C4—2511no—
139C2×M16—5021no—
140#123—5021no—
141C22⋊Q8—2511no—
142C22⋊Q8C2×D25111yescharacteristic
143#121—2511no—
144#121—2511no—
145#120—5021no—
146C4×D4—5021no—
147C4×D4—5021no—
148#117—2511no—
149C2×C4⋊C4—5021no—
150C2×C22⋊C4—5021no—
151C22⋊C8—5021no—
152C22⋊C8—2511no—
153C22⋊C8C2×D25111yescharacteristic
154#95—2511no—
155#95C2×D25111yescharacteristic
156C24—2511no—
157#41—10042no—
158C2×Q8C4×D25111yescharacteristic
159C2×D4—5021no—
160C2×D4—5021no—
161C2×D4—5021no—
162C2×D4—2511no—
163C22×C4—21no—
164C22×C4—10042no—
165C22×C4—5021no—
166C22×C4—5021no—
167C22×C4—2511no—
168C22×C4—2511no—
169C22×C4—2511no—
170C22×C4—2511no—
171C22×C4—2511no—
172C22×C4C22×D25111yescharacteristic
173M16—10042no—
174C2×C8—21no—
175C2×C8—5021no—
176C4⋊C4—21no—
177C4⋊C4—10042no—
178C4⋊C4—5021no—
179C4⋊C4—2511no—
180C4⋊C4C4×D25111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—10042no—
183C22⋊C4—5021no—
184C22⋊C4—5021no—
185C22⋊C4—5021no—
186C22⋊C4—5021no—
187C42—5021no—
188C23—10041no—
189C23—10041no—
190C23—5021no—
191C23—5021no—
192C23—5021no—
193C23—2511no—
194C23—2511no—
195C23D4×D25111yescharacteristic
196Q8—41no—
197D4—10041no—
198D4—10042no—
199D4—10042no—
200D4—10041no—
201D4—10041no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—10041no—
209C2×C4—10041no—
210C2×C4—10042no—
211C2×C4—10041no—
212C2×C4—10041no—
213C2×C4—10041no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5021no—
217C2×C4—5022no—
218C2×C4—2511no—
219C2×C4—2511no—
220C2×C4C2×C4×D25111yescharacteristic
221C2×C4D4×D25111yescharacteristic
222C8—41no—
223C8—10041no—
224C22—21no—
225C22—21no—
226C22—10042no—
227C22—10042no—
228C22—10041no—
229C22—10041no—
230C22—10041no—
231C22—5021no—
232C22—5021no—
233C22—5021no—
234C22—5021no—
235C22—5021no—
236C22—5021no—
237C22C22⋊C4×D25111yescharacteristic
238C22#2254874211yescharacteristic
239C22#2254874711yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—20081no—
246C4—10041no—
247C4—5021no—
248C4—2512no—
249C2—21no—
250C2—21no—
251C2—20081no—
252C2—10041no—
253C2—10041no—
254C2#99×D25111yescharacteristic
255C2#3063168711yescharacteristic
256C2#3063170211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632200C2
cosocleC22×C4×C251C23
hypercenter#328D251
hypocenterC251#1513
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #30631649C2semidirect
3. #30631546C2semidirect
4. #30631424C2semidirect
5. #30631422C2semidirect
6. #30631420C2semidirect
7. #328×C251C2semidirect
8. C2×#16860811C4semidirect
12. #22548691C4semidirect
13. #22548690C4semidirect
18. #22548673C4semidirect
128. C251#1513semidirect
134. #328D251semidirect

Maximal subgroups

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

subgroupindex
#306322002
#306316492
#306315462
#306314242
#306314222
#306314202
#328×C2512
#1513251