Finite Group Databasebeta

← #30637712

Subgroups — #30637712

Order 16256.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550464C211yesmaximal, characteristic
2#22550034C211yesmaximal, characteristic, complemented
3#22550017C211yesmaximal, characteristic, complemented
4#22549788C211yesmaximal, characteristic, complemented
5#22549614C211yesmaximal, characteristic, complemented
6#22549611C211yesmaximal, characteristic, complemented
7#323×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C2×Dic508—21no—
10C2×#14523237—21no—
11#16861202—21no—
12#16861195—21no—
13#16861191—21no—
14#16861188C2211yescharacteristic
15#16861183C2211yescharacteristic
16#16861182—21no—
17#16861178—21no—
18C4×C22⋊D127—21no—
19C2×C22⋊Dic127—21no—
20#16861134—21no—
21#16861134—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—42no—
30C2×Dic254D411yescharacteristic
31C2×D508D411yescharacteristic
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127—21no—
37C2×C4×D127—21no—
38Dic508—42no—
39#14523237—42no—
40C4⋊Dic127D411yescharacteristic
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—42no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50C4×Dic127—21no—
51C24×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C22×D127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63Dic254—41no—
64D508—41no—
65C22⋊D127—41no—
66C22⋊D127—41no—
67C22⋊D127—81no—
68C22⋊D127—81no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C4×D127—81no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127C2×D411yescharacteristic
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C8×C127—41no—
91C8×C127—41no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—42no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—81no—
99C22×C127#12011yescharacteristic
100C22×C127#13511yescharacteristic
101C22×C127#13711yescharacteristic
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—41no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—41no—
108Dic127—41no—
109Dic127—41no—
110Dic127—42no—
111Dic127—41no—
112Dic127—81no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—81no—
116C2×C127#45211yescharacteristic
117C2×C127#45511yescharacteristic
118C2×C127B(4,𝔽2)11yescharacteristic
119D127—81no—
120#1606—1271nomaximal, Sylow
121C127#160611yescharacteristic, complemented, Sylow
122#538—1271no—
123#451—1271no—
124#448—1271no—
125#393—1271no—
126#328—1271no—
127#327—1271no—
128#323D12711yescharacteristic, complemented
129C22×Q8—1271no—
130C2×Q16—2541no—
131C2×SD16—2541no—
132#126—2541no—
133#124—2541no—
134#123—2541no—
135C22⋊Q8—1271no—
136#121—1271no—
137#121—2541no—
138#120—2541no—
139C4×D4—2541no—
140C2×C22⋊C4—2541no—
141C2×C22⋊C4C2×D12711yescharacteristic
142#103—2541no—
143#103—2541no—
144C22⋊C8C2×D12711yescharacteristic
145C22⋊C8C2×D12711yescharacteristic
146#95—1271no—
147#95—2541no—
148C24D50811yescharacteristic
149C2×Q8—1271no—
150C2×Q8—5082no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×D4—1271no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4—2541no—
160C22×C4D50811yescharacteristic
161C22×C4C22×D12711yescharacteristic
162Q16—5082no—
163SD16—5082no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—1271no—
167C4⋊C4—1271no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—5082no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C42—2541no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—2541no—
185C23C2×D50811yescharacteristic
186Q8—5081no—
187Q8—5081no—
188Q8—5081no—
189Q8—2542no—
190Q8—2542no—
191D4—10161no—
192D4—10161no—
193D4—5081no—
194D4—5081no—
195D4—5081no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—10161no—
200C2×C4—5081no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2542no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C2×C4D4×D12711yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—5081no—
221C22—5081no—
222C22—5081no—
223C22#1686117911yescharacteristic
224C22#1686123111yescharacteristic
225C22#1686124211yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—10161no—
230C4—5081no—
231C4—5081no—
232C4—5082no—
233C4—5081no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—10161no—
238C2#2255004011yescharacteristic
239C2#2255005611yescharacteristic
240C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
prime radical#22550464C2
hypercenter#323D127
hypocenterC127#1606
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
3. #22550017C2semidirect
4. #22549788C2semidirect
5. #22549614C2semidirect
6. #22549611C2semidirect
7. #323×C127C2semidirect
121. C127#1606semidirect
128. #323D127semidirect

Maximal subgroups

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

subgroupindex
#225504642
#225500342
#225500172
#225497882
#225496142
#225496112
#323×C1272
#1606127