Finite Group Databasebeta

← #30640679

Subgroups — #30640679

Order 16256.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#22550673C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic, complemented
3#22550131C211yesmaximal, characteristic, complemented
4#22549687C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic, complemented
6#22549657C211yesmaximal, characteristic, complemented
7#510×C127C211yesmaximal, characteristic, complemented
8#41×D127—21no—
9#16861241—21no—
10#16861238—21no—
11#16861236—21no—
12#16861233—21no—
13#16861231C2211yescharacteristic
14#16861229—21no—
15C2×#14523231C2211yescharacteristic
16#16861185—21no—
17#16861155—21no—
18#16861152—21no—
19#16861149C2211yescharacteristic, complemented
20#16861141C2211yescharacteristic, complemented
21#16861133—21no—
22#135×C127C2211yescharacteristic
23C2×Q16×C127C2211yescharacteristic, complemented
24Q32×C127—21no—
25SD32×C127—21no—
26M32×C127C2211yescharacteristic, complemented
27#14523249—21no—
28#14523248—21no—
29#14523248—41no—
30#14523247—21no—
31#14523247—41no—
32Q8×D127—21no—
33D4×D127—21no—
34D4×D127—41no—
35C2×C4×D127—41no—
36C2×C4×D127D411yescharacteristic
37Dic508D411yescharacteristic
38#14523238—21no—
39#14523237—21no—
40#14523236—21no—
41#14523235—21no—
42D1016D411yescharacteristic
43#14523233—21no—
44#14523231—21no—
45#14523231—21no—
46C2×C127⋊C8D411yescharacteristic
47C4⋊Dic127—21no—
48Dic127⋊C4—41no—
49(C2×D127)⋊C4—41no—
50C127⋊C16—21no—
51#41×C127—21no—
52C2×Q8×C127—21no—
53Q16×C127—21no—
54Q16×C127—21no—
55Q16×C127D411yescharacteristic
56SD16×C127—21no—
57D8×C127D411yescharacteristic
58C2×C8×C127C2311yescharacteristic
59C16×C127—21no—
60C22×D127—41no—
61C22×D127D811yescharacteristic
62Dic254—21no—
63Dic254—41no—
64D508—21no—
65D508—41no—
66C22⋊D127—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127D811yescharacteristic
72C4×D127—21no—
73C4×D127—21no—
74C4×D127—41no—
75C4×D127—41no—
76C4×D127—41no—
77C127⋊C8—21no—
78Q8×C127—21no—
79Q8×C127—41no—
80Q8×C127—41no—
81D4×C127—21no—
82D4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C2×D411yescharacteristic
86C8×C127C2×D411yescharacteristic
87C8×C127C2×D411yescharacteristic
88C22×C127—41no—
89C22×C127C2×D811yescharacteristic
90C2×D127—21no—
91C2×D127—41no—
92C2×D127—41no—
93C2×D127—82no—
94C4×C127—41no—
95C4×C127—81no—
96C4×C127#12011yescharacteristic
97C4×C127AGL(1,ℤ/8)11yescharacteristic
98Dic127—21no—
99Dic127—41no—
100Dic127—81no—
101C2×C127—21no—
102C2×C127—81no—
103C2×C127#44711yescharacteristic
104D127—41no—
105D127—81no—
106#2191—1271nomaximal, Sylow
107C127#219111yescharacteristic, complemented, Sylow
108#575—1271no—
109#510—1271no—
110#510D12711yescharacteristic, complemented
111#470—1271no—
112#360—1271no—
113#360—1271no—
114#349—1271no—
115C2×#41—2541no—
116#137—2541no—
117#137—2541no—
118AGL(1,ℤ/8)—2541no—
119AGL(1,ℤ/8)—2541no—
120#135—1271no—
121#135—2541no—
122#135C2×D12711yescharacteristic
123C2×Q16C2×D12711yescharacteristic, complemented
124C2×M16—1271no—
125C22⋊Q8—2541no—
126Q32—21no—
127Q32—2541no—
128SD32—21no—
129SD32—2541no—
130M32—1271no—
131M32C2×D12711yescharacteristic, complemented
132#106—1271no—
133#103—2541no—
134#41—21no—
135#41—5081no—
136#41—5081no—
137#41—2541no—
138#41—2541no—
139#41—2541no—
140C2×Q8—21no—
141C2×Q8—2541no—
142C2×D4—5081no—
143C2×D4—2541no—
144C22×C4—1271no—
145C22×C4—5081no—
146Q16—21no—
147Q16—21no—
148Q16—1271no—
149Q16—2541no—
150Q16C22⋊D12711yescharacteristic
151SD16—21no—
152SD16—2541no—
153SD16—2541no—
154SD16—2541no—
155D8—1271no—
156D8—2541no—
157D8C22⋊D12711yescharacteristic
158M16—2541no—
159M16—2541no—
160C2×C8—1271no—
161C2×C8C22×D12711yescharacteristic
162C4⋊C4—5081no—
163C4⋊C4—2541no—
164C22⋊C4—5081no—
165C16—21no—
166C16—2541no—
167C23—1271no—
168C23—5081no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—5081no—
173Q8—2541no—
174D4—21no—
175D4—41no—
176D4—5081no—
177D4—5081no—
178D4—5081no—
179D4—5081no—
180D4—2541no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—1271no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—5081no—
187C2×C4—5081no—
188C2×C4—5081no—
189C2×C4—2541no—
190C2×C4—2541no—
191C2×C4D4×D12711yescharacteristic
192C8—2541no—
193C8D4×D12711yescharacteristic
194C8C2×C22⋊D12711yescharacteristic
195C22—41no—
196C22—10162no—
197C22—5081no—
198C22—5081no—
199C22—2541no—
200C22D8×D12711yescharacteristic
201C4—41no—
202C4—81no—
203C4—10161no—
204C4—5081no—
205C4—2541no—
206C4#1686117711yescharacteristic
207C4#1686123311yescharacteristic
208C2—21no—
209C2—81no—
210C2—10161no—
211C2—5081no—
212C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#510×C127C2
socleC2×C127#447
prime radical#22550673C2
hypercenter#510D127
hypocenterC127#2191
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. #22550131C2semidirect
4. #22549687C2semidirect
5. #22549686C2semidirect
6. #22549657C2semidirect
7. #510×C127C2semidirect
19. #16861149C22semidirect
20. #16861141C22semidirect
23. C2×Q16×C127C22semidirect
26. M32×C127C22semidirect
107. C127#2191semidirect
110. #510D127semidirect
123. C2×Q16C2×D127semidirect
131. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506732
#225503172
#225501312
#225496872
#225496862
#225496572
#510×C1272
#2191127