Finite Group Databasebeta

← #30640375

Subgroups — #30640375

Order 16256.

280 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550437C211yesmaximal, characteristic
2#100×D127C211yesmaximal, characteristic, complemented
3#99×D127C211yesmaximal, characteristic, complemented
4#22549666C211yesmaximal, characteristic, complemented
5#22549664C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8C22×D508C411yescharacteristic, complemented
9C2×D4×D127C2211yescharacteristic
10M16×D127—21no—
11#16861194—21no—
12#16861192C411yescharacteristic, complemented
13#16861184—21no—
14#16861182—21no—
15#16861179—21no—
16#16861179C411yescharacteristic, complemented
17#16861177C2211yescharacteristic
18C22⋊C4×D127—21no—
19C22⋊C4×D127C411yescharacteristic, complemented
20#16861126C2211yescharacteristic
21#16861125—21no—
22#16861124C2211yescharacteristic
23#16861123—21no—
24#123×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C23×D127C2×C411yescharacteristic, complemented
29C23×D127D411yescharacteristic
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—41no—
34C2×D508—41no—
35C2×D508D411yescharacteristic
36C2×C22⋊D127—21no—
37C2×C22⋊D127C2×C411yescharacteristic
38C2×C22⋊D127D411yescharacteristic
39D4×D127—41no—
40D4×D127—41no—
41C2×C4×D127—21no—
42C2×C4×D127—41no—
43C2×C4×D127D411yescharacteristic
44#14523232—21no—
45#14523231—41no—
46C8×D127—41no—
47Dic127⋊C4—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127C2×C411yescharacteristic, complemented
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53(C2×D127)⋊C4—21no—
54(C2×D127)⋊C4—41no—
55C2×D4×C127C2311yescharacteristic
56C22×C4×C127C2×C411yescharacteristic, complemented
57M16×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127C2×C411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—41no—
68C22×D127—81no—
69C22×D127—81no—
70C22×D127—81no—
71C22×D127C22⋊C411yescharacteristic
72C22×D127C22⋊C411yescharacteristic
73C22×D127C22⋊C411yescharacteristic
74D508—41no—
75D508—42no—
76D508—41no—
77D508—41no—
78D508—81no—
79C22⋊D127—41no—
80C22⋊D127—41no—
81C22⋊D127—81no—
82C2×Dic127—21no—
83C2×Dic127—41no—
84C2×Dic127C22⋊C411yescharacteristic
85C4×D127—21no—
86C4×D127—21no—
87C4×D127—81no—
88C4×D127—82no—
89C127⋊C8—41no—
90C23×C127C22×C411yescharacteristic
91C23×C127C2×D411yescharacteristic
92D4×C127—41no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C8×C127—41no—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127C2×C22⋊C411yescharacteristic
104C2×D127—21no—
105C2×D127—21no—
106C2×D127—41no—
107C2×D127—42no—
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—81no—
114C2×D127—81no—
115C2×D127—81no—
116C2×D127—82no—
117C2×D127#9911yescharacteristic
118C2×D127#9911yescharacteristic
119C4×C127—21no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127—81no—
123Dic127—21no—
124Dic127—41no—
125Dic127—81no—
126C2×C127—21no—
127C2×C127—41no—
128C2×C127—41no—
129C2×C127C2×#9911yescharacteristic
130D127—21no—
131D127—21no—
132D127—41no—
133D127—41no—
134D127—81no—
135#2121—1271nomaximal, Sylow
136C127#212111yescharacteristic, complemented, Sylow
137#534—1271no—
138C2×#100—1271no—
139C2×#99—1271no—
140#352—1271no—
141#352D12711yescharacteristic, complemented
142AΣL(1,𝔽16)—1271no—
143AΣL(1,𝔽16)—1271no—
144C22×D4—1271no—
145C22×D4—1271no—
146C2×M16—2541no—
147#123—1271no—
148#123—2541no—
149#123C2×D12711yescharacteristic
150#121—2541no—
151#121—2541no—
152#120—1271no—
153#120—1271no—
154#120—2541no—
155C2×C22⋊C4—1271no—
156C2×C22⋊C4—2541no—
157#100—1271no—
158#100—2541no—
159#100C2×D12711yescharacteristic
160#99—1271no—
161#99—2541no—
162#99C2×D12711yescharacteristic
163C24—1271no—
164C24—1271no—
165C24—1271no—
166C2×D4—1271no—
167C2×D4—1271no—
168C2×D4—1271no—
169C2×D4—5081no—
170C2×D4—5081no—
171C2×D4—5081no—
172C2×D4—5081no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4C22×D12711yescharacteristic
178C22×C4—1271no—
179C22×C4—5081no—
180C22×C4—2541no—
181C22×C4C4×D12711yescharacteristic, complemented
182M16—21no—
183M16—5081no—
184M16—2541no—
185C2×C8—5081no—
186C4⋊C4—21no—
187C4⋊C4—2541no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—1271no—
191C22⋊C4—5081no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C22⋊C4—2541no—
197C22⋊C4C4×D12711yescharacteristic, complemented
198C23—1271no—
199C23—1271no—
200C23—1271no—
201C23—10161no—
202C23—10161no—
203C23—10161no—
204C23—5081no—
205C23—2541no—
206C23—2541no—
207C23—2541no—
208C23—2541no—
209C23—2541no—
210C23C2×C4×D12711yescharacteristic
211C23D4×D12711yescharacteristic
212D4—41no—
213D4—10161no—
214D4—10161no—
215D4—5081no—
216D4—5081no—
217D4—5081no—
218D4—5082no—
219D4—5081no—
220D4—5081no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1271no—
226C2×C4—10161no—
227C2×C4—10162no—
228C2×C4—5081no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C8—41no—
234C8—5081no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—1271no—
240C22—1271no—
241C22—10162no—
242C22—10161no—
243C22—10161no—
244C22—10161no—
245C22—5081no—
246C22—5081no—
247C22—5081no—
248C22—5081no—
249C22—5081no—
250C22—5082no—
251C22—5081no—
252C22—2541no—
253C22—2541no—
254C22C22⋊C4×D12711yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—10161no—
260C4—5081no—
261C4—2541no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—10161no—
266C2—5081no—
267C2—5081no—
268C2—2541no—
269C2—2541no—
270C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127C2×#99
prime radical#22550437C2
cosocleC2×D4×C127C23
hypercenter#352D127
hypocenterC127#2121
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D127C2semidirect
3. #99×D127C2semidirect
4. #22549666C2semidirect
5. #22549664C2semidirect
6. #22549662C2semidirect
7. #352×C127C2semidirect
8. C22×D508C4semidirect
12. #16861192C4semidirect
16. #16861179C4semidirect
19. C22⋊C4×D127C4semidirect
28. C23×D127C2×C4semidirect
49. C22⋊Dic127C2×C4semidirect
56. C22×C4×C127C2×C4semidirect
61. C22⋊C4×C127C2×C4semidirect
136. C127#2121semidirect
141. #352D127semidirect
181. C22×C4C4×D127semidirect
197. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225504372
#100×D1272
#99×D1272
#225496662
#225496642
#225496622
#352×C1272
#2121127