Finite Group Databasebeta

← #30640356

Subgroups — #30640356

Order 16256.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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#131×D127C211yesmaximal, characteristic
2C2×#16861148C211yesmaximal, characteristic, complemented
3M32×D127C211yesmaximal, characteristic, complemented
4#22549661C211yesmaximal, characteristic, complemented
5#22549660C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7#350×C127C211yesmaximal, characteristic, complemented
8#41×D127C411yescharacteristic
9#16861217—21no—
10#16861217C411yescharacteristic
11#16861216C411yescharacteristic
12#16861215—21no—
13#16861215C2211yescharacteristic
14M16×D127—21no—
15M16×D127C411yescharacteristic
16C2×C8×D127—21no—
17C2×C8×D127C2211yescharacteristic
18#16861149C2211yescharacteristic, complemented
19#16861148—21no—
20#16861148—21no—
21#16861148—21no—
22C2×C127⋊C16C2211yescharacteristic, complemented
23C16×D127—21no—
24#131×C127C2211yescharacteristic
25M32×C127C2211yescharacteristic, complemented
26C2×C16×C127C2211yescharacteristic, complemented
27#14523249C811yescharacteristic
28#14523248—21no—
29#14523248C2×C411yescharacteristic
30#14523247—21no—
31#14523247C811yescharacteristic
32Q8×D127C811yescharacteristic
33D4×D127—21no—
34D4×D127C811yescharacteristic
35C2×C4×D127—21no—
36C2×C4×D127C2×C411yescharacteristic
37#14523232—21no—
38#14523232C2×C411yescharacteristic
39#14523231—21no—
40#14523231—21no—
41#14523231—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8C2×C411yescharacteristic
44C8×D127—21no—
45C8×D127—21no—
46C8×D127—21no—
47C8×D127D411yescharacteristic
48C8×D127D411yescharacteristic
49C8×D127D411yescharacteristic
50C8×D127D411yescharacteristic
51C127⋊C16—21no—
52C127⋊C16—21no—
53#41×C127C2×C411yescharacteristic
54M16×C127—21no—
55M16×C127C2×C411yescharacteristic
56C2×C8×C127—21no—
57C2×C8×C127C2311yescharacteristic
58C16×C127—21no—
59C16×C127—21no—
60C22×D127—21no—
61C22×D127M1611yescharacteristic
62Dic254—21no—
63Dic254C2×C811yescharacteristic
64D508—21no—
65D508C2×C811yescharacteristic
66C22⋊D127—21no—
67C22⋊D127—21no—
68C22⋊D127—21no—
69C2×Dic127—21no—
70C2×Dic127M1611yescharacteristic
71C4×D127—21no—
72C4×D127—21no—
73C4×D127—21no—
74C4×D127C22⋊C411yescharacteristic
75C4×D127C22⋊C411yescharacteristic
76C4×D127C2×C811yescharacteristic
77C4×D127C2×C811yescharacteristic
78C127⋊C8—21no—
79C127⋊C8C22⋊C411yescharacteristic
80C127⋊C8C22⋊C411yescharacteristic
81Q8×C127C2×C811yescharacteristic
82D4×C127—21no—
83D4×C127C2×C811yescharacteristic
84C2×C4×C127—21no—
85C2×C4×C127C22×C411yescharacteristic
86C8×C127—21no—
87C8×C127C2×D411yescharacteristic
88C8×C127C2×D411yescharacteristic
89C22×C127—21no—
90C22×C127C2×M1611yescharacteristic
91C2×D127—21no—
92C2×D127—41no—
93C2×D127—42no—
94C2×D127C22⋊C811yescharacteristic
95C2×D127C22⋊C811yescharacteristic
96C4×C127—21no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C22×C811yescharacteristic
99Dic127—21no—
100Dic127C22⋊C811yescharacteristic
101Dic127C22⋊C811yescharacteristic
102C2×C127—21no—
