Finite Group Databasebeta

← #44549345

Subgroups — #44549345

Order 32128.

260 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 53 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#30631842C211yesmaximal, characteristic, complemented
2#30631825C211yesmaximal, characteristic, complemented
3#30631599C211yesmaximal, characteristic, complemented
4#30631424C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631411C211yesmaximal, characteristic, complemented
7#538×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9C2×#16860800—21no—
10C2×#16860798—21no—
11C2×C22⋊Dic251—21no—
12#22548638—21no—
13#22548638—21no—
14C22⋊(C251⋊C8)C2211yescharacteristic
15C22⋊(C251⋊C8)C2211yescharacteristic
16C251⋊#95—21no—
17C251⋊#95C2211yescharacteristic
18#126×C251—21no—
19#124×C251—21no—
20#123×C251—21no—
21C22⋊Q8×C251C2211yescharacteristic
22#121×C251—21no—
23#121×C251C2211yescharacteristic
24#120×C251—21no—
25C4×D4×C251—21no—
26C2×C22⋊C4×C251C2211yescharacteristic
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502—42no—
30C22×Dic251—21no—
31C22×Dic251—21no—
32#16860800—42no—
33#16860798—42no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8—21no—
36C22⋊Dic251—42no—
37C24×C251D411yescharacteristic
38C2×Q8×C251D411yescharacteristic
39C2×D4×C251—21no—
40C2×D4×C251—21no—
41C2×D4×C251—21no—
42C2×D4×C251D411yescharacteristic
43C22×C4×C251—21no—
44C22×C4×C251D411yescharacteristic
45C22×C4×C251C2311yescharacteristic
46C4⋊C4×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251D411yescharacteristic
49C4⋊C4×C251D411yescharacteristic
50C22⋊C4×C251—21no—
51C22⋊C4×C251—21no—
52C22⋊C4×C251—21no—
53C22⋊C4×C251—21no—
54C22⋊C4×C251—21no—
55C22⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C42×C251—21no—
58Dic502—22no—
59Dic502—22no—
60Dic502—41no—
61Dic502—41no—
62C2×Dic251—22no—
63C2×Dic251—41no—
64C2×Dic251—41no—
65C2×Dic251—41no—
66C2×Dic251—41no—
67C2×Dic251—41no—
68C251⋊C8—41no—
69C251⋊C8—41no—
70C23×C251—21no—
71C23×C251—21no—
72C23×C251—21no—
73C23×C251—21no—
74C23×C251—41no—
75C23×C251—41no—
76C23×C251C2×D411yescharacteristic
77Q8×C251—41no—
78D4×C251—41no—
79D4×C251—41no—
80D4×C251—41no—
81D4×C251—81no—
82D4×C251—81no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—41no—
88C2×C4×C251—41no—
89C2×C4×C251—81no—
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2×D411yescharacteristic
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—42no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251—81no—
102C22×C251#12011yescharacteristic
103C22×C251#13511yescharacteristic
104C22×C251#13711yescharacteristic
105C4×C251—21no—
106C4×C251—21no—
107C4×C251—41no—
108C4×C251—41no—
109C4×C251—41no—
110C4×C251—41no—
111Dic251—42no—
112Dic251—81no—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251—81no—
116C2×C251—81no—
117C2×C251#45211yescharacteristic
118C2×C251#45511yescharacteristic
119C2×C251B(4,𝔽2)11yescharacteristic
120C251#160611yescharacteristic, complemented, Sylow
121#1606—2511nomaximal, Sylow
122#538D25111yescharacteristic
123#451—2511no—
124#448—2511no—
125#393—2511no—
126#328—2511no—
127#327—2511no—
128#323—2511no—
129C22×Q8—2511no—
130C2×Q16—5021no—
131C2×SD16—5021no—
132#126—21no—
133#124—21no—
134#123—21no—
135C22⋊Q8C2×D25111yescharacteristic
136#121—21no—
137#121C2×D25111yescharacteristic
138#120—21no—
139C4×D4—21no—
140C2×C22⋊C4—5021no—
141C2×C22⋊C4C2×D25111yescharacteristic
142#103—5021no—
143#103—5021no—
144C22⋊C8—2511no—
145C22⋊C8—2511no—
146#95—5021no—
147#95—2511no—
148C24C22⋊D25111yescharacteristic
149C2×Q8—10042no—
150C2×Q8—5021no—
151C2×Q8—5021no—
152C2×Q8C22⋊D25111yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C22⋊D25111yescharacteristic
157C22×C4—21no—
158C22×C4—5021no—
159C22×C4—5021no—
160C22×C4C22⋊D25111yescharacteristic
161C22×C4C22×D25111yescharacteristic
162Q16—10042no—
163SD16—10042no—
164C2×C8—5021no—
165C2×C8—5021no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4C22⋊D25111yescharacteristic
169C4⋊C4C22⋊D25111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—10042no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23C2×C22⋊D25111yescharacteristic
186Q8—41no—
187Q8—10041no—
188Q8—10041no—
189Q8—5022no—
190Q8—5022no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—10041no—
206C2×C4—10041no—
207C2×C4—10041no—
208C2×C4—5022no—
209C2×C4C2×C22⋊D25111yescharacteristic
210C2×C4C2×C22⋊D25111yescharacteristic
211C8—10041no—
212C8—10041no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22#2254868411yescharacteristic
224C22#2254873511yescharacteristic
225C22#2254874511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—20081no—
233C4—10042no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#3063184911yescharacteristic
239C2#3063186511yescharacteristic
240C2#3063187411yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting, prime radical#538×C251C2
socleC22×C251#120
hypercenter#538D251
hypocenterC251#1606
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #30631842C2semidirect
2. #30631825C2semidirect
3. #30631599C2semidirect
4. #30631424C2semidirect
5. #30631421C2semidirect
6. #30631411C2semidirect
120. C251#1606semidirect

Maximal subgroups

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

subgroupindex
#306318422
#306318252
#306315992
#306314242
#306314212
#306314112
#538×C2512
#1606251