Finite Group Databasebeta

← #30639177

Subgroups — #30639177

Order 16256.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550388C211yesmaximal, characteristic
2C2×#16861166C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549839C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×C4×D127C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861193—21no—
12#16861189C412yescomplemented
13#16861184C412yescomplemented
14#16861179—21no—
15C4×D508—22no—
16#16861166—22no—
17#16861166—22no—
18C2×Dic127⋊C4—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×C4×Dic127C2211yescharacteristic
21#16861137—22no—
22#16861135—22no—
23(C2×D127)⋊C8—22no—
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32C2×Dic254C2×C411yescharacteristic
33C2×D508—22no—
34C2×D508C2×C411yescharacteristic
35C2×C22⋊D127—21no—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C4×D127—21no—
40C2×C4×D127—21no—
41C2×C4×D127—21no—
42C2×C4×D127—21no—
43C2×C4×D127C2×C412yes—
44C2×C4×D127D412yes—
45#14523232—21no—
46#14523232—21no—
47C2×C127⋊C8—21no—
48C4⋊Dic127—22no—
49Dic127⋊C4—24no—
50(C2×D127)⋊C4—24no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—21no—
54C4×Dic127D412yescomplemented
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57M16×C127—21no—
58M16×C127—21no—
59C2×C8×C127—21no—
60C4⋊C4×C127C2×C412yes—
61C22⋊C4×C127—21no—
62C42×C127—21no—
63C42×C127—21no—
64C42×C127D412yescomplemented
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—42no—
69C22×D127C22⋊C412yes—
70Dic254—21no—
71Dic254—21no—
72D508—21no—
73D508—21no—
74D508—44no—
75C22⋊D127—41no—
76C22⋊D127—41no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—42no—
82C2×Dic127C22⋊C412yes—
83C4×D127—22no—
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—22no—
87C4×D127—41no—
88C4×D127—41no—
89C127⋊C8—41no—
90C23×C127C2×D411yescharacteristic
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—42no—
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—41no—
101C22×C127—22no—
102C22×C127C2×C22⋊C411yescharacteristic
103C22×C127#12011yescharacteristic
104C22×C127#12311yescharacteristic
105C2×D127—22no—
106C2×D127—22no—
107C2×D127—22no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—41no—
111C2×D127—42no—
112C4×C127—42no—
113C4×C127—42no—
114C4×C127C4×D411yescharacteristic
115C4×C127C4×D411yescharacteristic
116C4×C127#12111yescharacteristic
117C4×C127#12111yescharacteristic
118Dic127—22no—
119Dic127—42no—
120Dic127—41no—
121C2×C127—21no—
122C2×C127—21no—
123C2×C127#38611yescharacteristic
124C2×C127#42112yes—
125D127—42no—
126D127—81no—
127#1897—1271nomaximal, Sylow
128C127#189711yescharacteristic, complemented, Sylow
129#525—1271no—
130C2×#117—1271no—
131C2×#104—1271no—
132C2×#104—1271no—
133#407—1271no—
134#339—1271no—
135#339D12711yescharacteristic, complemented
136C2×#41—1271no—
137C23×C4—1271no—
138C2×M16—1271no—
139C2×M16C2×D12711yescharacteristic
140#123—2541no—
141C22⋊Q8—1272no—
142#121—1272no—
143#120—2541no—
144C4×D4—2542no—
145#117—2542no—
146#117—2542no—
147#117C2×D12711yescharacteristic
148C2×C4⋊C4—2541no—
149C2×C22⋊C4—2541no—
150C2×C42—1271no—
151C2×C42C2×D12711yescharacteristic
152#104—2542no—
153#104—2542no—
154C22⋊C8—2542no—
155C24—1271no—
156#41—2541no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160C2×Q8—1271no—
161C2×D4—1271no—
162C2×D4—2542no—
163C2×D4—2541no—
164C22×C4—21no—
165C22×C4—1271no—
166C22×C4—1272no—
167C22×C4—1272no—
168C22×C4—2541no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4—2541no—
174C22×C4C22×D12711yescharacteristic
175M16—21no—
176M16—21no—
177M16—2541no—
178M16—2541no—
179C2×C8—21no—
180C2×C8—2541no—
181C4⋊C4—2542no—
182C4⋊C4—2544no—
183C4⋊C4C4×D12712yes—
184C22⋊C4—21no—
185C22⋊C4—2542no—
186C22⋊C4—2542no—
187C22⋊C4—2544no—
188C42—21no—
189C42—21no—
190C42—1272no—
191C42—2541no—
192C42D50812yescomplemented
193C23—1272no—
194C23—5082no—
195C23—2541no—
196C23—2541no—
197C23—2541no—
198C23D4×D12711yescharacteristic
199Q8—2541no—
200Q8—2541no—
201D4—5084no—
202D4—5081no—
203D4—5081no—
204D4—2541no—
205D4—2541no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1272no—
210C2×C4—5082no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—2541no—
214C2×C4—2542no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4C2×C4×D12711yescharacteristic
222C2×C4D4×D12711yescharacteristic
223C2×C4D4×D12711yescharacteristic
224C2×C4C2×D50811yescharacteristic
225C2×C4#1452324811yescharacteristic
226C2×C4#1452324911yescharacteristic
227C8—41no—
228C8—5081no—
229C22—22no—
230C22—5082no—
231C22—5082no—
232C22—5082no—
233C22—5081no—
234C22—2542no—
235C22—2542no—
236C22—2542no—
237C22C22⋊C4×D12711yescharacteristic
238C22#1686117911yescharacteristic
239C22#1686119411yescharacteristic
240C4—42no—
241C4—42no—
242C4—5082no—
243C4—5081no—
244C4—2542no—
245C4#1686117111yescharacteristic
246C4C4×D50811yescharacteristic
247C4#1686118211yescharacteristic
248C4#1686118411yescharacteristic
249C2—21no—
250C2—21no—
251C2—10161no—
252C2—5082no—
253C2#2254975411yescharacteristic
254C2#2254989112yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550388C2
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1897
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
2. C2×#16861166C2semidirect
3. C2×#16861137C2semidirect
4. C2×#16861135C2semidirect
5. #22549839C2semidirect
6. #22549635C2semidirect
7. #339×C127C2semidirect
12. #16861189C4semidirect
13. #16861184C4semidirect
54. C4×Dic127D4semidirect
64. C42×C127D4semidirect
128. C127#1897semidirect
135. #339D127semidirect
192. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225503882
C2×#168611662
C2×#168611372
C2×#168611352
#225498392
#225496352
#339×C1272
#1897127