Finite Group Databasebeta

← #38744656

Subgroups — #38744656

Order 25216.

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#27161876C211yesmaximal, characteristic, complemented
2#27161515C211yesmaximal, characteristic, complemented
3#27161329C211yesmaximal, characteristic, complemented
4#27160890C211yesmaximal, characteristic, complemented
5#27160889C211yesmaximal, characteristic, complemented
6#27160860C211yesmaximal, characteristic, complemented
7#509×C197C211yesmaximal, characteristic, complemented
8#41×D197—21no—
9#20263358—21no—
10#20263355—21no—
11#20263353—21no—
12#20263350—21no—
13#20263348C2211yescharacteristic
14#20263346—21no—
15C2×#15691341C2211yescharacteristic
16#20263297—21no—
17#20263270—21no—
18#20263267—21no—
19#20263266C2211yescharacteristic, complemented
20#20263258C2211yescharacteristic, complemented
21#20263247—21no—
22#135×C197C2211yescharacteristic
23C2×D8×C197C2211yescharacteristic, complemented
24SD32×C197—21no—
25D16×C197—21no—
26M32×C197C2211yescharacteristic, complemented
27#15691359—21no—
28#15691358—21no—
29#15691358—41no—
30#15691357—21no—
31#15691357—41no—
32Q8×D197—21no—
33C2×C22⋊D197—41no—
34D4×D197—21no—
35D4×D197—41no—
36C2×C4×D197—41no—
37C2×C4×D197D411yescharacteristic
38Dic788D411yescharacteristic, complemented
39#15691348—21no—
40#15691347—21no—
41#15691346—21no—
42#15691345—21no—
43D1576D411yescharacteristic, complemented
44#15691343—21no—
45#15691341—21no—
46#15691341—21no—
47C2×#13096802D411yescharacteristic
48C4⋊Dic197—21no—
49C22⋊Dic197—41no—
50#15691333—21no—
51#41×C197—21no—
52C2×D4×C197—21no—
53Q16×C197D411yescharacteristic, complemented
54SD16×C197—21no—
55D8×C197—21no—
56D8×C197—21no—
57D8×C197D411yescharacteristic, complemented
58C2×C8×C197C2311yescharacteristic
59C16×C197—21no—
60C22×D197—41no—
61C22×D197D811yescharacteristic
62Dic394—21no—
63Dic394—41no—
64D788—21no—
65D788—41no—
66C22⋊D197—41no—
67C22⋊D197—41no—
68C22⋊D197—41no—
69C22⋊D197—82no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197D811yescharacteristic
73C4×D197—21no—
74C4×D197—21no—
75C4×D197—41no—
76C4×D197—41no—
77C4×D197—41no—
78#13096802—21no—
79C23×C197—41no—
80Q8×C197—21no—
81D4×C197—21no—
82D4×C197—41no—
83D4×C197—41no—
84D4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197C2×D411yescharacteristic
87C8×C197C2×D411yescharacteristic
88C8×C197C2×D411yescharacteristic
89C22×C197—41no—
90C22×C197—81no—
91C22×C197—81no—
92C22×C197—81no—
93C22×C197C2×D811yescharacteristic
94C2×D197—21no—
95C2×D197—41no—
96C2×D197—41no—
97C2×D197—82no—
98C4×C197—41no—
99C4×C197#12011yescharacteristic
100C4×C197AGL(1,ℤ/8)11yescharacteristic
101Dic197—21no—
102Dic197—41no—
103Dic197—81no—
104C2×C197—21no—
105C2×C197—81no—
106C2×C197—161no—
107C2×C197#44711yescharacteristic
108D197—41no—
109D197—81no—
110C197#219011yescharacteristic, complemented, Sylow
111#2190—1971nomaximal, Sylow
112#575—1971no—
113#509—1971no—
114#509D19711yescharacteristic, complemented
115#469—1971no—
116#360—1971no—
117#360—1971no—
118#349—1971no—
119C2×#41—3941no—
120#137—3941no—
121#137—3941no—
122AGL(1,ℤ/8)—3941no—
123AGL(1,ℤ/8)—3941no—
124#135—3941no—
125#135—1971no—
126#135C2×D19711yescharacteristic
127C2×D8C2×D19711yescharacteristic, complemented
128C2×M16—1971no—
129#121—3941no—
130SD32—21no—
131SD32—3941no—
132D16—21no—
133D16—3941no—
134M32—1971no—
135M32C2×D19711yescharacteristic, complemented
136#106—1971no—
137#102—3941no—
138#41—21no—
139#41—7881no—
140#41—7881no—
141#41—3941no—
142#41—3941no—
143#41—3941no—
144C2×Q8—3941no—
145C2×D4—21no—
146C2×D4—7881no—
147C2×D4—7881no—
148C2×D4—3941no—
149C22×C4—7881no—
150C22×C4—1971no—
151Q16—3941no—
152Q16—1971no—
153Q16C22⋊D19711yescharacteristic, complemented
154SD16—21no—
155SD16—3941no—
156SD16—3941no—
157SD16—3941no—
158D8—21no—
159D8—21no—
160D8—3941no—
161D8—1971no—
162D8C22⋊D19711yescharacteristic, complemented
163M16—3941no—
164M16—3941no—
165C2×C8—1971no—
166C2×C8C22×D19711yescharacteristic
167C4⋊C4—3941no—
168C22⋊C4—7881no—
169C16—21no—
170C16—3941no—
171C23—41no—
172C23—7881no—
173C23—1971no—
174Q8—21no—
175Q8—7881no—
176Q8—3941no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7881no—
182D4—7881no—
183D4—7881no—
184D4—7881no—
185D4—15762no—
186D4—3941no—
187C2×C4—41no—
188C2×C4—7881no—
189C2×C4—7881no—
190C2×C4—7881no—
191C2×C4—7881no—
192C2×C4—7881no—
193C2×C4—3941no—
194C2×C4—3941no—
195C2×C4—1971no—
196C2×C4D4×D19711yescharacteristic
197C8—3941no—
198C8D4×D19711yescharacteristic
199C8C2×C22⋊D19711yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—7881no—
205C22—7881no—
206C22—15762no—
207C22—3941no—
208C22D8×D19711yescharacteristic
209C4—41no—
210C4—7881no—
211C4—15761no—
212C4—3941no—
213C4#2026329411yescharacteristic
214C4#2026335011yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—7881no—
219C2—15761no—
220C2#2716121111yescharacteristic
subgroupG/H
centerC2#27161211
commutator, cosocleC2×C8×C197C23
FrattiniC2×C8C22×D197
Fitting#509×C197C2
socleC2×C197#447
hypercenter#509D197
hypocenterC197#2190
LCSG ▷ C2×C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #27161876C2semidirect
2. #27161515C2semidirect
3. #27161329C2semidirect
4. #27160890C2semidirect
5. #27160889C2semidirect
6. #27160860C2semidirect
7. #509×C197C2semidirect
19. #20263266C22semidirect
20. #20263258C22semidirect
23. C2×D8×C197C22semidirect
26. M32×C197C22semidirect
38. Dic788D4semidirect
43. D1576D4semidirect
53. Q16×C197D4semidirect
57. D8×C197D4semidirect
110. C197#2190semidirect
114. #509D197semidirect
127. C2×D8C2×D197semidirect
135. M32C2×D197semidirect
153. Q16C22⋊D197semidirect
162. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618762
#271615152
#271613292
#271608902
#271608892
#271608602
#509×C1972
#2190197