Finite Group Databasebeta

← #38744659

Subgroups — #38744659

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#27161517C211yesmaximal, characteristic, complemented
3#27161514C211yesmaximal, characteristic, complemented
4#27161333C211yesmaximal, characteristic, complemented
5#27160888C211yesmaximal, characteristic, complemented
6#27160860C211yesmaximal, characteristic, complemented
7#360×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×D1576C2211yescharacteristic, complemented
16C2×#15691341C2211yescharacteristic
17#20263301—21no—
18#20263271—21no—
19#20263269—21no—
20D3152—21no—
21#20263267—21no—
22#20263266C2211yescharacteristic, complemented
23#20263246—21no—
24#135×C197C2211yescharacteristic
25M32×C197C2211yescharacteristic, complemented
26#106×C197C2211yescharacteristic, complemented
27#15691359—21no—
28#15691358—21no—
29#15691358—41no—
30#15691357—21no—
31#15691357—41no—
32Q8×D197—21no—
33C2×D788—21no—
34C2×D788—41no—
35D4×D197—21no—
36D4×D197—41no—
37C2×C4×D197—41no—
38C2×C4×D197D411yescharacteristic
39Dic788D411yescharacteristic, complemented
40#15691348—21no—
41#15691347—21no—
42#15691346—21no—
43#15691345—21no—
44D1576—21no—
45D1576—21no—
46D1576D411yescharacteristic, complemented
47#15691343—21no—
48#15691341—21no—
49#15691341—21no—
50C2×#13096802D411yescharacteristic
51#15691335—41no—
52#15691333—21no—
53#41×C197—21no—
54Q16×C197D411yescharacteristic, complemented
55SD16×C197—21no—
56D8×C197D411yescharacteristic, complemented
57C2×C8×C197C2311yescharacteristic
58C4⋊C4×C197—21no—
59C16×C197—21no—
60C22×D197—41no—
61C22×D197—41no—
62C22×D197D811yescharacteristic
63Dic394—21no—
64Dic394—41no—
65D788—21no—
66D788—41no—
67D788—41no—
68D788—41no—
69D788—82no—
70C22⋊D197—41no—
71C22⋊D197—41no—
72C22⋊D197—41no—
73C2×Dic197—41no—
74C2×Dic197D811yescharacteristic
75C4×D197—21no—
76C4×D197—21no—
77C4×D197—41no—
78C4×D197—41no—
79C4×D197—41no—
80#13096802—21no—
81Q8×C197—21no—
82D4×C197—21no—
83D4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197C2×D411yescharacteristic
87C8×C197C2×D411yescharacteristic
88C8×C197C2×D411yescharacteristic
89C22×C197—41no—
90C22×C197C2×D811yescharacteristic
91C2×D197—21no—
92C2×D197—41no—
93C2×D197—41no—
94C2×D197—82no—
95C2×D197—81no—
96C2×D197—81no—
97C2×D197—81no—
98C4×C197—41no—
99C4×C197—81no—
100C4×C197#12011yescharacteristic
101C4×C197AGL(1,ℤ/8)11yescharacteristic
102Dic197—21no—
103Dic197—41no—
104C2×C197—21no—
105C2×C197—81no—
106C2×C197#44711yescharacteristic
107D197—41no—
108D197—81no—
109D197—161no—
110C197#219011yescharacteristic, complemented, Sylow
111#2190—1971nomaximal, Sylow
112#575—1971no—
113#509—1971no—
114#509—1971no—
115#469—1971no—
116#360—1971no—
117#360D19711yescharacteristic, complemented
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×D8—1971no—
128C2×M16—1971no—
129#121—3941no—
130SD32—3941no—
131SD32—3941no—
132D16—3941no—
133D16—3941no—
134M32—1971no—
135M32C2×D19711yescharacteristic, complemented
136#106C2×D19711yescharacteristic, complemented
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—7881no—
146C2×D4—7881no—
147C2×D4—3941no—
148C2×D4—3941no—
149C22×C4—7881no—
150C22×C4—1971no—
151Q16—3941no—
152Q16—1971no—
153Q16D78811yescharacteristic, complemented
154SD16—21no—
155SD16—3941no—
156SD16—3941no—
157SD16—3941no—
158D8—3941no—
159D8—3941no—
160D8—3941no—
161D8—1971no—
162D8D78811yescharacteristic, complemented
163M16—3941no—
164M16—3941no—
165C2×C8—1971no—
166C2×C8C22×D19711yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—7881no—
169C16—21no—
170C16—3941no—
171C23—7881no—
172C23—7881no—
173C23—1971no—
174Q8—21no—
175Q8—7881no—
176Q8—3941no—
177D4—21no—
178D4—41no—
179D4—7881no—
180D4—7881no—
181D4—7881no—
182D4—7881no—
183D4—7881no—
184D4—7881no—
185D4—15762no—
186D4—3941no—
187C2×C4—41no—
188C2×C4—41no—
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×D78811yescharacteristic
200C22—41no—
201C22—7881no—
202C22—7881no—
203C22—15761no—
204C22—15761no—
205C22—15762no—
206C22—15761no—
207C22—3941no—
208C22D8×D19711yescharacteristic
209C4—41no—
210C4—81no—
211C4—7881no—
212C4—3941no—
213C4#2026329611yescharacteristic
214C4#2026335311yescharacteristic
215C2—21no—
216C2—81no—
217C2—7881no—
218C2—15761no—
219C2—31521no—
220C2#2716121411yescharacteristic
subgroupG/H
centerC2#27161214
commutator, cosocleC2×C8×C197C23
FrattiniC2×C8C22×D197
Fitting#360×C197C2
socleC2×C197#447
hypercenter#360D197
hypocenterC197#2190
LCSG ▷ C2×C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #27161876C2semidirect
2. #27161517C2semidirect
3. #27161514C2semidirect
4. #27161333C2semidirect
5. #27160888C2semidirect
6. #27160860C2semidirect
7. #360×C197C2semidirect
15. C2×D1576C22semidirect
22. #20263266C22semidirect
25. M32×C197C22semidirect
26. #106×C197C22semidirect
39. Dic788D4semidirect
46. D1576D4semidirect
54. Q16×C197D4semidirect
56. D8×C197D4semidirect
110. C197#2190semidirect
117. #360D197semidirect
135. M32C2×D197semidirect
136. #106C2×D197semidirect
153. Q16D788semidirect
162. D8D788semidirect

Maximal subgroups

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

subgroupindex
#271618762
#271615172
#271615142
#271613332
#271608882
#271608602
#360×C1972
#2190197