Finite Group Databasebeta

← #30640700

Subgroups — #30640700

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#22550677C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic, complemented
3#22550148C211yesmaximal, characteristic, complemented
4#22549689C211yesmaximal, characteristic, complemented
5#22549684C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7C2×SD32×C127C211yesmaximal, characteristic, complemented
8#16861254—21no—
9#16861240—21no—
10#16861235—21no—
11#16861233—22no—
12#16861231C2211yescharacteristic
13#16861228—21no—
14D8×D127—21no—
15#16861215C2211yescharacteristic
16#16861155—21no—
17#16861152—21no—
18#16861149C2211yescharacteristic, complemented
19#16861145C2211yescharacteristic, complemented
20#16861128—21no—
21C2×Q16×C127C2211yescharacteristic, complemented
22C2×D8×C127C2211yescharacteristic
23SD32×C127—22no—
24C2×C16×C127C2211yescharacteristic, complemented
25#14523248—21no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—41no—
29C2×C22⋊D127—42no—
30D4×D127—22no—
31D4×D127—41no—
32Dic508D411yescharacteristic
33#14523238—41no—
34#14523237—21no—
35#14523236—22no—
36#14523235—41no—
37D1016D411yescharacteristic
38#14523233—22no—
39#14523232—21no—
40#14523232D411yescharacteristic
41#14523231—21no—
42C8×D127—21no—
43C127⋊C16—21no—
44C2×Q8×C127—21no—
45C2×D4×C127—21no—
46Q16×C127—21no—
47Q16×C127—21no—
48D8×C127—21no—
49D8×C127D412yes—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—42no—
53Dic254—21no—
54Dic254D811yescharacteristic
55D508—21no—
56D508D811yescharacteristic
57C22⋊D127—21no—
58C22⋊D127—42no—
59C22⋊D127—42no—
60C22⋊D127—41no—
61C2×Dic127—42no—
62C4×D127—21no—
63C4×D127—41no—
64C127⋊C8—21no—
65C127⋊C8—41no—
66C23×C127—41no—
67Q8×C127—41no—
68Q8×C127—41no—
69D4×C127—22no—
70D4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—42no—
76C22×C127—161no—
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—21no—
79C2×D127—41no—
80C2×D127—162no—
81C4×C127—81no—
82C4×C127#12011yescharacteristic
83C4×C127C2×D811yescharacteristic
84Dic127—21no—
85Dic127—41no—
86C2×C127—21no—
87C2×C127—82no—
88C2×C127#44711yescharacteristic
89D127—41no—
90D127—81no—
91#2194—1271nomaximal, Sylow
92C127#219411yescharacteristic, complemented, Sylow
93#576—1271no—
94#510—1271no—
95C2×SD32D12711yescharacteristic, complemented
96#473—1271no—
97#361—1271no—
98#359—1271no—
99#350—1271no—
100H(5,𝔽2)—2541no—
101#137—2541no—
102AGL(1,ℤ/8)—2541no—
103AGL(1,ℤ/8)—2542no—
104#135—1271no—
105#135—2541no—
106C2×Q16C2×D12711yescharacteristic, complemented
107C2×D8—2541no—
108C2×D8C2×D12711yescharacteristic
109#131—1271no—
110Q32—2541no—
111SD32—22no—
112SD32—2541no—
113M32—1271no—
114C2×C16C2×D12711yescharacteristic, complemented
115#108—1271no—
116#101—2541no—
117#41—1271no—
118#41—5081no—
119#41—2542no—
120#41—2541no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—5081no—
124C2×D4—5082no—
125C2×D4—2542no—
126Q16—21no—
127Q16—21no—
128Q16—1271no—
129Q16—5081no—
130SD16—5081no—
131SD16—2542no—
132SD16—2541no—
133D8—21no—
134D8—1271no—
135D8—2542no—
136D8C22⋊D12712yes—
137M16—1271no—
138M16—2541no—
139M16—2541no—
140C2×C8—2541no—
141C2×C8C22×D12711yescharacteristic
142C16—21no—
143C16—2541no—
144C23—41no—
145C23—5082no—
146Q8—41no—
147Q8—41no—
148Q8—1271no—
149Q8—2541no—
150D4—22no—
151D4—41no—
152D4—1271no—
153D4—5082no—
154D4—5081no—
155D4—5082no—
156D4—2541no—
157D4—2541no—
158C2×C4—41no—
159C2×C4—5082no—
160C2×C4—5081no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C8—5081no—
164C8—2541no—
165C8D4×D12711yescharacteristic
166C8C2×C22⋊D12711yescharacteristic
167C22—42no—
168C22—161no—
169C22—20322no—
170C22—5081no—
171C22—2541no—
172C22#1686123311yescharacteristic
173C4—81no—
174C4—5081no—
175C4—2541no—
176C4#1686117711yescharacteristic
177C4D8×D12711yescharacteristic
178C2—21no—
179C2—82no—
180C2—10161no—
181C2—5081no—
182C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×SD32×C127C2
socleC2×C127#447
prime radical#22550677C2
hypercenterC2×SD32D127
hypocenterC127#2194
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. #22550148C2semidirect
4. #22549689C2semidirect
5. #22549684C2semidirect
6. #22549659C2semidirect
7. C2×SD32×C127C2semidirect
18. #16861149C22semidirect
19. #16861145C22semidirect
21. C2×Q16×C127C22semidirect
24. C2×C16×C127C22semidirect
92. C127#2194semidirect
95. C2×SD32D127semidirect
106. C2×Q16C2×D127semidirect
114. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506772
#225503172
#225501482
#225496892
#225496842
#225496592
C2×SD32×C1272
#2194127