Finite Group Databasebeta

← #30639110

Subgroups — #30639110

Order 16256.

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

normal subgroups only

This group has 332 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#22550642C211yesmaximal, characteristic, complemented
2#22550422C211yesmaximal, characteristic
3#22549893C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12M16×D127—22no—
13#16861207C411yescharacteristic, complemented
14#16861205C411yescharacteristic, complemented
15#16861198C411yescharacteristic, complemented
16#16861196C411yescharacteristic, complemented
17#16861181—22no—
18C4×C22⋊D127—22no—
19#16861168C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21#16861136—22no—
22#16861123—22no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26#104×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127D412yes—
35C22×Dic127—21no—
36C2×C4×D127D412yes—
37#14523232—21no—
38#14523232D412yescomplemented
39#14523231—24no—
40C2×C127⋊C8—21no—
41C8×D127—24no—
42C4⋊Dic127—21no—
43Dic127⋊C4—22no—
44C22⋊Dic127—22no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—22no—
47C4×Dic127—21no—
48C4×Dic127—21no—
49C4×Dic127C2×C411yescharacteristic
50C4×Dic127C2×C411yescharacteristic
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54#41×C127—21no—
55C2×Q8×C127C2×C411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127C2×C411yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—21no—
61M16×C127D412yescomplemented
62C2×C8×C127—21no—
63C4⋊C4×C127—21no—
64C22⋊C4×C127—21no—
65C42×C127—21no—
66C22×D127C22⋊C412yes—
67Dic254—21no—
68Dic254—21no—
69D508—21no—
70D508—21no—
71C22⋊D127—22no—
72C22⋊D127—22no—
73C22⋊D127—44no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—42no—
77C2×Dic127—41no—
78C2×Dic127C22⋊C412yes—
79C4×D127—22no—
80C4×D127—22no—
81C127⋊C8—22no—
82C23×C127—21no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—21no—
85Q8×C127—21no—
86D4×C127—21no—
87D4×C127—21no—
88D4×C127—41no—
89D4×C127—41no—
90C2×C4×C127—21no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—22no—
101C22×C127—41no—
102C22×C127—42no—
103C22×C127—41no—
104C22×C127C2×C22⋊C411yescharacteristic
105C22×C127#12111yescharacteristic
106C22×C127#12111yescharacteristic
107C2×D127—22no—
108C2×D127—42no—
109C4×C127—41no—
110C4×C127—81no—
111C4×C127C4×D411yescharacteristic
112C4×C127C4×D411yescharacteristic
113C4×C127#12011yescharacteristic
114C4×C127#12311yescharacteristic
115Dic127—22no—
116Dic127—22no—
117Dic127—41no—
118Dic127—81no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—81no—
123C2×C127#38611yescharacteristic
124D127—42no—
125#1884—1271nomaximal, Sylow
126C127#188411yescharacteristic, complemented, Sylow
127#568—1271no—
128#532—1271no—
129#421—1271no—
130#421D12711yescharacteristic, complemented
131#410—1271no—
132#339—1271no—
133#339—1271no—
134C2×#41—1271no—
135C2×#41C2×D12711yescharacteristic
136#131—2542no—
137#131—2542no—
138C2×M16—1271no—
139C2×M16—2542no—
140C2×M16C2×D12711yescharacteristic
141C4⋊Q8—1271no—
142#127—1271no—
143#124—1271no—
144#124—1271no—
145#121—2542no—
146C4×D4—2542no—
147#117—1271no—
148#117C2×D12711yescharacteristic
149C2×C42—1271no—
150#104—22no—
151#104—2542no—
152#99—2542no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160#41—2541no—
161C2×Q8—1271no—
162C2×Q8C4×D12711yescharacteristic
163C2×D4—21no—
164C2×D4—1272no—
165C2×D4—1271no—
166C2×D4—2542no—
167C2×D4C4×D12711yescharacteristic
168C22×C4—21no—
169C22×C4—1272no—
170C22×C4—2541no—
171C22×C4C22×D12711yescharacteristic
172M16—21no—
173M16—1272no—
174M16—2544no—
175M16—2541no—
176M16C22⋊D12712yescomplemented
177C2×C8—21no—
178C2×C8—2544no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—2542no—
182C4⋊C4—2541no—
183C22⋊C4—21no—
184C22⋊C4—2542no—
185C22⋊C4—2542no—
186C22⋊C4—2541no—
187C42—21no—
188C42—1271no—
189C42—1271no—
190C42—2541no—
191C42—2541no—
192C23—21no—
193C23—1272no—
194C23D4×D12711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—2541no—
198Q8—2541no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—5084no—
204D4—2542no—
205D4—2542no—
206D4—2541no—
207D4—2541no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—1272no—
213C2×C4—5081no—
214C2×C4—5082no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4C2×C4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C8—22no—
226C8—2542no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—5082no—
231C22—2542no—
232C22C22⋊C4×D12711yescharacteristic
233C22#1686118011yescharacteristic
234C22#1686118211yescharacteristic
235C4—41no—
236C4—81no—
237C4—10161no—
238C4—5081no—
239C4—2542no—
240C4—2542no—
241C4(C22⋊D127)⋊C411yescharacteristic
242C4C4×C22⋊D12711yescharacteristic
243C4#1686117711yescharacteristic
244C4#1686119111yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—5082no—
250C2#2254975211yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radical#22550422C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1884
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549893C2semidirect
4. #22549849C2semidirect
5. #22549637C2semidirect
6. #22549635C2semidirect
7. #421×C127C2semidirect
13. #16861207C4semidirect
14. #16861205C4semidirect
15. #16861198C4semidirect
16. #16861196C4semidirect
38. #14523232D4semidirect
61. M16×C127D4semidirect
126. C127#1884semidirect
130. #421D127semidirect
176. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504222
#225498932
#225498492
#225496372
#225496352
#421×C1272
#1884127