Finite Group Databasebeta

← #30649128

Subgroups — #30649128

Order 16256.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550422C211yesmaximal, characteristic
2C2×#16861200C211yesmaximal, characteristic, complemented
3#22550243C212yesmaximal, complemented
4#22550239C212yesmaximal, complemented
5#22550232C212yesmaximal, complemented
6#22550229C212yesmaximal, complemented
7#22549878C211yesmaximal, characteristic, complemented
8#22549877C211yesmaximal, characteristic, complemented
9#22549875C211yesmaximal, characteristic, complemented
10M16×Dic127C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861207C2211yescharacteristic
15#16861205C2211yescharacteristic
16#16861200—21no—
17#16861200—21no—
18#16861200C2212yes—
19#16861200C2212yes—
20#16861198C2211yescharacteristic
21#16861196C2211yescharacteristic
22#16861181—21no—
23#16861181—21no—
24C4×C22⋊D127—21no—
25C4×C22⋊D127—21no—
26C2×C4⋊Dic127—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861134C2212yes—
31#16861133C2212yes—
32#16861132C2212yes—
33#16861131C2212yes—
34#16861130C2212yes—
35#16861129C2212yes—
36#16861119C2212yes—
37C8×Dic127C2212yes—
38C2×#41×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yes—
42#102×C127C2212yes—
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C2×C22⋊D127—21no—
50C22×Dic127—21no—
51C22×Dic127D411yescharacteristic
52C22×Dic127D411yescharacteristic
53C2×C4×D127—21no—
54#14523232—22no—
55C2×C127⋊C8C2312yes—
56C4⋊Dic127—22no—
57C4⋊Dic127—22no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64C22⋊Dic127—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127—21no—
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127C2311yescharacteristic
80M16×C127—22no—
81C2×C8×C127C2312yes—
82C4⋊C4×C127—21no—
83C4⋊C4×C127C2312yes—
84C22×D127—21no—
85Dic254—21no—
86Dic254—21no—
87D508—21no—
88D508—21no—
89C22⋊D127—42no—
90C22⋊D127—42no—
91C22⋊D127—42no—
92C2×Dic127—21no—
93C2×Dic127—22no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C2×Dic127—42no—
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C2×Dic127C2×D411yescharacteristic
101C4×D127—42no—
102C127⋊C8—22no—
103C23×C127—21no—
104C23×C127C2×D411yescharacteristic
105Q8×C127—21no—
106Q8×C127—21no—
107D4×C127—21no—
108D4×C127—21no—
109D4×C127—42no—
110C2×C4×C127—22no—
111C2×C4×C127—21no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127#4112yes—
116C2×C4×C127#4112yes—
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127#12412yes—
125C22×C127C22×D411yescharacteristic
126C2×D127—41no—
127C2×D127—41no—
128C2×D127—41no—
129C4×C127—41no—
130C4×C127—42no—
131C4×C127#12412yes—
132C4×C127C2×#4111yescharacteristic
133C4×C127C2×#4111yescharacteristic
134Dic127—21no—
135Dic127—21no—
136Dic127—21no—
137Dic127—21no—
138Dic127—41no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127—81no—
142C2×C127C2×#12411yescharacteristic
143C2×C127#57511yescharacteristic
144C2×C127#57511yescharacteristic
145D127—81no—
146#3142—1271nomaximal, Sylow
147C127#314211yescharacteristic, complemented, Sylow
148#532—1271no—
149C2×#125—1271no—
150#491—1272no—
151#490—1272no—
152#488—1272no—
153#488—1272no—
154#417—1271no—
