Finite Group Databasebeta

← #38770794

Subgroups — #38770794

Order 25472.

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

normal subgroups only

This group has 226 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#27164815C211yesmaximal, characteristic, complemented
2#27164454C211yesmaximal, characteristic, complemented
3#27164268C211yesmaximal, characteristic, complemented
4#27163829C211yesmaximal, characteristic, complemented
5#27163828C211yesmaximal, characteristic, complemented
6#27163799C211yesmaximal, characteristic, complemented
7#509×C199C211yesmaximal, characteristic, complemented
8#41×D199—21no—
9#20263987—21no—
10#20263984—21no—
11#20263982—21no—
12#20263979—21no—
13#20263977C2211yescharacteristic
14#20263975—21no—
15C2×#15691869C2211yescharacteristic
16#20263926—21no—
17#20263899—21no—
18#20263896—21no—
19#20263895C2211yescharacteristic, complemented
20#20263887C2211yescharacteristic, complemented
21#20263876—21no—
22#135×C199C2211yescharacteristic
23C2×D8×C199C2211yescharacteristic, complemented
24SD32×C199—21no—
25D16×C199—21no—
26M32×C199C2211yescharacteristic, complemented
27#15691887—21no—
28#15691886—21no—
29#15691886—41no—
30#15691885—21no—
31#15691885—41no—
32Q8×D199—21no—
33C2×C22⋊D199—41no—
34D4×D199—21no—
35D4×D199—41no—
36C2×C4×D199—41no—
37C2×C4×D199D411yescharacteristic
38Dic796D411yescharacteristic, complemented
39#15691876—21no—
40#15691875—21no—
41#15691874—21no—
42#15691873—21no—
43D1592D411yescharacteristic, complemented
44#15691871—21no—
45#15691869—21no—
46#15691869—21no—
47C2×C199⋊C8D411yescharacteristic
48C4⋊Dic199—21no—
49C22⋊Dic199—41no—
50C199⋊C16—21no—
51#41×C199—21no—
52C2×D4×C199—21no—
53Q16×C199D411yescharacteristic, complemented
54SD16×C199—21no—
55D8×C199—21no—
56D8×C199—21no—
57D8×C199D411yescharacteristic, complemented
58C2×C8×C199C2311yescharacteristic
59C16×C199—21no—
60C22×D199—41no—
61C22×D199D811yescharacteristic
62Dic398—21no—
63Dic398—41no—
64D796—21no—
65D796—41no—
66C22⋊D199—41no—
67C22⋊D199—41no—
68C22⋊D199—41no—
69C22⋊D199—82no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C2×Dic199D811yescharacteristic
73C4×D199—21no—
74C4×D199—21no—
75C4×D199—41no—
76C4×D199—41no—
77C4×D199—41no—
78C199⋊C8—21no—
79C23×C199—41no—
80Q8×C199—21no—
81D4×C199—21no—
82D4×C199—41no—
83D4×C199—41no—
84D4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199C2×D411yescharacteristic
87C8×C199C2×D411yescharacteristic
88C8×C199C2×D411yescharacteristic
89C22×C199—41no—
90C22×C199—81no—
91C22×C199—81no—
92C22×C199—81no—
93C22×C199C2×D811yescharacteristic
94C2×D199—21no—
95C2×D199—41no—
96C2×D199—41no—
97C2×D199—82no—
98C4×C199—41no—
99C4×C199#12011yescharacteristic
100C4×C199AGL(1,ℤ/8)11yescharacteristic
101Dic199—21no—
102Dic199—41no—
103Dic199—81no—
104C2×C199—21no—
105C2×C199—81no—
106C2×C199—161no—
107C2×C199#44711yescharacteristic
108D199—41no—
109D199—81no—
110C199#219011yescharacteristic, complemented, Sylow
111#2190—1991nomaximal, Sylow
112#575—1991no—
113#509—1991no—
114#509D19911yescharacteristic, complemented
115#469—1991no—
116#360—1991no—
117#360—1991no—
118#349—1991no—
119C2×#41—3981no—
120#137—3981no—
121#137—3981no—
122AGL(1,ℤ/8)—3981no—
123AGL(1,ℤ/8)—3981no—
124#135—3981no—
125#135—1991no—
126#135C2×D19911yescharacteristic
127C2×D8C2×D19911yescharacteristic, complemented
128C2×M16—1991no—
129#121—3981no—
130SD32—21no—
131SD32—3981no—
132D16—21no—
133D16—3981no—
134M32—1991no—
135M32C2×D19911yescharacteristic, complemented
136#106—1991no—
137#102—3981no—
138#41—21no—
139#41—7961no—
140#41—7961no—
141#41—3981no—
142#41—3981no—
143#41—3981no—
144C2×Q8—3981no—
145C2×D4—21no—
146C2×D4—7961no—
147C2×D4—7961no—
148C2×D4—3981no—
149C22×C4—7961no—
150C22×C4—1991no—
151Q16—3981no—
152Q16—1991no—
153Q16C22⋊D19911yescharacteristic, complemented
154SD16—21no—
155SD16—3981no—
156SD16—3981no—
157SD16—3981no—
158D8—21no—
159D8—21no—
160D8—3981no—
161D8—1991no—
162D8C22⋊D19911yescharacteristic, complemented
163M16—3981no—
164M16—3981no—
165C2×C8—1991no—
166C2×C8C22×D19911yescharacteristic
167C4⋊C4—3981no—
168C22⋊C4—7961no—
169C16—21no—
170C16—3981no—
171C23—41no—
172C23—7961no—
173C23—1991no—
174Q8—21no—
175Q8—7961no—
176Q8—3981no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7961no—
182D4—7961no—
183D4—7961no—
184D4—7961no—
185D4—15922no—
186D4—3981no—
187C2×C4—41no—
188C2×C4—7961no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—3981no—
194C2×C4—3981no—
195C2×C4—1991no—
196C2×C4D4×D19911yescharacteristic
197C8—3981no—
198C8D4×D19911yescharacteristic
199C8C2×C22⋊D19911yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—7961no—
205C22—7961no—
206C22—15922no—
207C22—3981no—
208C22D8×D19911yescharacteristic
209C4—41no—
210C4—7961no—
211C4—15921no—
212C4—3981no—
213C4#2026392311yescharacteristic
214C4#2026397911yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—7961no—
219C2—15921no—
220C2#2716415011yescharacteristic
subgroupG/H
centerC2#27164150
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#509×C199C2
socleC2×C199#447
hypercenter#509D199
hypocenterC199#2190
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. #27164815C2semidirect
2. #27164454C2semidirect
3. #27164268C2semidirect
4. #27163829C2semidirect
5. #27163828C2semidirect
6. #27163799C2semidirect
7. #509×C199C2semidirect
19. #20263895C22semidirect
20. #20263887C22semidirect
23. C2×D8×C199C22semidirect
26. M32×C199C22semidirect
38. Dic796D4semidirect
43. D1592D4semidirect
53. Q16×C199D4semidirect
57. D8×C199D4semidirect
110. C199#2190semidirect
114. #509D199semidirect
127. C2×D8C2×D199semidirect
135. M32C2×D199semidirect
153. Q16C22⋊D199semidirect
162. D8C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271648152
#271644542
#271642682
#271638292
#271638282
#271637992
#509×C1992
#2190199