Finite Group Databasebeta

← #30640370

Subgroups — #30640370

Order 16256.

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

normal subgroups only

This group has 264 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#22550384C211yesmaximal, characteristic
2#22549857C211yesmaximal, characteristic, complemented
3#22549848C211yesmaximal, characteristic, complemented
4#22549666C211yesmaximal, characteristic, complemented
5#22549664C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C2211yescharacteristic
9C22×C4×D127C411yescharacteristic, complemented
10M16×D127—21no—
11#16861192C411yescharacteristic, complemented
12#16861187—21no—
13#16861182—21no—
14#16861179C411yescharacteristic, complemented
15#16861177C2211yescharacteristic
16#16861171—21no—
17(C22⋊D127)⋊C4—21no—
18#16861165—21no—
19#16861165C411yescharacteristic, complemented
20#16861127—21no—
21#16861126C2211yescharacteristic
22#16861124C2211yescharacteristic
23#16861123—21no—
24#123×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28#14523247—41no—
29#14523247—41no—
30C2×Dic254D411yescharacteristic
31C2×D508—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127C2×C411yescharacteristic
34C2×C22⋊D127D411yescharacteristic
35C22×Dic127C2×C411yescharacteristic
36C22×Dic127D411yescharacteristic
37C2×C4×D127—21no—
38C2×C4×D127—21no—
39C2×C4×D127—41no—
40C2×C4×D127—41no—
41C2×C4×D127D411yescharacteristic
42#14523232—21no—
43#14523231—41no—
44C8×D127—41no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47Dic127⋊C4—41no—
48C22⋊Dic127—21no—
49C22⋊Dic127C2×C411yescharacteristic
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C4×Dic127—21no—
54C4×Dic127—41no—
55C2×D4×C127C2311yescharacteristic
56C22×C4×C127C2×C411yescharacteristic
57M16×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127C2×C411yescharacteristic
62C22×D127—21no—
63C22×D127—42no—
64C22×D127—41no—
65C22×D127C22⋊C411yescharacteristic
66Dic254—41no—
67D508—42no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—42no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—41no—
77C2×Dic127C22⋊C411yescharacteristic
78C2×Dic127C22⋊C411yescharacteristic
79C2×Dic127C22⋊C411yescharacteristic
80C4×D127—21no—
81C4×D127—21no—
82C4×D127—42no—
83C4×D127—41no—
84C4×D127—41no—
85C127⋊C8—41no—
86C23×C127C22×C411yescharacteristic
87C23×C127C2×D411yescharacteristic
88D4×C127—41no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C2×D411yescharacteristic
94C8×C127—41no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127C2×C22⋊C411yescharacteristic
100C2×D127—22no—
101C2×D127—21no—
102C2×D127—41no—
103C2×D127—42no—
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—42no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—81no—
111Dic127—21no—
112Dic127—21no—
113Dic127—41no—
114Dic127—81no—
115Dic127#9911yescharacteristic
116Dic127#9911yescharacteristic
117C2×C127—21no—
118C2×C127—41no—
119C2×C127—41no—
120C2×C127C2×#9911yescharacteristic
121D127—41no—
122D127—42no—
123#2120—1271nomaximal, Sylow
124C127#212011yescharacteristic, complemented, Sylow
125#525—1271no—
126#413—1271no—
127#410—1271no—
128#352—1271no—
129#352D12711yescharacteristic, complemented
130AΣL(1,𝔽16)—1271no—
131AΣL(1,𝔽16)—1271no—
132C2×#41—1271no—
133C23×C4—1271no—
134C2×M16—2541no—
135#123—1271no—
136#123C2×D12711yescharacteristic
137C22⋊Q8—2541no—
138#121—2541no—
139#120—1271no—
140#120—1271no—
141C4×D4—2541no—
142C4×D4—2541no—
143#117—1271no—
144#117—2541no—
145#101—2541no—
146#100—1271no—
147#100C2×D12711yescharacteristic
148#99—1271no—
149#99—2541no—
150#99C2×D12711yescharacteristic
151C24—1271no—
152#41—5081no—
153#41—5081no—
154C2×Q8—1271no—
155C2×D4—1271no—
156C2×D4—1271no—
157C2×D4—2541no—
158C2×D4—2541no—
159C2×D4C22×D12711yescharacteristic
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—1271no—
163C22×C4—5081no—
164C22×C4—5081no—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4C4×D12711yescharacteristic
168M16—21no—
169M16—5081no—
170M16—2541no—
171C2×C8—5081no—
172C4⋊C4—21no—
173C4⋊C4—5081no—
174C4⋊C4—2541no—
175C4⋊C4—2541no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—1271no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C22⋊C4C4×D12711yescharacteristic
184C42—5081no—
185C42—2541no—
186C23—1271no—
187C23—5081no—
188C23—5082no—
189C23—2541no—
190C23C2×C4×D12711yescharacteristic
191C23D4×D12711yescharacteristic
192Q8—5081no—
193D4—41no—
194D4—5082no—
195D4—5082no—
196D4—5081no—
197D4—5081no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1271no—
203C2×C4—1271no—
204C2×C4—1271no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—5082no—
208C2×C4—5081no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4D4×D12711yescharacteristic
217C8—41no—
218C8—5081no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—5081no—
224C22—5081no—
225C22—5082no—
226C22—5082no—
227C22—5081no—
228C22—2542no—
229C22—2541no—
230C22C22⋊C4×D12711yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—1271no—
236C4—1271no—
237C4—10161no—
238C4—5081no—
239C4—2541no—
240C4—2541no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—5082no—
245C2—5081no—
246C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127C2×#99
prime radical#22550384C2
cosocleC2×D4×C127C23
hypercenter#352D127
hypocenterC127#2120
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. #22549857C2semidirect
3. #22549848C2semidirect
4. #22549666C2semidirect
5. #22549664C2semidirect
6. #22549662C2semidirect
7. #352×C127C2semidirect
9. C22×C4×D127C4semidirect
11. #16861192C4semidirect
14. #16861179C4semidirect
19. #16861165C4semidirect
124. C127#2120semidirect
129. #352D127semidirect

Maximal subgroups

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

subgroupindex
#225503842
#225498572
#225498482
#225496662
#225496642
#225496622
#352×C1272
#2120127