Finite Group Databasebeta

← #30648383

Subgroups — #30648383

Order 16256.

460 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×Q16×D127C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic
3#22550449C212yesmaximal
4#22550135C211yesmaximal, characteristic, complemented
5#22550134C211yesmaximal, characteristic, complemented
6#22550131C211yesmaximal, characteristic, complemented
7#22550116C212yesmaximal, complemented
8#22550115C212yesmaximal, complemented
9#22549911C211yesmaximal, characteristic, complemented
10#22549883C212yesmaximal, complemented
11#470×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×Q8×D127C2212yes—
14C2×Dic508C2211yescharacteristic
15C2×#14523238C2212yes—
16Q16×D127—24no—
17#16861217—24no—
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861208—22no—
24#16861195—22no—
25#16861194—22no—
26#16861191—22no—
27#16861189C2212yes—
28#16861188C2212yes—
29#16861185C2212yes—
30#16861167C2212yes—
31#16861142C2212yes—
32#16861141C2211yescharacteristic
33#16861134C2212yes—
34#16861133C2212yes—
35#16861132C2212yes—
36C2×Q16×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C22⋊Q8×C127C2212yes—
39#106×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—22no—
44C2×Dic254D411yescharacteristic
45C2×Dic254C2312yes—
46Q8×D127—24no—
47Q8×D127—24no—
48C2×D508D411yescharacteristic
49C2×C22⋊D127D411yescharacteristic
50C2×C22⋊D127D411yescharacteristic
51C2×C4×D127—22no—
52C2×C4×D127C2311yescharacteristic
53C2×C4×D127C2311yescharacteristic
54Dic508—22no—
55#14523238—24no—
56#14523231—22no—
57#14523231—22no—
58C2×C127⋊C8—22no—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—22no—
62C8×D127D414yes—
63C4⋊Dic127—22no—
64C4⋊Dic127C2312yes—
65Dic127⋊C4—22no—
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C4×Dic127—22no—
72C2×Q8×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—22no—
75M16×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127C2×D411yescharacteristic
82C22×D127C2×D411yescharacteristic
83Dic254—22no—
84Dic254—22no—
85Dic254—22no—
86Dic254—44no—
87D508—22no—
88C22⋊D127—22no—
89C22⋊D127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C4×D127—21no—
95C4×D127—21no—
96C4×D127—44no—
97C4×D127C2×D412yes—
98C4×D127#4112yes—
99C127⋊C8—21no—
100C127⋊C8C2×D412yes—
101C23×C127C2×D411yescharacteristic
102Q8×C127—22no—
103Q8×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C8×C127—21no—
111C8×C127C2×D412yes—
112C22×C127—21no—
113C22×C127—22no—
114C22×C127C22×D411yescharacteristic
115C2×D127—22no—
116C2×D127—21no—
117C2×D127—22no—
118C2×D127#12112yes—
119C4×C127—21no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127C22×D411yescharacteristic
123C4×C127C2×#4111yescharacteristic
124Dic127—21no—
125Dic127—42no—
126Dic127—42no—
127Dic127#12112yes—
128C2×C127—41no—
129C2×C127C2×#12111yescharacteristic
130C2×C127#57812yes—
131D127—22no—
132D127—41no—
133#3059—1271nomaximal, Sylow
134C127#305911yescharacteristic, complemented, Sylow
135C22×Q16—1271no—
136C2×#131—1271no—
137#536—1272no—
138#470—1271no—
139#470—1271no—
140#470—1271no—
141#470D12711yescharacteristic, complemented
142#467—1272no—
143#467—1272no—
144#427—1271no—
145#419—1272no—
146C2×#41—1271no—
147C22×Q8—1272no—
148C2×Q16—1272no—
149C2×Q16—1271no—
150C2×Q16—2544no—
151C2×Q16C2×D12711yescharacteristic
152#131—2544no—
153C2×M16—1271no—
154C2×M16—1271no—
155C2×M16C2×D12711yescharacteristic
156C22×C8—1271no—
157C22×C8—1271no—
158C22×C8—1271no—
159C4⋊Q8—2542no—
160#124—2542no—
161#123—2542no—
162#123—2542no—
163C22⋊Q8—1272no—
164C22⋊Q8—1272no—
165C22⋊Q8—1272no—
166C22⋊Q8C2×D12712yes—
167#117—1272no—
168#107—1272no—
169#106—1271no—
170#106C2×D12711yescharacteristic
171#103—1272no—
172#103—1272no—
173#103—1272no—
174#103C2×D12712yes—
175#41—2542no—
176#41—2542no—
177C2×Q8—1271no—
178C2×Q8—1272no—
179C2×Q8—2542no—
180C2×Q8—2544no—
181C2×Q8—2544no—
182C2×Q8C22×D12712yes—
183C2×D4—1271no—
184C2×D4—1271no—
185C2×D4—1271no—
186C22×C4—1271no—
187C22×C4—1271no—
188C22×C4—2542no—
189C22×C4C22×D12711yescharacteristic
190Q16—22no—
191Q16—2544no—
192Q16—2542no—
193M16—22no—
194M16—2542no—
195M16—2542no—
196C2×C8—1271no—
197C2×C8—1274no—
198C2×C8—1271no—
199C2×C8—2542no—
200C2×C8—2542no—
201C2×C8C22×D12711yescharacteristic
202C2×C8C22×D12711yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—1272no—
205C4⋊C4—2542no—
206C4⋊C4—2542no—
207C4⋊C4—2542no—
208C4⋊C4C22×D12712yes—
209C22⋊C4—22no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C42—2542no—
215C23—1271no—
216C23—1271no—
217C23D4×D12711yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—5084no—
221Q8—2542no—
222Q8—2542no—
223Q8—2542no—
224D4—2542no—
225D4—2542no—
226D4—2542no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—1271no—
232C2×C4—1272no—
233C2×C4—1272no—
234C2×C4—1271no—
235C2×C4—5084no—
236C2×C4—2542no—
237C2×C4—2542no—
238C2×C4—2541no—
239C2×C4—2541no—
240C2×C4D4×D12711yescharacteristic
241C2×C4C23×D12711yescharacteristic
242C8—21no—
243C8—1272no—
244C8—2541no—
245C8D4×D12712yes—
246C22—21no—
247C22—22no—
248C22—1272no—
249C22—2541no—
250C22—2542no—
251C22—2542no—
252C22C2×D4×D12711yescharacteristic
253C4—21no—
254C4—42no—
255C4—42no—
256C4—1272no—
257C4—5082no—
258C4—5082no—
259C4—2541no—
260C4C2×D4×D12711yescharacteristic
261C4#41×D12711yescharacteristic
262C2—41no—
263C2—5081no—
264C2—2542no—
265C2#121×D12711yescharacteristic
266C2#2255069212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#470D127
hypocenterC127#3059
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D127C2semidirect
4. #22550135C2semidirect
5. #22550134C2semidirect
6. #22550131C2semidirect
7. #22550116C2semidirect
8. #22550115C2semidirect
9. #22549911C2semidirect
10. #22549883C2semidirect
11. #470×C127C2semidirect
134. C127#3059semidirect
141. #470D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1272
C2×#168612172
#225504492
#225501352
#225501342
#225501312
#225501162
#225501152
#225499112
#225498832
#470×C1272
#3059127