Finite Group Databasebeta

← #38770804

Subgroups — #38770804

Order 25472.

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#27164820C211yesmaximal, characteristic, complemented
2C2×#20263896C211yesmaximal, characteristic, complemented
3#27164285C211yesmaximal, characteristic, complemented
4#27163832C211yesmaximal, characteristic, complemented
5#27163825C211yesmaximal, characteristic, complemented
6#27163803C211yesmaximal, characteristic, complemented
7#509×C199C211yesmaximal, characteristic, complemented
8#20264001—21no—
9#20263982—22no—
10#20263981—21no—
11#20263980—21no—
12#20263978—21no—
13C2×D1592C2211yescharacteristic
14D8×D199—21no—
15#20263961C2211yescharacteristic
16#20263896—22no—
17C2×C199⋊C16C2211yescharacteristic, complemented
18#20263891C2211yescharacteristic, complemented
19#20263872—21no—
20#135×C199C2211yescharacteristic
21C2×D8×C199C2211yescharacteristic, complemented
22SD32×C199—21no—
23D16×C199—21no—
24M32×C199C2211yescharacteristic, complemented
25#15691887—22no—
26#15691886—41no—
27#15691886D411yescharacteristic
28C2×D796—21no—
29C2×D796—41no—
30D4×D199—22no—
31D4×D199—42no—
32#15691874—41no—
33#15691873—22no—
34D1592—21no—
35D1592D412yes—
36#15691871—22no—
37#15691871—41no—
38#15691870—21no—
39#15691870D411yescharacteristic
40#15691869—21no—
41C8×D199—21no—
42C199⋊C16—21no—
43#41×C199—21no—
44C2×D4×C199—21no—
45Q16×C199D411yescharacteristic
46SD16×C199—21no—
47D8×C199—21no—
48D8×C199—21no—
49D8×C199D411yescharacteristic
50C2×C8×C199C2311yescharacteristic
51C16×C199—21no—
52C22×D199—42no—
53C22×D199—41no—
54Dic398D811yescharacteristic
55D796—22no—
56D796—42no—
57D796—41no—
58D796D811yescharacteristic
59C22⋊D199—21no—
60C22⋊D199—42no—
61C4×D199—21no—
62C4×D199—42no—
63C199⋊C8—21no—
64C199⋊C8—41no—
65C23×C199—41no—
66Q8×C199—21no—
67D4×C199—21no—
68D4×C199—41no—
69D4×C199—41no—
70D4×C199—41no—
71C2×C4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C8×C199C2×D411yescharacteristic
74C8×C199C2×D411yescharacteristic
75C22×C199—41no—
76C22×C199—81no—
77C22×C199—161no—
78C22×C199AGL(1,ℤ/8)11yescharacteristic
79C2×D199—21no—
80C2×D199—42no—
81C2×D199—162no—
82C2×D199—161no—
83C4×C199—41no—
84C4×C199#12011yescharacteristic
85C4×C199C2×D811yescharacteristic
86Dic199—21no—
87C2×C199—21no—
88C2×C199—81no—
89C2×C199—161no—
90C2×C199#44711yescharacteristic
91D199—41no—
92D199—82no—
93C199#219211yescharacteristic, complemented, Sylow
94#2192—1991nomaximal, Sylow
95#576—1991no—
96#509D19911yescharacteristic, complemented
97C2×D16—1991no—
98#472—1991no—
99#361—1991no—
100#359—1991no—
101#350—1991no—
102H(5,𝔽2)—3981no—
103AGL(1,ℤ/8)—3981no—
104AGL(1,ℤ/8)—3982no—
105AGL(1,ℤ/8)—3981no—
106#135—3981no—
107#135C2×D19911yescharacteristic
108C2×D8—3981no—
109C2×D8—1991no—
110C2×D8C2×D19911yescharacteristic, complemented
111#131—1991no—
112SD32—21no—
113D16—21no—
114D16—3982no—
115M32C2×D19911yescharacteristic, complemented
116C2×C16—1991no—
117#108—1991no—
118#100—3981no—
119#41—21no—
120#41—7961no—
121#41—3982no—
122#41—1991no—
123C2×D4—21no—
124C2×D4—7961no—
125C2×D4—7962no—
126C2×D4—3982no—
127C2×D4—3981no—
128Q16C22⋊D19911yescharacteristic
129SD16—21no—
130SD16—7961no—
131SD16—3982no—
132D8—21no—
133D8—21no—
134D8—7961no—
135D8—3982no—
136D8—3981no—
137D8—1992no—
138D8C22⋊D19911yescharacteristic
139M16—3981no—
140M16—3981no—
141M16—1991no—
142C2×C8—3981no—
143C2×C8C22×D19911yescharacteristic
144C16—21no—
145C16—3981no—
146C23—41no—
147C23—7962no—
148C23—7961no—
149Q8—21no—
150Q8—1991no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—7962no—
156D4—7962no—
157D4—7961no—
158D4—3981no—
159D4—3982no—
160D4—1991no—
161C2×C4—41no—
162C2×C4—7962no—
163C2×C4—3981no—
164C2×C4D4×D19911yescharacteristic
165C8—7961no—
166C8—3981no—
167C8D4×D19911yescharacteristic
168C8C2×C22⋊D19911yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—7962no—
173C22—31841no—
174C22—31842no—
175C22—3981no—
176C22#2026397911yescharacteristic
177C4—41no—
178C4—3981no—
179C4#2026392311yescharacteristic
180C4D8×D19911yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—7961no—
185C2—15922no—
186C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#509×C199C2
socleC2×C199#447
hypercenter#509D199
hypocenterC199#2192
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #27164820C2semidirect
2. C2×#20263896C2semidirect
3. #27164285C2semidirect
4. #27163832C2semidirect
5. #27163825C2semidirect
6. #27163803C2semidirect
7. #509×C199C2semidirect
17. C2×C199⋊C16C22semidirect
18. #20263891C22semidirect
21. C2×D8×C199C22semidirect
24. M32×C199C22semidirect
93. C199#2192semidirect
96. #509D199semidirect
110. C2×D8C2×D199semidirect
115. M32C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648202
C2×#202638962
#271642852
#271638322
#271638252
#271638032
#509×C1992
#2192199