Finite Group Databasebeta

← #30640734

Subgroups — #30640734

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#22550616C211yesmaximal, characteristic, complemented
2#22550063C211yesmaximal, characteristic, complemented
3#22550044C211yesmaximal, characteristic, complemented
4#22549668C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7#458×C127C211yesmaximal, characteristic, complemented
8#16861234—21no—
9#16861205C2211yescharacteristic
10#16861204—21no—
11#16861191—21no—
12#16861184—21no—
13#16861183—21no—
14#16861182—21no—
15#16861179—21no—
16#16861177—21no—
17#16861177C2211yescharacteristic
18#16861136—21no—
19#16861126C2211yescharacteristic
20#16861124—21no—
21#16861124C2211yescharacteristic
22H(5,𝔽2)×C127C2211yescharacteristic
23#123×C127—21no—
24#123×C127C2211yescharacteristic
25#99×C127—21no—
26#99×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508D411yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—41no—
35C2×C4×D127—21no—
36#14523235—41no—
37#14523233—41no—
38#14523232—21no—
39C4⋊Dic127—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127D411yescharacteristic
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—41no—
48C4×Dic127D411yescharacteristic, complemented
49#41×C127—21no—
50#41×C127—41no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—41no—
54C2×D4×C127C2311yescharacteristic
55C22×C4×C127D411yescharacteristic, complemented
56C4⋊C4×C127—21no—
57C4⋊C4×C127—41no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—41no—
61C22⋊C4×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—41no—
65C22×D127—81no—
66D508—41no—
67D508—81no—
68D508—81no—
69C22⋊D127—81no—
70C22⋊D127—81no—
71C22⋊D127—81no—
72C22⋊D127—82no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—41no—
76C4×D127—81no—
77C127⋊C8—41no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82Q8×C127—21no—
83D4×C127—21no—
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—81no—
102C22×C127—81no—
103C22×C127#12011yescharacteristic
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—41no—
107C2×D127—41no—
108C2×D127—42no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—81no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—41no—
116C4×C127—81no—
117Dic127—41no—
118Dic127—41no—
119Dic127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—81no—
124C2×C127B(4,𝔽2)11yescharacteristic
125D127—81no—
126D127—81no—
127#2200—1271nomaximal, Sylow
128C127#220011yescharacteristic, complemented, Sylow
129#560—1271no—
130#458D12711yescharacteristic, complemented
131B(4,𝔽2)—1271no—
132#453—1271no—
133#353—1271no—
134#352—1271no—
135AΣL(1,𝔽16)—1271no—
136H(5,𝔽2)C2×D12711yescharacteristic
137AGL(1,ℤ/8)—2541no—
138#127—1271no—
139#126—2541no—
140#123—21no—
141#123—2541no—
142#123C2×D12711yescharacteristic
143#121—2541no—
144#121—2541no—
145#121—2541no—
146#120—1271no—
147#120—2541no—
148#120—2541no—
149#104—2541no—
150#100—1271no—
151#99—21no—
152#99—1271no—
153#99—2541no—
154#99C2×D12711yescharacteristic
155C24—1271no—
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—1271no—
162C2×D4—5081no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C2×D4C22×D12711yescharacteristic
169C22×C4—2541no—
170C22×C4C22⋊D12711yescharacteristic, complemented
171SD16—5081no—
172D8—5081no—
173M16—2541no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—41no—
181C22⋊C4—1271no—
182C22⋊C4—5081no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4C22⋊D12711yescharacteristic, complemented
189C42—1271no—
190C23—21no—
191C23—21no—
192C23—10161no—
193C23—5081no—
194C23—2541no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197C23C2×C22⋊D12711yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—10161no—
205D4—10161no—
206D4—10162no—
207D4—10161no—
208D4—10161no—
209D4—10161no—
210D4—5081no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—10161no—
219C2×C4—5081no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4D4×D12711yescharacteristic
223C8—5081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—81no—
230C22—81no—
231C22—10161no—
232C22—5081no—
233C22—5081no—
234C22—5081no—
235C22—5081no—
236C22—5082no—
237C22—5081no—
238C22—5081no—
239C22#1686117711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—10161no—
246C4—5081no—
247C4—5081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—10161no—
253C2—10161no—
254C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#458×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#458D127
hypocenterC127#2200
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #22550616C2semidirect
2. #22550063C2semidirect
3. #22550044C2semidirect
4. #22549668C2semidirect
5. #22549666C2semidirect
6. #22549662C2semidirect
7. #458×C127C2semidirect
30. C2×D508D4semidirect
48. C4×Dic127D4semidirect
55. C22×C4×C127D4semidirect
61. C22⋊C4×C127D4semidirect
128. C127#2200semidirect
130. #458D127semidirect
170. C22×C4C22⋊D127semidirect
188. C22⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506162
#225500632
#225500442
#225496682
#225496662
#225496622
#458×C1272
#2200127