Finite Group Databasebeta

← #30648708

Subgroups — #30648708

Order 16256.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550454C211yesmaximal, characteristic
2#22550217C212yesmaximal, complemented
3#22550211C212yesmaximal, complemented
4#22550200C212yesmaximal, complemented
5#22550189C212yesmaximal, complemented
6#22549919C211yesmaximal, characteristic, complemented
7C2×#16861142C211yesmaximal, characteristic, complemented
8#22549875C211yesmaximal, characteristic, complemented
9#22549870C211yesmaximal, characteristic, complemented
10#22549843C211yesmaximal, characteristic, complemented
11#520×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861188C2212yes—
16#16861185—21no—
17#16861183C2212yes—
18#16861181—22no—
19#16861180—21no—
20C2×C4⋊Dic127C2211yescharacteristic
21#16861142C2212yes—
22#16861142C2212yes—
23#16861142C2212yes—
24#16861140C2212yes—
25#16861132C2211yescharacteristic
26#16861132C2211yescharacteristic
27#16861132C2212yes—
28#16861129C2211yescharacteristic
29#16861129C2211yescharacteristic
30#16861129C2212yes—
31C22⋊(C127⋊C8)C2212yes—
32C22⋊(C127⋊C8)C2212yes—
33C2×#41×C127C2211yescharacteristic
34C4×Q8×C127—21no—
35C4×D4×C127—21no—
36C4×D4×C127—22no—
37#117×C127C2211yescharacteristic
38#117×C127C2212yes—
39C2×C4⋊C4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254C2311yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46C2×C22⋊D127—22no—
47C22×Dic127—21no—
48C2×C4×D127—21no—
49#14523232—22no—
50C2×C127⋊C8—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2312yes—
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57C22⋊Dic127—22no—
58(C2×D127)⋊C4—22no—
59#41×C127—22no—
60#41×C127—22no—
61C2×Q8×C127D411yescharacteristic
62C2×D4×C127D411yescharacteristic
63C2×D4×C127D412yes—
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127C2312yes—
67C22×C4×C127C2311yescharacteristic
68C4⋊C4×C127—22no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—22no—
74C22⋊C4×C127—21no—
75C42×C127—22no—
76C42×C127—21no—
77C22×D127—21no—
78Dic254—21no—
79Dic254—21no—
80D508—21no—
81D508—21no—
82C22⋊D127—42no—
83C22⋊D127—82no—
84C2×Dic127—22no—
85C2×Dic127—21no—
86C2×Dic127—81no—
87C4×D127—42no—
88C127⋊C8—21no—
89C127⋊C8—21no—
90C127⋊C8—22no—
91C23×C127C2×D412yes—
92C23×C127C2×D411yescharacteristic
93Q8×C127—22no—
94D4×C127—24no—
95D4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—81no—
102C2×C4×C127—82no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4112yes—
106C2×C4×C127#4112yes—
107C2×C4×C127C2411yescharacteristic
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127#12312yes—
114C22×C127C22×D411yescharacteristic
115C2×D127—41no—
116C2×D127—41no—
117C2×D127—41no—
118C4×C127—22no—
119C4×C127—42no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127#12312yes—
123C4×C127C2×#4111yescharacteristic
124C4×C127C2×#4111yescharacteristic
125Dic127—41no—
126Dic127—42no—
127C2×C127—22no—
128C2×C127—42no—
129C2×C127C2×#12311yescharacteristic
130C2×C127#57611yescharacteristic
131C2×C127#57811yescharacteristic
132D127—81no—
133#3098—1271nomaximal, Sylow
134C127#309811yescharacteristic, complemented, Sylow
135#537—1271no—
136#520D12711yescharacteristic, complemented
137#485—1272no—
138#484—1272no—
139#482—1272no—
140#480—1272no—
141#428—1271no—
142C2×#107—1271no—
143#417—1271no—
144#416—1271no—
145#408—1271no—
146C2×#41—1271no—
147C2×#41C2×D12711yescharacteristic
148C2×M16—1271no—
149C22×C8—1271no—
150C22⋊Q8—1272no—
151C22⋊Q8—2541no—
152#121—1272no—
153#121—2541no—
154#121—2542no—
155C4×Q8—21no—
156C4×D4—22no—
157C4×D4—21no—
158#117C2×D12711yescharacteristic
159#117C2×D12712yes—
160C2×C4⋊C4—1271no—
161C2×C4⋊C4C2×D12711yescharacteristic
162C2×C4⋊C4C2×D12712yes—
163#107—1272no—
164#107—1272no—
165#107—1272no—
166#106—1272no—
167#103—1271no—
168#103—1271no—
169#103—1272no—
170#102—1271no—
171#102—1271no—
172#102—1272no—
173C22⋊C8—1272no—
174C22⋊C8—1272no—
175#41—22no—
176#41—22no—
177#41—2542no—
178#41—2542no—
179C2×Q8—1271no—
180C2×Q8C22⋊D12711yescharacteristic
181C2×D4—1271no—
182C2×D4—2542no—
183C2×D4—2541no—
184C2×D4C22⋊D12712yes—
185C2×D4C22⋊D12711yescharacteristic
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—2541no—
189C22×C4—2541no—
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192M16—2542no—
193C2×C8—1272no—
194C2×C8—1271no—
195C2×C8—1271no—
196C2×C8—2542no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—1272no—
200C4⋊C4—2542no—
201C4⋊C4—2541no—
202C4⋊C4C22×D12711yescharacteristic
203C4⋊C4C22×D12711yescharacteristic
204C4⋊C4C22×D12712yes—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—2542no—
208C22⋊C4—2542no—
209C42—22no—
210C42—21no—
211C23—2541no—
212C23C2×C22⋊D12712yes—
213C23C2×C22⋊D12711yescharacteristic
214Q8—22no—
215Q8—2541no—
216Q8—2541no—
217D4—24no—
218D4—22no—
219D4—10162no—
220D4—5082no—
221D4—2541no—
222D4—2541no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4—10161no—
231C2×C4—5082no—
232C2×C4—2542no—
233C2×C4—2541no—
234C2×C4C2×C22⋊D12712yes—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4#1452324812yes—
237C2×C4#1452324812yes—
238C2×C4C23×D12711yescharacteristic
239C8—2541no—
240C8—2541no—
241C8—2542no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—5081no—
248C22—5081no—
249C22—5081no—
250C22#1686119212yes—
251C22C22×C22⋊D12711yescharacteristic
252C4—22no—
253C4—42no—
254C4—41no—
255C4—41no—
256C4—5081no—
257C4—5082no—
258C4#1686119212yes—
259C4C2×#1452324811yescharacteristic
260C4C2×#1452324811yescharacteristic
261C2—22no—
262C2—42no—
263C2—10161no—
264C2C2×#1686119211yescharacteristic
265C2#2255067911yescharacteristic
266C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#520×C127C2
socleC22×C127C22×D4
prime radical#22550454C2
hypercenter#520D127
hypocenterC127#3098
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #22550217C2semidirect
3. #22550211C2semidirect
4. #22550200C2semidirect
5. #22550189C2semidirect
6. #22549919C2semidirect
7. C2×#16861142C2semidirect
8. #22549875C2semidirect
9. #22549870C2semidirect
10. #22549843C2semidirect
11. #520×C127C2semidirect
134. C127#3098semidirect
136. #520D127semidirect

Maximal subgroups

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

subgroupindex
#225504542
#225502172
#225502112
#225502002
#225501892
#225499192
C2×#168611422
#225498752
#225498702
#225498432
#520×C1272
#3098127