Finite Group Databasebeta

← #30640686

Subgroups — #30640686

Order 16256.

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

normal subgroups only

This group has 220 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, complemented
2#22550312C211yesmaximal, characteristic, complemented
3#22550143C211yesmaximal, characteristic, complemented
4#22549689C211yesmaximal, characteristic, complemented
5#22549684C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7C2×D16×C127C211yesmaximal, characteristic, complemented
8#16861254—21no—
9#16861235—21no—
10#16861235—21no—
11#16861233—22no—
12#16861231C2211yescharacteristic
13#16861228—21no—
14D8×D127—21no—
15#16861215C2211yescharacteristic
16#16861153—21no—
17#16861150—21no—
18#16861149C2211yescharacteristic, complemented
19#16861145C2211yescharacteristic, complemented
20#16861126—21no—
21C2×D8×C127C2211yescharacteristic
22C2×D8×C127C2211yescharacteristic, complemented
23D16×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#14523237—21no—
34#14523236—22no—
35#14523236—41no—
36D1016D411yescharacteristic
37#14523233—22no—
38#14523233—41no—
39#14523232—21no—
40#14523232D411yescharacteristic
41#14523231—21no—
42C8×D127—21no—
43C127⋊C16—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46D8×C127—21no—
47D8×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—
67C23×C127—41no—
68D4×C127—22no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—42no—
76C22×C127—81no—
77C22×C127—161no—
78C22×C127—161no—
79C22×C127AGL(1,ℤ/8)11yescharacteristic
80C2×D127—21no—
81C2×D127—41no—
82C2×D127—162no—
83C4×C127#12011yescharacteristic
84C4×C127C2×D811yescharacteristic
85Dic127—21no—
86Dic127—41no—
87C2×C127—21no—
88C2×C127—82no—
89C2×C127—161no—
90C2×C127#44711yescharacteristic
91D127—41no—
92D127—81no—
93#2192—1271nomaximal, Sylow
94C127#219211yescharacteristic, complemented, Sylow
95#576—1271no—
96#509—1271no—
97C2×D16D12711yescharacteristic, complemented
98#472—1271no—
99#361—1271no—
100#359—1271no—
101#350—1271no—
102H(5,𝔽2)—2541no—
103AGL(1,ℤ/8)—2541no—
104AGL(1,ℤ/8)—2541no—
105AGL(1,ℤ/8)—2542no—
106#135—1271no—
107#135—2541no—
108C2×D8—2541no—
109C2×D8C2×D12711yescharacteristic
110C2×D8C2×D12711yescharacteristic, complemented
111#131—1271no—
112SD32—2541no—
113D16—22no—
114D16—2541no—
115M32—1271no—
116C2×C16C2×D12711yescharacteristic, complemented
117#108—1271no—
118#100—2541no—
119#41—1271no—
120#41—5081no—
121#41—2542no—
122#41—2541no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—5081no—
126C2×D4—5082no—
127C2×D4—2542no—
128Q16—1271no—
129SD16—5081no—
130SD16—2542no—
131SD16—2541no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—1271no—
136D8—5081no—
137D8—2542no—
138D8C22⋊D12712yes—
139M16—1271no—
140M16—2541no—
141M16—2541no—
142C2×C8—2541no—
143C2×C8C22×D12711yescharacteristic
144C16—21no—
145C16—2541no—
146C23—41no—
147C23—41no—
148C23—5082no—
149Q8—1271no—
150Q8—2541no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—1271no—
156D4—5082no—
157D4—5081no—
158D4—5082no—
159D4—2541no—
160D4—2541no—
161C2×C4—5082no—
162C2×C4—5081no—
163C2×C4—2541no—
164C2×C4D4×D12711yescharacteristic
165C8—5081no—
166C8—2541no—
167C8D4×D12711yescharacteristic
168C8C2×C22⋊D12711yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—20322no—
174C22—5081no—
175C22—2541no—
176C22#1686123311yescharacteristic
177C4—5081no—
178C4—2541no—
179C4#1686117711yescharacteristic
180C4D8×D12711yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—10161no—
185C2—5081no—
186C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×D16×C127C2
socleC2×C127#447
hypercenterC2×D16D127
hypocenterC127#2192
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22550677C2semidirect
2. #22550312C2semidirect
3. #22550143C2semidirect
4. #22549689C2semidirect
5. #22549684C2semidirect
6. #22549659C2semidirect
7. C2×D16×C127C2semidirect
18. #16861149C22semidirect
19. #16861145C22semidirect
22. C2×D8×C127C22semidirect
24. C2×C16×C127C22semidirect
94. C127#2192semidirect
97. C2×D16D127semidirect
110. C2×D8C2×D127semidirect
116. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506772
#225503122
#225501432
#225496892
#225496842
#225496592
C2×D16×C1272
#2192127