Finite Group Databasebeta

← #30640729

Subgroups — #30640729

Order 16256.

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

normal subgroups only

This group has 270 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#22550619C211yesmaximal, characteristic, complemented
2#22550063C211yesmaximal, characteristic, complemented
3#22550054C211yesmaximal, characteristic, complemented
4#22549670C211yesmaximal, characteristic, complemented
5#22549663C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C127C211yesmaximal, characteristic, complemented
8#16861239—21no—
9#16861196—21no—
10#16861196C2211yescharacteristic
11#16861195—21no—
12#16861195—21no—
13#16861195—21no—
14#16861178—21no—
15#16861177—21no—
16#16861177—21no—
17#16861177C2211yescharacteristic
18#16861136—21no—
19#16861126C2211yescharacteristic
20#16861124—21no—
21#16861124C2211yescharacteristic
22H(5,𝔽2)×C127C2211yescharacteristic
23#120×C127—21no—
24#120×C127C2211yescharacteristic
25#99×C127—21no—
26#99×C127C2211yescharacteristic
27C23×D127D411yescharacteristic, complemented
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic, complemented
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—41no—
34#14523238—41no—
35#14523236—41no—
36#14523232—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127D411yescharacteristic, complemented
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—41no—
47C4×Dic127—21no—
48C4×Dic127D411yescharacteristic, complemented
49C24×C127D411yescharacteristic, complemented
50#41×C127—21no—
51#41×C127—41no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127C2311yescharacteristic
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—41no—
61C22⋊C4×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—81no—
65Dic254—41no—
66Dic254—81no—
67C22⋊D127—81no—
68C22⋊D127—81no—
69C22⋊D127—81no—
70C22⋊D127—82no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127—41no—
80C23×C127—81no—
81C23×C127C2×D411yescharacteristic
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84D4×C127—21no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89D4×C127—81no—
90D4×C127—82no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C2×D411yescharacteristic
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—81no—
107C22×C127—81no—
108C22×C127—81no—
109C22×C127#12011yescharacteristic
110C2×D127—41no—
111C2×D127—41no—
112C2×D127—42no—
113C2×D127—41no—
114C2×D127—81no—
115C4×C127—21no—
116C4×C127—41no—
117C4×C127—41no—
118C4×C127—81no—
119Dic127—41no—
120Dic127—41no—
121Dic127—41no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—41no—
125C2×C127—41no—
126C2×C127—81no—
127C2×C127—81no—
128C2×C127B(4,𝔽2)11yescharacteristic
129D127—81no—
130#2199—1271nomaximal, Sylow
131C127#219911yescharacteristic, complemented, Sylow
132H(3,𝔽4)—1271no—
133B(4,𝔽2)—1271no—
134B(4,𝔽2)D12711yescharacteristic, complemented
135#455—1271no—
136#354—1271no—
137AΣL(1,𝔽16)—1271no—
138AΣL(1,𝔽16)—1271no—
139H(5,𝔽2)C2×D12711yescharacteristic
140#137—2541no—
141#124—1271no—
142#124—2541no—
143#124—2541no—
144#124—2541no—
145#124—2541no—
146#120—21no—
147#120—1271no—
148#120—2541no—
149#120—2541no—
150#120—2541no—
151#120C2×D12711yescharacteristic
152#104—2541no—
153#100—1271no—
154#99—21no—
155#99—1271no—
156#99—2541no—
157#99C2×D12711yescharacteristic
158C24—1271no—
159C24C22⋊D12711yescharacteristic, complemented
160#41—21no—
161#41—41no—
162C2×Q8—1271no—
163C2×Q8—2541no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—5081no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4C22×D12711yescharacteristic
174Q16—5081no—
175SD16—5081no—
176M16—2541no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—1271no—
181C22⋊C4—5081no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C22⋊C4C22⋊D12711yescharacteristic, complemented
191C42—1271no—
192C42—2541no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—10161no—
199C23—5081no—
200C23—2541no—
201C23D4×D12711yescharacteristic
202C23C2×C22⋊D12711yescharacteristic
203Q8—21no—
204Q8—10161no—
205Q8—5081no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—10162no—
214D4—10161no—
215D4—10161no—
216D4—10161no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—5081no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4D4×D12711yescharacteristic
227C8—5081no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—10161no—
241C22—5081no—
242C22—5081no—
243C22—5082no—
244C22—5081no—
245C22#1686117711yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—10161no—
251C4—5081no—
252C4—5081no—
253C4—5081no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—10161no—
260C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingB(4,𝔽2)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterB(4,𝔽2)D127
hypocenterC127#2199
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550619C2semidirect
2. #22550063C2semidirect
3. #22550054C2semidirect
4. #22549670C2semidirect
5. #22549663C2semidirect
6. #22549662C2semidirect
7. B(4,𝔽2)×C127C2semidirect
27. C23×D127D4semidirect
29. C2×Dic254D4semidirect
42. C22⋊Dic127D4semidirect
48. C4×Dic127D4semidirect
49. C24×C127D4semidirect
61. C22⋊C4×C127D4semidirect
131. C127#2199semidirect
134. B(4,𝔽2)D127semidirect
159. C24C22⋊D127semidirect
190. C22⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506192
#225500632
#225500542
#225496702
#225496632
#225496622
B(4,𝔽2)×C1272
#2199127