Finite Group Databasebeta

← #38770806

Subgroups — #38770806

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#27164819C211yesmaximal, characteristic, complemented
2#27164454C211yesmaximal, characteristic, complemented
3#27164285C211yesmaximal, characteristic, complemented
4#27163831C211yesmaximal, characteristic, complemented
5#27163826C211yesmaximal, characteristic, complemented
6#27163801C211yesmaximal, characteristic, complemented
7C2×D16×C199C211yesmaximal, characteristic, complemented
8#20264000—21no—
9#20263981—21no—
10#20263981—21no—
11#20263979—22no—
12#20263977C2211yescharacteristic
13#20263974—21no—
14D8×D199—21no—
15#20263961C2211yescharacteristic
16#20263899—21no—
17#20263896—21no—
18#20263895C2211yescharacteristic, complemented
19#20263891C2211yescharacteristic, complemented
20#20263872—21no—
21C2×D8×C199C2211yescharacteristic
22C2×D8×C199C2211yescharacteristic, complemented
23D16×C199—22no—
24C2×C16×C199C2211yescharacteristic, complemented
25#15691886—21no—
26#15691886D411yescharacteristic
27#15691885—22no—
28#15691885—41no—
29C2×C22⋊D199—42no—
30D4×D199—22no—
31D4×D199—41no—
32Dic796D411yescharacteristic
33#15691875—21no—
34#15691874—22no—
35#15691874—41no—
36D1592D411yescharacteristic
37#15691871—22no—
38#15691871—41no—
39#15691870—21no—
40#15691870D411yescharacteristic
41#15691869—21no—
42C8×D199—21no—
43C199⋊C16—21no—
44C2×D4×C199—21no—
45C2×D4×C199—21no—
46D8×C199—21no—
47D8×C199—21no—
48D8×C199—21no—
49D8×C199D412yes—
50C2×C8×C199C2311yescharacteristic
51C16×C199—21no—
52C22×D199—42no—
53Dic398—21no—
54Dic398D811yescharacteristic
55D796—21no—
56D796D811yescharacteristic
57C22⋊D199—21no—
58C22⋊D199—42no—
59C22⋊D199—42no—
60C22⋊D199—41no—
61C2×Dic199—42no—
62C4×D199—21no—
63C4×D199—41no—
64C199⋊C8—21no—
65C199⋊C8—41no—
66C23×C199—41no—
67C23×C199—41no—
68D4×C199—22no—
69D4×C199—41no—
70D4×C199—41no—
71D4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C8×C199C2×D411yescharacteristic
74C8×C199C2×D411yescharacteristic
75C22×C199—42no—
76C22×C199—81no—
77C22×C199—161no—
78C22×C199—161no—
79C22×C199AGL(1,ℤ/8)11yescharacteristic
80C2×D199—21no—
81C2×D199—41no—
82C2×D199—162no—
83C4×C199#12011yescharacteristic
84C4×C199C2×D811yescharacteristic
85Dic199—21no—
86Dic199—41no—
87C2×C199—21no—
88C2×C199—82no—
89C2×C199—161no—
90C2×C199#44711yescharacteristic
91D199—41no—
92D199—81no—
93C199#219211yescharacteristic, complemented, Sylow
94#2192—1991nomaximal, Sylow
95#576—1991no—
96#509—1991no—
97C2×D16D19911yescharacteristic, complemented
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#135—1991no—
108C2×D8—3981no—
109C2×D8C2×D19911yescharacteristic
110C2×D8C2×D19911yescharacteristic, complemented
111#131—1991no—
112SD32—3981no—
113D16—22no—
114D16—3981no—
115M32—1991no—
116C2×C16C2×D19911yescharacteristic, complemented
117#108—1991no—
118#100—3981no—
119#41—7961no—
120#41—3982no—
121#41—3981no—
122#41—1991no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—7961no—
126C2×D4—7962no—
127C2×D4—3982no—
128Q16—1991no—
129SD16—7961no—
130SD16—3982no—
131SD16—3981no—
132D8—21no—
133D8—21no—
134D8—21no—
135D8—7961no—
136D8—3982no—
137D8—1991no—
138D8C22⋊D19912yes—
139M16—3981no—
140M16—3981no—
141M16—1991no—
142C2×C8—3981no—
143C2×C8C22×D19911yescharacteristic
144C16—21no—
145C16—3981no—
146C23—41no—
147C23—41no—
148C23—7962no—
149Q8—3981no—
150Q8—1991no—
151D4—22no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—7962no—
156D4—7961no—
157D4—7962no—
158D4—3981no—
159D4—3981no—
160D4—1991no—
161C2×C4—7962no—
162C2×C4—7961no—
163C2×C4—3981no—
164C2×C4D4×D19911yescharacteristic
165C8—7961no—
166C8—3981no—
167C8D4×D19911yescharacteristic
168C8C2×C22⋊D19911yescharacteristic
169C22—42no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22—7961no—
174C22—31842no—
175C22—3981no—
176C22#2026397911yescharacteristic
177C4—7961no—
178C4—3981no—
179C4#2026392311yescharacteristic
180C4D8×D19911yescharacteristic
181C2—21no—
182C2—82no—
183C2—161no—
184C2—7961no—
185C2—15921no—
186C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
FittingC2×D16×C199C2
socleC2×C199#447
hypercenterC2×D16D199
hypocenterC199#2192
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #27164819C2semidirect
2. #27164454C2semidirect
3. #27164285C2semidirect
4. #27163831C2semidirect
5. #27163826C2semidirect
6. #27163801C2semidirect
7. C2×D16×C199C2semidirect
18. #20263895C22semidirect
19. #20263891C22semidirect
22. C2×D8×C199C22semidirect
24. C2×C16×C199C22semidirect
93. C199#2192semidirect
97. C2×D16D199semidirect
110. C2×D8C2×D199semidirect
116. C2×C16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648192
#271644542
#271642852
#271638312
#271638262
#271638012
C2×D16×C1992
#2192199