Finite Group Databasebeta

← #38770820

Subgroups — #38770820

Order 25472.

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

normal subgroups only

This group has 216 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
2#27164459C211yesmaximal, characteristic, complemented
3#27164290C211yesmaximal, characteristic, complemented
4#27163831C211yesmaximal, characteristic, complemented
5#27163826C211yesmaximal, characteristic, complemented
6#27163801C211yesmaximal, characteristic, complemented
7C2×SD32×C199C211yesmaximal, characteristic, complemented
8#20264000—21no—
9#20263986—21no—
10#20263981—21no—
11#20263979—22no—
12#20263977C2211yescharacteristic
13#20263974—21no—
14D8×D199—21no—
15#20263961C2211yescharacteristic
16#20263901—21no—
17#20263898—21no—
18#20263895C2211yescharacteristic, complemented
19#20263891C2211yescharacteristic, complemented
20#20263874—21no—
21C2×Q16×C199C2211yescharacteristic, complemented
22C2×D8×C199C2211yescharacteristic
23SD32×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#15691876—41no—
34#15691875—21no—
35#15691874—22no—
36#15691873—41no—
37D1592D411yescharacteristic
38#15691871—22no—
39#15691870—21no—
40#15691870D411yescharacteristic
41#15691869—21no—
42C8×D199—21no—
43C199⋊C16—21no—
44C2×Q8×C199—21no—
45C2×D4×C199—21no—
46Q16×C199—21no—
47Q16×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—
67Q8×C199—41no—
68Q8×C199—41no—
69D4×C199—22no—
70D4×C199—41no—
71C2×C4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C8×C199C2×D411yescharacteristic
74C8×C199C2×D411yescharacteristic
75C22×C199—42no—
76C22×C199—161no—
77C22×C199AGL(1,ℤ/8)11yescharacteristic
78C2×D199—21no—
79C2×D199—41no—
80C2×D199—162no—
81C4×C199—81no—
82C4×C199#12011yescharacteristic
83C4×C199C2×D811yescharacteristic
84Dic199—21no—
85Dic199—41no—
86C2×C199—21no—
87C2×C199—82no—
88C2×C199#44711yescharacteristic
89D199—41no—
90D199—81no—
91C199#219411yescharacteristic, complemented, Sylow
92#2194—1991nomaximal, Sylow
93#576—1991no—
94#510—1991no—
95C2×SD32D19911yescharacteristic, complemented
96#473—1991no—
97#361—1991no—
98#359—1991no—
99#350—1991no—
100H(5,𝔽2)—3981no—
101#137—3981no—
102AGL(1,ℤ/8)—3982no—
103AGL(1,ℤ/8)—3981no—
104#135—3981no—
105#135—1991no—
106C2×Q16C2×D19911yescharacteristic, complemented
107C2×D8—3981no—
108C2×D8C2×D19911yescharacteristic
109#131—1991no—
110Q32—3981no—
111SD32—22no—
112SD32—3981no—
113M32—1991no—
114C2×C16C2×D19911yescharacteristic, complemented
115#108—1991no—
116#101—3981no—
117#41—7961no—
118#41—3982no—
119#41—3981no—
120#41—1991no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—7961no—
124C2×D4—7962no—
125C2×D4—3982no—
126Q16—21no—
127Q16—21no—
128Q16—7961no—
129Q16—1991no—
130SD16—7961no—
131SD16—3982no—
132SD16—3981no—
133D8—21no—
134D8—3982no—
135D8—1991no—
136D8C22⋊D19912yes—
137M16—3981no—
138M16—3981no—
139M16—1991no—
140C2×C8—3981no—
141C2×C8C22×D19911yescharacteristic
142C16—21no—
143C16—3981no—
144C23—41no—
145C23—7962no—
146Q8—41no—
147Q8—41no—
148Q8—3981no—
149Q8—1991no—
150D4—22no—
151D4—41no—
152D4—7962no—
153D4—7961no—
154D4—7962no—
155D4—3981no—
156D4—3981no—
157D4—1991no—
158C2×C4—41no—
159C2×C4—7962no—
160C2×C4—7961no—
161C2×C4—3981no—
162C2×C4D4×D19911yescharacteristic
163C8—7961no—
164C8—3981no—
165C8D4×D19911yescharacteristic
166C8C2×C22⋊D19911yescharacteristic
167C22—42no—
168C22—161no—
169C22—7961no—
170C22—31842no—
171C22—3981no—
172C22#2026397911yescharacteristic
173C4—81no—
174C4—7961no—
175C4—3981no—
176C4#2026392311yescharacteristic
177C4D8×D19911yescharacteristic
178C2—21no—
179C2—82no—
180C2—7961no—
181C2—15921no—
182C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
FittingC2×SD32×C199C2
socleC2×C199#447
prime radical#27164819C2
hypercenterC2×SD32D199
hypocenterC199#2194
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #27164459C2semidirect
3. #27164290C2semidirect
4. #27163831C2semidirect
5. #27163826C2semidirect
6. #27163801C2semidirect
7. C2×SD32×C199C2semidirect
18. #20263895C22semidirect
19. #20263891C22semidirect
21. C2×Q16×C199C22semidirect
24. C2×C16×C199C22semidirect
91. C199#2194semidirect
95. C2×SD32D199semidirect
106. C2×Q16C2×D199semidirect
114. C2×C16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648192
#271644592
#271642902
#271638312
#271638262
#271638012
C2×SD32×C1992
#2194199