Finite Group Databasebeta

← #352×D127 #30640364

Subgroups — #352×D127 #30640364

Order 16256.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D127C211yesmaximal, characteristic
2#100×D127C211yesmaximal, characteristic
3#99×D127C211yesmaximal, characteristic
4#22549667C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549665C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127C2211yescharacteristic
9C22×C4×D127C411yescharacteristic, complemented
10M16×D127—21no—
11#16861194—21no—
12#16861193C411yescharacteristic, complemented
13#16861192C411yescharacteristic, complemented
14#16861191—21no—
15#16861190C2211yescharacteristic
16C4⋊C4×D127—21no—
17C22⋊C4×D127—21no—
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×D127D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C22⋊D127C2×C411yescharacteristic
31C2×C22⋊D127D411yescharacteristic
32D4×D127—41no—
33D4×D127—41no—
34C22×Dic127C2×C411yescharacteristic, complemented
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127—21no—
38C2×C4×D127—41no—
39C2×C4×D127—41no—
40C2×C4×D127—41no—
41C2×C4×D127D411yescharacteristic
42#14523232—21no—
43#14523231—41no—
44C8×D127—41no—
45C4⋊Dic127—21no—
46Dic127⋊C4—21no—
47Dic127⋊C4—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127C2×C411yescharacteristic, complemented
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—42no—
67C22×D127—42no—
68C22×D127C22⋊C411yescharacteristic
69C22×D127C22⋊C411yescharacteristic
70C22×D127C22⋊C411yescharacteristic
71D508—41no—
72C22⋊D127—41no—
73C22⋊D127—41no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127C22⋊C411yescharacteristic
79C4×D127—21no—
80C4×D127—21no—
81C4×D127—42no—
82C4×D127—42no—
83C4×D127—41no—
84C4×D127—41no—
85C4×D127—82no—
86C127⋊C8—41no—
87C23×C127C22×C411yescharacteristic
88C23×C127C2×D411yescharacteristic
89D4×C127—41no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127C2×C22⋊C411yescharacteristic
101C2×D127—22no—
102C2×D127—21no—
103C2×D127—21no—
104C2×D127—42no—
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—81no—
110C2×D127#9911yescharacteristic
111C2×D127#9911yescharacteristic
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—81no—
116Dic127—21no—
117Dic127—41no—
118Dic127—41no—
119Dic127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127C2×#9911yescharacteristic
124D127—21no—
125D127—41no—
126D127—41no—
127D127#35212yesdirect product factor
128C2×#352—1271nomaximal, Sylow
129C127C2×#35211yescharacteristic, complemented, Sylow
130C2×#123—1271no—
131C2×#100—1271no—
132C2×#99—1271no—
133#352—1271no—
134#352—1271no—
135#352—1271no—
136#352D12711yescharacteristic, direct product factor
137C22×D4—1271no—
138C23×C4—1271no—
139C2×M16—2541no—
140#123—1271no—
141#123—1271no—
142#123—1271no—
143#123—2541no—
144#123—2541no—
145#123C2×D12711yescharacteristic
146C2×C4⋊C4—2541no—
147C2×C22⋊C4—1271no—
148C2×C22⋊C4—2541no—
149C2×C22⋊C4—2541no—
150#100—1271no—
151#100—2541no—
152#100C2×D12711yescharacteristic
153#99—1271no—
154#99—2541no—
155#99C2×D12711yescharacteristic
156C24—1271no—
157C24—1271no—
158C2×D4—1271no—
159C2×D4—1271no—
160C2×D4—1271no—
161C2×D4—5081no—
162C2×D4—5081no—
163C2×D4C22×D12711yescharacteristic
164C22×C4—1271no—
165C22×C4—1271no—
166C22×C4—5081no—
167C22×C4—5081no—
168C22×C4—5081no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4—2541no—
172C22×C4C4×D12711yescharacteristic, complemented
173M16—21no—
174M16—5081no—
175M16—2541no—
176C2×C8—5081no—
177C4⋊C4—21no—
178C4⋊C4—2541no—
179C4⋊C4—2541no—
180C4⋊C4—2541no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—1271no—
184C22⋊C4—5081no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C22⋊C4C4×D12711yescharacteristic, complemented
191C23—1271no—
192C23—1271no—
193C23—1271no—
194C23—5082no—
195C23—5082no—
196C23—2541no—
197C23—2541no—
198C23—2541no—
199C23—2541no—
200C23C2×C4×D12711yescharacteristic
201C23D4×D12711yescharacteristic
202D4—41no—
203D4—5081no—
204D4—5081no—
205D4—5081no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1271no—
211C2×C4—10162no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5082no—
215C2×C4—5082no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4D4×D12711yescharacteristic
223C8—41no—
224C8—5081no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—1271no—
230C22—1271no—
231C22—10161no—
232C22—5081no—
233C22—5081no—
234C22—5082no—
235C22—5082no—
236C22—5082no—
237C22—2542no—
238C22—2541no—
239C22—2541no—
240C22C22⋊C4×D12711yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—10161no—
246C4—5081no—
247C4—5081no—
248C4—2541no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—1272no—
253C2—5081no—
254C2—5081no—
255C2—2541no—
256C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127C2×#99
prime radicalC2×D4×D127C22
cosocleC2×D4×C127C23
hypercenter#352D127
hypocenterC127C2×#352
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
4. #22549667C2semidirect
5. #22549666C2semidirect
6. #22549665C2semidirect
7. #352×C127C2semidirect
9. C22×C4×D127C4semidirect
12. #16861193C4semidirect
13. #16861192C4semidirect
19. C22⋊C4×D127C4semidirect
34. C22×Dic127C2×C4semidirect
50. C22⋊Dic127C2×C4semidirect
56. C22×C4×C127C2×C4semidirect
61. C22⋊C4×C127C2×C4semidirect
127. D127#352direct (N × H)
129. C127C2×#352semidirect
136. #352D127direct (N × H)
172. C22×C4C4×D127semidirect
190. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
#123×D1272
#100×D1272
#99×D1272
#225496672
#225496662
#225496652
#352×C1272
C2×#352127