103C2×C127—41no—
104C2×C127C2×C22⋊C811yescharacteristic
105D127—21no—
106D127—21no—
107D127—41no—
108#2117—1271nomaximal, Sylow
109C127#211711yescharacteristic, complemented, Sylow
110C2×#131—1271no—
111C2×M32—1271no—
112C2×M32—1271no—
113#350—1271no—
114#350—1271no—
115#350—1271no—
116#350D12711yescharacteristic, complemented
117C2×#41—1271no—
118#131—1271no—
119#131—1271no—
120#131—1271no—
121#131—2541no—
122#131—2541no—
123#131C2×D12711yescharacteristic
124C2×M16—1271no—
125C2×M16—2541no—
126C22×C8—1271no—
127C22×C8—2541no—
128M32—1271no—
129M32—2541no—
130M32—2541no—
131M32—2541no—
132M32C2×D12711yescharacteristic, complemented
133C2×C16—1271no—
134C2×C16—2541no—
135C2×C16C2×D12711yescharacteristic, complemented
136#41—1271no—
137#41—1271no—
138#41—1271no—
139#41—2541no—
140#41—2541no—
141#41C4×D12711yescharacteristic
142C2×Q8—1271no—
143C2×D4—1271no—
144C2×D4—2541no—
145C22×C4—1271no—
146C22×C4—2541no—
147M16—21no—
148M16—1271no—
149M16—2541no—
150M16—2541no—
151M16—2541no—
152M16—2541no—
153M16C4×D12711yescharacteristic
154C2×C8—21no—
155C2×C8—1271no—
156C2×C8—1271no—
157C2×C8—1271no—
158C2×C8—1271no—
159C2×C8—1271no—
160C2×C8—2541no—
161C2×C8—2541no—
162C2×C8—2541no—
163C2×C8—2541no—
164C2×C8C22×D12711yescharacteristic
165C16—21no—
166C16—21no—
167C16—2541no—
168C16—2541no—
169C23—1271no—
170C23—2541no—
171Q8—1271no—
172Q8—2541no—
173Q8C8×D12711yescharacteristic
174D4—21no—
175D4—1271no—
176D4—2541no—
177D4—2541no—
178D4—2541no—
179D4—2541no—
180D4C8×D12711yescharacteristic
181C2×C4—21no—
182C2×C4—1271no—
183C2×C4—1271no—
184C2×C4—1271no—
185C2×C4—1271no—
186C2×C4—1271no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2541no—
191C2×C4C2×C4×D12711yescharacteristic
192C8—21no—
193C8—1271no—
194C8—1271no—
195C8—2541no—
196C8D4×D12711yescharacteristic
197C8D4×D12711yescharacteristic
198C22—21no—
199C22—1271no—
200C22—1271no—
201C22—5081no—
202C22—5082no—
203C22—2541no—
204C22M16×D12711yescharacteristic
205C4—21no—
206C4—1271no—
207C4—1271no—
208C4—2541no—
209C4C22⋊C4×D12711yescharacteristic
210C4C2×C8×D12711yescharacteristic
211C2—21no—
212C2—41no—
213C2—5081no—
214C2—2541no—
215C2—2541no—
216C2C22⋊C8×D12711yescharacteristic
subgroupG/H
centerC8D4×D127
commutatorC4×C127C22×C8
FrattiniC2×C8C22×D127
Fitting#350×C127C2
socleC2×C127C2×C22⋊C8
prime radical#41×D127C4
cosocleC2×C8×C127C23
hypercenter#350D127
hypocenterC127#2117
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C8 ◁ C2×C8 ◁ #350

Complements

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

NH ≅ G/Ntype
2. C2×#16861148C2semidirect
3. M32×D127C2semidirect
4. #22549661C2semidirect
5. #22549660C2semidirect
6. #22549659C2semidirect
7. #350×C127C2semidirect
18. #16861149C22semidirect
22. C2×C127⋊C16C22semidirect
25. M32×C127C22semidirect
26. C2×C16×C127C22semidirect
109. C127#2117semidirect
116. #350D127semidirect
132. M32C2×D127semidirect
135. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#131×D1272
C2×#168611482
M32×D1272
#225496612
#225496602
#225496592
#350×C1272
#2117127