Finite Group Databasebeta

← #30639145

Subgroups — #30639145

Order 16256.

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

normal subgroups only

This group has 342 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#22550387C211yesmaximal, characteristic
3#22549858C211yesmaximal, characteristic, complemented
4#22549840C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7#407×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861192C412yes—
15#16861188—21no—
16#16861183—21no—
17#16861178C412yes—
18C4×C22⋊D127—22no—
19#16861168C2211yescharacteristic
20#16861127—21no—
21#16861125—21no—
22C22⋊(C127⋊C8)—22no—
23C23×C4×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C22⋊C8×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127C2×C412yes—
35C2×C4×D127D412yes—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523231—24no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8D412yescomplemented
43C8×D127—24no—
44C4⋊Dic127—21no—
45Dic127⋊C4—22no—
46C22⋊Dic127—22no—
47C22⋊Dic127C2×C412yes—
48(C2×D127)⋊C4—22no—
49C4×Dic127—21no—
50C24×C127C2×C411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2×C411yescharacteristic
58C22×C4×C127C2311yescharacteristic
59M16×C127—21no—
60M16×C127—21no—
61M16×C127—21no—
62M16×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yescomplemented
65C22×D127C22⋊C412yes—
66Dic254—21no—
67Dic254—21no—
68D508—21no—
69D508—21no—
70C22⋊D127—22no—
71C22⋊D127—22no—
72C22⋊D127—44no—
73C2×Dic127—22no—
74C2×Dic127C22⋊C412yes—
75C4×D127—22no—
76C4×D127—22no—
77C127⋊C8—22no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—42no—
83C23×C127C22×C411yescharacteristic
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C8×C127—22no—
98C8×C127—41no—
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127C2×C22⋊C411yescharacteristic
108C22×C127C4×D411yescharacteristic
109C22×C127C4×D411yescharacteristic
110C2×D127—22no—
111C2×D127—42no—
112C4×C127—41no—
113C4×C127#12011yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117Dic127—22no—
118Dic127—42no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—42no—
123C2×C127#38611yescharacteristic
124D127—42no—
125#1891—1271nomaximal, Sylow
126C127#189111yescharacteristic, complemented, Sylow
127#568—1271no—
128#525—1271no—
129#413—1271no—
130#407—1271no—
131#407D12711yescharacteristic, complemented
132#344—1271no—
133#344—1271no—
134C2×#41—1271no—
135C23×C4C2×D12711yescharacteristic
136#131—2542no—
137#131—2542no—
138C2×M16—1271no—
139C2×M16—1271no—
140C2×M16—2542no—
141C2×M16C2×D12711yescharacteristic
142C2×M16C2×D12711yescharacteristic
143#123—1272no—
144C22⋊Q8—2541no—
145#121—2541no—
146#120—1272no—
147C4×D4—2542no—
148#117—1271no—
149#101—2541no—
150#100—2541no—
151C22⋊C8—22no—
152C22⋊C8—2542no—
153C24C4×D12711yescharacteristic
154#41—2541no—
155#41—2541no—
156#41—2541no—
157#41—2541no—
158C2×Q8—1271no—
159C2×D4—1272no—
160C2×D4—1271no—
161C2×D4—2542no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1272no—
169C22×C4C4×D12711yescharacteristic
170C22×C4C22×D12711yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—2544no—
176M16—2541no—
177M16—2541no—
178M16—2541no—
179M16—2541no—
180C2×C8—21no—
181C2×C8—1272no—
182C2×C8—2544no—
183C2×C8—2541no—
184C2×C8C22⋊D12712yescomplemented
185C4⋊C4—2542no—
186C4⋊C4—2541no—
187C22⋊C4—1272no—
188C22⋊C4—2542no—
189C22⋊C4—2542no—
190C42—2541no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—1272no—
196C23C2×C4×D12711yescharacteristic
197Q8—2541no—
198Q8—2541no—
199D4—5084no—
200D4—2542no—
201D4—2542no—
202D4—2541no—
203D4—2541no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1272no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4D4×D12711yescharacteristic
216C2×C4D4×D12711yescharacteristic
217C2×C4D4×D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4#1452324711yescharacteristic
220C2×C4#1452324811yescharacteristic
221C8—22no—
222C8—41no—
223C8—5081no—
224C8—2542no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—5082no—
234C22—2542no—
235C22C22⋊C4×D12711yescharacteristic
236C22(C22⋊D127)⋊C411yescharacteristic
237C22C4×C22⋊D12711yescharacteristic
238C4—41no—
239C4—5082no—
240C4—2542no—
241C4#1686117711yescharacteristic
242C4#1686118011yescharacteristic
243C4#1686118211yescharacteristic
244C4#1686119111yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—5082no—
250C2#2254975211yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC2×C127#386
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1891
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549858C2semidirect
4. #22549840C2semidirect
5. #22549648C2semidirect
6. #22549647C2semidirect
7. #407×C127C2semidirect
42. C2×C127⋊C8D4semidirect
64. C2×C8×C127D4semidirect
126. C127#1891semidirect
131. #407D127semidirect
184. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225503872
#225498582
#225498402
#225496482
#225496472
#407×C1272
#1891127