155#417—1271no—
156#417—1271no—
157#417D12711yescharacteristic, complemented
158C4×M16—1271no—
159C2×#41—1271no—
160C2×#41C2×D12711yescharacteristic
161C2×M16—1271no—
162C2×M16C2×D12711yescharacteristic
163C4⋊Q8—1271no—
164#127—1271no—
165#125—1272no—
166#125—1272no—
167#125—2541no—
168#125—2541no—
169#124—1271no—
170#124—1271no—
171#121—2541no—
172#121—2541no—
173C4×D4—2541no—
174C4×D4—2541no—
175C2×C4⋊C4—1271no—
176C2×C4⋊C4—2541no—
177C2×C4⋊C4—2541no—
178C2×C4⋊C4C2×D12711yescharacteristic
179C2×C42—1271no—
180#103—1272no—
181#103—1272no—
182#103—1272no—
183#103C2×D12712yes—
184#102—1272no—
185#102—1272no—
186#102—1272no—
187#102C2×D12712yes—
188#97—1272no—
189C4×C8—1272no—
190#41—22no—
191#41—22no—
192#41—2542no—
193#41—2542no—
194C2×Q8—1271no—
195C2×Q8C22×D12711yescharacteristic
196C2×D4—21no—
197C2×D4—1271no—
198C2×D4—2541no—
199C2×D4—2541no—
200C2×D4—2541no—
201C2×D4C22×D12711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1271no—
205C22×C4—1271no—
206C22×C4—2541no—
207C22×C4—2541no—
208C22×C4C22×D12711yescharacteristic
209M16—22no—
210M16—2542no—
211C2×C8—1272no—
212C2×C8C22×D12712yes—
213C4⋊C4—21no—
214C4⋊C4—1272no—
215C4⋊C4—2542no—
216C4⋊C4—2542no—
217C4⋊C4—2541no—
218C4⋊C4—2541no—
219C4⋊C4—2542no—
220C4⋊C4—2542no—
221C4⋊C4—2541no—
222C4⋊C4C22×D12712yes—
223C22⋊C4—2541no—
224C22⋊C4—2541no—
225C22⋊C4—2541no—
226C22⋊C4—2541no—
227C42—1271no—
228C42—1271no—
229C42—1271no—
230C42—1271no—
231C23—21no—
232C23—2541no—
233C23D4×D12711yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—2541no—
237Q8—2541no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—5082no—
242D4—5082no—
243D4—5082no—
244D4—2541no—
245D4—2541no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—1271no—
251C2×C4—1271no—
252C2×C4—1271no—
253C2×C4—1271no—
254C2×C4—5082no—
255C2×C4—5082no—
256C2×C4—5081no—
257C2×C4—5081no—
258C2×C4—2541no—
259C2×C4—2542no—
260C2×C4D4×D12711yescharacteristic
261C2×C4#1452324712yes—
262C2×C4#1452324812yes—
263C2×C4C23×D12711yescharacteristic
264C8—22no—
265C8—2542no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—5081no—
272C22—5081no—
273C22—5081no—
274C22#1686119512yes—
275C22C2×D4×D12711yescharacteristic
276C4—41no—
277C4—42no—
278C4—5081no—
279C4—5082no—
280C4—2541no—
281C4—2541no—
282C4—2541no—
283C4—2541no—
284C4#1686119512yes—
285C4C2×#1452324711yescharacteristic
286C4C2×#1452324811yescharacteristic
287C2—22no—
288C2—81no—
289C2—10161no—
290C2C2×#1686119511yescharacteristic
291C2#2255067211yescharacteristic
292C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550422C2
hypercenter#417D127
hypocenterC127#3142
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16861200C2semidirect
3. #22550243C2semidirect
4. #22550239C2semidirect
5. #22550232C2semidirect
6. #22550229C2semidirect
7. #22549878C2semidirect
8. #22549877C2semidirect
9. #22549875C2semidirect
10. M16×Dic127C2semidirect
11. #417×C127C2semidirect
147. C127#3142semidirect
157. #417D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
C2×#168612002
#225502432
#225502392
#225502322
#225502292
#225498782
#225498772
#225498752
M16×Dic1272
#417×C1272
#3142127