Finite Group Databasebeta

← #30638828

Subgroups — #30638828

Order 16256.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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#117×D127C211yesmaximal, characteristic
2#22549839C211yesmaximal, characteristic
3C4×(C2×D127)⋊C4C211yesmaximal, characteristic
4#22549637C211yesmaximal, characteristic, complemented
5#22549636C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861168C2211yescharacteristic
11#16861167C411yescharacteristic, complemented
12#16861167C411yescharacteristic, complemented
13#16861166—21no—
14#16861165—21no—
15C4⋊C4×D127C411yescharacteristic, complemented
16C4⋊C4×D127C411yescharacteristic, complemented
17C2×(C2×D127)⋊C4—21no—
18C22⋊C4×D127—21no—
19C2×C4×Dic127C2211yescharacteristic
20C42×D127—21no—
21(C2×D127)⋊C8—21no—
22(C2×D127)⋊C8—21no—
23C127⋊#95—21no—
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C22×Dic127—21no—
29C22×Dic127D411yescharacteristic
30C2×C4×D127—21no—
31C2×C4×D127—21no—
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127C2×C411yescharacteristic
37C2×C4×D127C2×C411yescharacteristic
38C2×C4×D127D411yescharacteristic
39C2×C4×D127D411yescharacteristic
40#14523232—21no—
41#14523232—21no—
42C2×C127⋊C8—21no—
43C4⋊Dic127C2×C411yescharacteristic
44C4⋊Dic127C2×C411yescharacteristic
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—24no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51C4×Dic127—21no—
52C4×Dic127—21no—
53C4×Dic127—21no—
54C4×Dic127—21no—
55C4×Dic127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58M16×C127—21no—
59M16×C127—21no—
60C2×C8×C127—21no—
61C4⋊C4×C127C2×C411yescharacteristic
62C4⋊C4×C127C2×C411yescharacteristic
63C22⋊C4×C127—21no—
64C42×C127—21no—
65C42×C127—21no—
66C42×C127—21no—
67C22×D127—21no—
68C22×D127—21no—
69C22×D127—22no—
70C22×D127—41no—
71C22×D127C22⋊C411yescharacteristic
72C22×D127C22⋊C411yescharacteristic
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127C22⋊C411yescharacteristic
81C2×Dic127C22⋊C411yescharacteristic
82C4×D127—21no—
83C4×D127—21no—
84C4×D127—21no—
85C4×D127—21no—
86C4×D127—21no—
87C4×D127—21no—
88C4×D127—21no—
89C4×D127—21no—
90C4×D127—42no—
91C4×D127—42no—
92C127⋊C8—41no—
93C23×C127C2×D411yescharacteristic
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127C22×C411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C8×C127—41no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127C2×C22⋊C411yescharacteristic
109C22×C127#12311yescharacteristic
110C22×C127#12311yescharacteristic
111C2×D127—21no—
112C2×D127—21no—
113C2×D127—21no—
114C2×D127—21no—
115C2×D127—21no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127#10412yes—
119C4×C127—42no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127#11711yescharacteristic
123C4×C127#11711yescharacteristic
124C4×C127#12311yescharacteristic
125C4×C127#12311yescharacteristic
126Dic127—21no—
127Dic127—21no—
128Dic127—42no—
129Dic127—41no—
130Dic127—41no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127#38111yescharacteristic
134C2×C127C2×#10411yescharacteristic
135C2×C127#42111yescharacteristic
136D127—22no—
137D127—41no—
138#1826—1271nomaximal, Sylow
139C127#182611yescharacteristic, complemented, Sylow
140C2×#117—1271no—
141#407—1271no—
142C4×C22⋊C4—1271no—
143#339—1271no—
144#339—1271no—
145#339—1271no—
146#339D12711yescharacteristic, complemented
147C23×C4—1271no—
148C2×M16—1271no—
149C2×M16C2×D12711yescharacteristic
150#117—1271no—
151#117—1271no—
152#117—1271no—
153#117—2541no—
154#117—2541no—
155#117C2×D12711yescharacteristic
156C2×C4⋊C4—1271no—
157C2×C4⋊C4—1271no—
158C2×C22⋊C4—2541no—
159C2×C22⋊C4—2541no—
160C2×C42—1271no—
161C2×C42—2541no—
162C2×C42C2×D12711yescharacteristic
163C22⋊C8—2541no—
164C22⋊C8—2541no—
165#95—2541no—
166C24—1271no—
167C22×C4—21no—
168C22×C4—1271no—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—1271no—
172C22×C4—1271no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4C22×D12711yescharacteristic
181M16—21no—
182M16—21no—
183M16—2541no—
184M16—2541no—
185C2×C8—21no—
186C2×C8—2541no—
187C4⋊C4—1271no—
188C4⋊C4—1271no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4C4×D12711yescharacteristic
192C4⋊C4C4×D12711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C22⋊C4—2544no—
197C22⋊C4—2541no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—2541no—
202C42—2541no—
203C42—2541no—
204C42—2541no—
205C42—2541no—
206C23—1271no—
207C23—1271no—
208C23—5081no—
209C23—2542no—
210C23—2541no—
211C23—2541no—
212C23D4×D12711yescharacteristic
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—1271no—
219C2×C4—1271no—
220C2×C4—5082no—
221C2×C4—5082no—
222C2×C4—5081no—
223C2×C4—5081no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4—2541no—
227C2×C4—2541no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2541no—
233C2×C4—2541no—
234C2×C4—2541no—
235C2×C4—2541no—
236C2×C4—2541no—
237C2×C4C2×C4×D12711yescharacteristic
238C2×C4D4×D12711yescharacteristic
239C2×C4#1452324711yescharacteristic
240C2×C4#1452324811yescharacteristic
241C2×C4#1452324811yescharacteristic
242C2×C4#1452324911yescharacteristic
243C8—41no—
244C8—5081no—
245C22—21no—
246C22—21no—
247C22—1272no—
248C22—5082no—
249C22—5082no—
250C22—2541no—
251C22—2541no—
252C22—2541no—
253C22—2541no—
254C22—2541no—
255C22C22⋊C4×D12711yescharacteristic
256C22#1686119111yescharacteristic
257C22#1686119411yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—5082no—
262C4—5081no—
263C4—5081no—
264C4—2541no—
265C4—2541no—
266C4#1686116611yescharacteristic
267C4#1686116711yescharacteristic
268C4#1686119111yescharacteristic
269C4#1686119411yescharacteristic
270C2—21no—
271C2—21no—
272C2—5081no—
273C2—2542no—
274C2#2254973811yescharacteristic
275C2#104×D12711yescharacteristic
276C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1826
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. #22549637C2semidirect
5. #22549636C2semidirect
6. #22549635C2semidirect
7. #339×C127C2semidirect
11. #16861167C4semidirect
12. #16861167C4semidirect
15. C4⋊C4×D127C4semidirect
16. C4⋊C4×D127C4semidirect
139. C127#1826semidirect
146. #339D127semidirect

Maximal subgroups

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

subgroupindex
#117×D1272
#225498392
C4×(C2×D127)⋊C42
#225496372
#225496362
#225496352
#339×C1272
#1826127