Finite Group Databasebeta

← #30640737

Subgroups — #30640737

Order 16256.

260 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550065C211yesmaximal, characteristic, complemented
2#22550062C211yesmaximal, characteristic, complemented
3#22550043C211yesmaximal, characteristic, complemented
4#22549669C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549663C211yesmaximal, characteristic, complemented
7#560×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861235—21no—
10#16861192—21no—
11#16861178—21no—
12#16861137—21no—
13#16861126C2211yescharacteristic
14#16861124C2211yescharacteristic
15#16861124C2211yescharacteristic
16#16861123—21no—
17#16861123—21no—
18#127×C127C2211yescharacteristic
19#126×C127—21no—
20#123×C127—21no—
21#123×C127C2211yescharacteristic
22#121×C127—21no—
23#121×C127—21no—
24#121×C127—21no—
25#120×C127—21no—
26#120×C127C2211yescharacteristic
27#14523248—21no—
28#14523247—41no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—41no—
32D4×D127—41no—
33#14523236—41no—
34#14523233—41no—
35#14523232—21no—
36Dic127⋊C4—41no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—41no—
40(C2×D127)⋊C4—41no—
41C24×C127D411yescharacteristic
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127D411yescharacteristic, complemented
48C2×D4×C127C2311yescharacteristic
49C22×C4×C127—21no—
50C22×C4×C127D411yescharacteristic, complemented
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127D411yescharacteristic
60C22⋊C4×C127D411yescharacteristic, complemented
61C42×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—21no—
64Dic254—21no—
65D508—21no—
66C22⋊D127—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C22⋊D127—82no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C4×D127—41no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—41no—
79C23×C127—81no—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—81no—
85D4×C127—81no—
86D4×C127—81no—
87D4×C127—81no—
88D4×C127—81no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—81no—
95C2×C4×C127C2×D411yescharacteristic
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—42no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—81no—
108C22×C127#12011yescharacteristic
109C2×D127—41no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—41no—
116C4×C127—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#560D12711yescharacteristic, complemented
130#458—1271no—
131B(4,𝔽2)—1271no—
132#453—1271no—
133#353—1271no—
134#352—1271no—
135AΣL(1,𝔽16)—1271no—
136H(5,𝔽2)—1271no—
137AGL(1,ℤ/8)—2541no—
138#127C2×D12711yescharacteristic
139#126—21no—
140#123—21no—
141#123—2541no—
142#123C2×D12711yescharacteristic
143#121—21no—
144#121—21no—
145#121—21no—
146#120—21no—
147#120—2541no—
148#120C2×D12711yescharacteristic
149#104—2541no—
150#100—1271no—
151#99—1271no—
152#99—1271no—
153#99—2541no—
154#99—2541no—
155C24C22⋊D12711yescharacteristic
156#41—5081no—
157#41—2541no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—5081no—
164C2×D4—5081no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4C22⋊D12711yescharacteristic, complemented
168C2×D4C22×D12711yescharacteristic
169C22×C4—21no—
170C22×C4C22⋊D12711yescharacteristic, complemented
171SD16—5081no—
172D8—5081no—
173M16—2541no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—5081no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—5081no—
184C22⋊C4—5081no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4C22⋊D12711yescharacteristic
188C22⋊C4C22⋊D12711yescharacteristic, complemented
189C42C22⋊D12711yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—41no—
193C23—81no—
194C23—2541no—
195C23—2541no—
196C23C2×C22⋊D12711yescharacteristic
197C23C2×C22⋊D12711yescharacteristic
198Q8—2541no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—81no—
204D4—81no—
205D4—81no—
206D4—10162no—
207D4—5081no—
208D4—5081no—
209D4—5081no—
210D4—2541no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4C2×C22⋊D12711yescharacteristic
223C8—5081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—81no—
236C22—10161no—
237C22—10161no—
238C22—5081no—
239C22#1686117811yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—10161no—
246C4—10161no—
247C4—5081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—10161no—
254C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#560×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#560D127
hypocenterC127#2200
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #560

Complements

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

NH ≅ G/Ntype
1. #22550065C2semidirect
2. #22550062C2semidirect
3. #22550043C2semidirect
4. #22549669C2semidirect
5. #22549666C2semidirect
6. #22549663C2semidirect
7. #560×C127C2semidirect
47. C2×D4×C127D4semidirect
50. C22×C4×C127D4semidirect
60. C22⋊C4×C127D4semidirect
61. C42×C127D4semidirect
128. C127#2200semidirect
129. #560D127semidirect
167. C2×D4C22⋊D127semidirect
170. C22×C4C22⋊D127semidirect
188. C22⋊C4C22⋊D127semidirect
189. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225500652
#225500622
#225500432
#225496692
#225496662
#225496632
#560×C1272
#2200127