Finite Group Databasebeta

← #30640736

Subgroups — #30640736

Order 16256.

260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550613C211yesmaximal, characteristic, complemented
2#22550066C211yesmaximal, characteristic, complemented
3#22550063C211yesmaximal, characteristic, complemented
4#22549669C211yesmaximal, characteristic, complemented
5#22549667C211yesmaximal, characteristic, complemented
6#22549664C211yesmaximal, characteristic, complemented
7#453×C127C211yesmaximal, characteristic, complemented
8#16861203—21no—
9#16861190—21no—
10#16861190—21no—
11#16861190C2211yescharacteristic
12#16861181—21no—
13#16861181—21no—
14#16861181—21no—
15#16861177—21no—
16#16861177—21no—
17#16861177C2211yescharacteristic
18#16861124—21no—
19#16861124—21no—
20#16861124C2211yescharacteristic
21#16861124C2211yescharacteristic
22H(5,𝔽2)×C127C2211yescharacteristic
23AGL(1,ℤ/8)×C127—21no—
24#127×C127C2211yescharacteristic
25#104×C127—21no—
26#100×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—41no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic, complemented
34Dic127⋊C4—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—41no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—41no—
43C22⋊Dic127D411yescharacteristic
44C22⋊Dic127D411yescharacteristic, complemented
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—41no—
49#41×C127—21no—
50#41×C127—41no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—41no—
56C2×D4×C127D411yescharacteristic, complemented
57C2×D4×C127C2311yescharacteristic
58SD16×C127—41no—
59D8×C127—41no—
60M16×C127—21no—
61C42×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—81no—
65C22⋊D127—81no—
66C22⋊D127—81no—
67C22⋊D127—81no—
68C22⋊D127—82no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—81no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D411yescharacteristic
81Q8×C127—21no—
82D4×C127—21no—
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—81no—
89D4×C127—81no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127#12011yescharacteristic
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—42no—
110C2×D127—41no—
111C2×D127—81no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115Dic127—41no—
116Dic127—41no—
117Dic127—41no—
118Dic127—81no—
119Dic127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—81no—
124C2×C127—81no—
125C2×C127B(4,𝔽2)11yescharacteristic
126D127—81no—
127#2200—1271nomaximal, Sylow
128C127#220011yescharacteristic, complemented, Sylow
129#560—1271no—
130#458—1271no—
131B(4,𝔽2)—1271no—
132#453D12711yescharacteristic, complemented
133#353—1271no—
134#352—1271no—
135AΣL(1,𝔽16)—1271no—
136H(5,𝔽2)C2×D12711yescharacteristic
137AGL(1,ℤ/8)—21no—
138#127C2×D12711yescharacteristic
139#126—2541no—
140#123—1271no—
141#123—2541no—
142#123—2541no—
143#121—2541no—
144#121—2541no—
145#121—2541no—
146#120—1271no—
147#120—2541no—
148#120—2541no—
149#104—21no—
150#100C2×D12711yescharacteristic
151#99—1271no—
152#99—1271no—
153#99—2541no—
154#99—2541no—
155C24—1271no—
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—5081no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4C22⋊D12711yescharacteristic, complemented
168C2×D4C22×D12711yescharacteristic
169C22×C4—1271no—
170C22×C4—2541no—
171SD16—41no—
172D8—41no—
173M16—21no—
174C4⋊C4—5081no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C22⋊C4—1271no—
179C22⋊C4—1271no—
180C22⋊C4—5081no—
181C22⋊C4—5081no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C42C22⋊D12711yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—21no—
193C23—10161no—
194C23—5081no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197C23D4×D12711yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—81no—
206D4—81no—
207D4—10162no—
208D4—10161no—
209D4—10161no—
210D4—10161no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—10161no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4C2×C22⋊D12711yescharacteristic
223C8—41no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22—81no—
234C22—10161no—
235C22—5081no—
236C22—5081no—
237C22—5082no—
238C22—5081no—
239C22#1686117711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—10161no—
244C4—10161no—
245C4—5081no—
246C4—5081no—
247C4—5081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—10161no—
254C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#453×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#453D127
hypocenterC127#2200
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550613C2semidirect
2. #22550066C2semidirect
3. #22550063C2semidirect
4. #22549669C2semidirect
5. #22549667C2semidirect
6. #22549664C2semidirect
7. #453×C127C2semidirect
33. C22×Dic127D4semidirect
44. C22⋊Dic127D4semidirect
56. C2×D4×C127D4semidirect
61. C42×C127D4semidirect
128. C127#2200semidirect
132. #453D127semidirect
167. C2×D4C22⋊D127semidirect
189. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506132
#225500662
#225500632
#225496692
#225496672
#225496642
#453×C1272
#2200127