Finite Group Databasebeta

← #30640685

Subgroups — #30640685

Order 16256.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550681C211yesmaximal, characteristic, complemented
2#22550312C211yesmaximal, characteristic, complemented
3C2×#16861150C211yesmaximal, characteristic, complemented
4#22549688C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#472×C127C211yesmaximal, characteristic, complemented
8#16861255—21no—
9#16861237—22no—
10#16861237—21no—
11#16861231—21no—
12#16861231C2211yescharacteristic
13C2×D1016—21no—
14C2×D1016C2211yescharacteristic
15#16861153—21no—
16#16861150—22no—
17#16861150—21no—
18#16861149C2211yescharacteristic, complemented
19C2×C127⋊C16C2211yescharacteristic, complemented
20AGL(1,ℤ/8)×C127—21no—
21C2×D8×C127C2211yescharacteristic, complemented
22#131×C127C2211yescharacteristic
23#108×C127C2211yescharacteristic, complemented
24#100×C127—21no—
25#14523249—41no—
26#14523248—22no—
27#14523248—21no—
28C2×D508—22no—
29C2×D508—21no—
30D4×D127—42no—
31D4×D127—41no—
32Dic508D411yescharacteristic
33#14523237—22no—
34#14523237—21no—
35D1016—22no—
36D1016—21no—
37D1016D411yescharacteristic
38D1016D412yes—
39C127⋊C16—21no—
40C127⋊C16—21no—
41#41×C127D411yescharacteristic
42C2×D4×C127—21no—
43SD16×C127—41no—
44D8×C127—21no—
45D8×C127—21no—
46D8×C127—41no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52C22×D127—42no—
53C22×D127—41no—
54Dic254—21no—
55D508—21no—
56D508—22no—
57D508—42no—
58D508—41no—
59C22⋊D127—42no—
60C22⋊D127—41no—
61C4×D127—42no—
62C4×D127—41no—
63C23×C127—41no—
64Q8×C127D811yescharacteristic
65D4×C127—21no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127D811yescharacteristic
69C2×C4×C127—21no—
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—21no—
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—81no—
77C22×C127—161no—
78C22×C127AGL(1,ℤ/8)11yescharacteristic
79C2×D127—42no—
80C2×D127—41no—
81C2×D127—161no—
82C2×D127—162no—
83C4×C127—21no—
84C4×C127#12011yescharacteristic
85C4×C127C2×D811yescharacteristic
86Dic127—41no—
87C2×C127—21no—
88C2×C127—41no—
89C2×C127—161no—
90C2×C127#44711yescharacteristic
91D127—82no—
92D127—81no—
93#2192—1271nomaximal, Sylow
94C127#219211yescharacteristic, complemented, Sylow
95#576—1271no—
96#509—1271no—
97C2×D16—1271no—
98#472D12711yescharacteristic, complemented
99#361—1271no—
100#359—1271no—
101#350—1271no—
102H(5,𝔽2)—2541no—
103AGL(1,ℤ/8)—21no—
104AGL(1,ℤ/8)—2542no—
105AGL(1,ℤ/8)—2541no—
106#135—1271no—
107#135—2541no—
108C2×D8—1271no—
109C2×D8—2541no—
110C2×D8C2×D12711yescharacteristic, complemented
111#131C2×D12711yescharacteristic
112SD32—2541no—
113D16—2542no—
114D16—2541no—
115M32—1271no—
116C2×C16—1271no—
117#108C2×D12711yescharacteristic, complemented
118#100—21no—
119#41—5081no—
120#41—2542no—
121#41—2541no—
122#41C22⋊D12711yescharacteristic
123C2×D4—21no—
124C2×D4—5082no—
125C2×D4—5081no—
126C2×D4—2542no—
127C2×D4—2541no—
128Q16—1271no—
129SD16—41no—
130SD16—2542no—
131SD16—2541no—
132D8—21no—
133D8—21no—
134D8—41no—
135D8—1272no—
136D8—1271no—
137D8—2542no—
138D8—2541no—
139M16—21no—
140M16—21no—
141M16C22⋊D12711yescharacteristic
142C2×C8—21no—
143C2×C8C22×D12711yescharacteristic
144C16—2541no—
145C16—2541no—
146C23—41no—
147C23—5082no—
148C23—5081no—
149Q8—2541no—
150Q8#1452323311yescharacteristic
151D4—21no—
152D4—41no—
153D4—41no—
154D4—5082no—
155D4—5082no—
156D4—5081no—
157D4—5081no—
158D4—2542no—
159D4—2541no—
160D4#1452323311yescharacteristic
161C2×C4—21no—
162C2×C4—5082no—
163C2×C4—5081no—
164C2×C4C2×C22⋊D12711yescharacteristic
165C8—21no—
166C8—41no—
167C8D4×D12711yescharacteristic
168C8D4×D12711yescharacteristic
169C22—21no—
170C22—81no—
171C22—161no—
172C22—20321no—
173C22—20322no—
174C22—5082no—
175C22—5081no—
176C22#1686123411yescharacteristic
177C4—21no—
178C4—5081no—
179C4#1686117711yescharacteristic
180C4C2×#1452323311yescharacteristic
181C2—21no—
182C2—41no—
183C2—161no—
184C2—10162no—
185C2—10161no—
186C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#472×C127C2
socleC2×C127#447
hypercenter#472D127
hypocenterC127#2192
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22550681C2semidirect
2. #22550312C2semidirect
3. C2×#16861150C2semidirect
4. #22549688C2semidirect
5. #22549682C2semidirect
6. #22549660C2semidirect
7. #472×C127C2semidirect
18. #16861149C22semidirect
19. C2×C127⋊C16C22semidirect
21. C2×D8×C127C22semidirect
23. #108×C127C22semidirect
94. C127#2192semidirect
98. #472D127semidirect
110. C2×D8C2×D127semidirect
117. #108C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506812
#225503122
C2×#168611502
#225496882
#225496822
#225496602
#472×C1272
#2192127