Finite Group Databasebeta

← #27143690

Subgroups — #27143690

Order 12416.

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#20261537C211yesmaximal, characteristic, complemented
2C2×#15690751C211yesmaximal, characteristic, complemented
3#20261002C211yesmaximal, characteristic, complemented
4#20260549C211yesmaximal, characteristic, complemented
5#20260542C211yesmaximal, characteristic, complemented
6#20260520C211yesmaximal, characteristic, complemented
7#509×C97C211yesmaximal, characteristic, complemented
8#15690856—21no—
9#15690837—22no—
10#15690836—21no—
11#15690835—21no—
12#15690833—21no—
13C2×D776C2211yescharacteristic
14D8×D97—21no—
15#15690816C2211yescharacteristic
16#15690751—22no—
17C2×#13095486C2211yescharacteristic, complemented
18#15690746C2211yescharacteristic, complemented
19#15690727—21no—
20#135×C97C2211yescharacteristic
21C2×D8×C97C2211yescharacteristic, complemented
22SD32×C97—21no—
23D16×C97—21no—
24M32×C97C2211yescharacteristic, complemented
25#13095512—22no—
26#13095511—41no—
27#13095511D411yescharacteristic
28C2×D388—21no—
29C2×D388—41no—
30D4×D97—22no—
31D4×D97—42no—
32#13095499—41no—
33#13095498—22no—
34D776—21no—
35D776D412yes—
36#13095496—22no—
37#13095496—41no—
38#13095495—21no—
39#13095495D411yescharacteristic
40#13095494—21no—
41C8×D97—21no—
42#13095486—21no—
43#41×C97—21no—
44C2×D4×C97—21no—
45Q16×C97D411yescharacteristic
46SD16×C97—21no—
47D8×C97—21no—
48D8×C97—21no—
49D8×C97D411yescharacteristic
50C2×C8×C97C2311yescharacteristic
51C16×C97—21no—
52C23×C97—41no—
53C22×D97—42no—
54C22×D97—41no—
55Q8×C97—21no—
56D4×C97—21no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97C2×D411yescharacteristic
62C22⋊D97—21no—
63C22⋊D97—42no—
64D388—22no—
65D388—42no—
66D388—41no—
67D388D811yescharacteristic
68C4×D97—21no—
69C4×D97—42no—
70Dic194D811yescharacteristic
71C8×C97C2×D411yescharacteristic
72C8×C97C2×D411yescharacteristic
73#11715895—21no—
74#11715895—41no—
75C22×C97—41no—
76C22×C97—81no—
77C22×C97—161no—
78C22×C97AGL(1,ℤ/8)11yescharacteristic
79C2×D97—21no—
80C2×D97—42no—
81C2×D97—162no—
82C2×D97—161no—
83C4×C97—41no—
84C4×C97#12011yescharacteristic
85C4×C97C2×D811yescharacteristic
86Dic97—21no—
87C2×C97—21no—
88C2×C97—81no—
89C2×C97—161no—
90C2×C97#44711yescharacteristic
91D97—41no—
92D97—82no—
93#2192—971nomaximal, Sylow
94C97#219211yescharacteristic, complemented, Sylow
95#576—971no—
96#509D9711yescharacteristic, complemented
97C2×D16—971no—
98#472—971no—
99#361—971no—
100#359—971no—
101#350—971no—
102H(5,𝔽2)—1941no—
103AGL(1,ℤ/8)—1941no—
104AGL(1,ℤ/8)—1942no—
105AGL(1,ℤ/8)—1941no—
106#135—1941no—
107#135C2×D9711yescharacteristic
108C2×D8—971no—
109C2×D8—1941no—
110C2×D8C2×D9711yescharacteristic, complemented
111#131—971no—
112SD32—21no—
113D16—21no—
114D16—1942no—
115M32C2×D9711yescharacteristic, complemented
116C2×C16—971no—
117#108—971no—
118#100—1941no—
119#41—21no—
120#41—971no—
121#41—3881no—
122#41—1942no—
123C2×D4—21no—
124C2×D4—3881no—
125C2×D4—3882no—
126C2×D4—1942no—
127C2×D4—1941no—
128Q16C22⋊D9711yescharacteristic
129SD16—21no—
130SD16—3881no—
131SD16—1942no—
132D8—21no—
133D8—21no—
134D8—972no—
135D8—3881no—
136D8—1942no—
137D8—1941no—
138D8C22⋊D9711yescharacteristic
139M16—971no—
140M16—1941no—
141M16—1941no—
142C2×C8—1941no—
143C2×C8C22×D9711yescharacteristic
144C16—21no—
145C16—1941no—
146C23—41no—
147C23—3882no—
148C23—3881no—
149Q8—21no—
150Q8—971no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—971no—
156D4—3882no—
157D4—3882no—
158D4—3881no—
159D4—1941no—
160D4—1942no—
161C2×C4—41no—
162C2×C4—3882no—
163C2×C4—1941no—
164C2×C4D4×D9711yescharacteristic
165C8—3881no—
166C8—1941no—
167C8D4×D9711yescharacteristic
168C8C2×C22⋊D9711yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—15521no—
173C22—15522no—
174C22—3882no—
175C22—1941no—
176C22#1569083411yescharacteristic
177C4—41no—
178C4—1941no—
179C4#1569077811yescharacteristic
180C4D8×D9711yescharacteristic
181C2—21no—
182C2—81no—
183C2—7762no—
184C2—161no—
185C2—3881no—
186C2#2026086711yescharacteristic
subgroupG/H
centerC2#20260867
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#509×C97C2
socleC2×C97#447
hypercenter#509D97
hypocenterC97#2192
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20261537C2semidirect
2. C2×#15690751C2semidirect
3. #20261002C2semidirect
4. #20260549C2semidirect
5. #20260542C2semidirect
6. #20260520C2semidirect
7. #509×C97C2semidirect
17. C2×#13095486C22semidirect
18. #15690746C22semidirect
21. C2×D8×C97C22semidirect
24. M32×C97C22semidirect
94. C97#2192semidirect
96. #509D97semidirect
110. C2×D8C2×D97semidirect
115. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615372
C2×#156907512
#202610022
#202605492
#202605422
#202605202
#509×C972
#219297