Finite Group Databasebeta

← #38741823

Subgroups — #38741823

Order 25216.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161803C211yesmaximal, characteristic, complemented
2#27161427C211yesmaximal, characteristic, complemented
3#27161280C211yesmaximal, characteristic, complemented
4#27161275C211yesmaximal, characteristic
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#332×C197C211yesmaximal, characteristic, complemented
8C2×#15691347—21no—
9C2×D1576—21no—
10#20263323C2211yescharacteristic
11#20263321—22no—
12#20263315—21no—
13#20263306—21no—
14Q8×Dic197—21no—
15C4×Dic394C2211yescharacteristic
16C4×D788C2211yescharacteristic
17#20263284—21no—
18#20263248—21no—
19#20263248—21no—
20#20263246—22no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394D411yescharacteristic
26C2×D788—21no—
27C2×D788—21no—
28C2×D788—21no—
29C2×D788D411yescharacteristic
30C2×C4×D197—21no—
31#15691347—42no—
32D1576—42no—
33C2×#13096802—21no—
34C4⋊Dic197—22no—
35C4⋊Dic197D411yescharacteristic
36C4⋊Dic197D411yescharacteristic
37#15691337—21no—
38#15691335—22no—
39#15691335—22no—
40#15691335—21no—
41C4×Dic197—22no—
42C4×Dic197—21no—
43C2×Q8×C197—21no—
44C2×C8×C197—21no—
45C2×C8×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197D412yes—
48C42×C197C2311yescharacteristic
49C22×D197—21no—
50C22×D197—41no—
51Dic394—21no—
52Dic394—21no—
53D788—21no—
54D788—21no—
55D788—42no—
56D788—41no—
57D788—41no—
58D788—41no—
59D788—41no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C4×D197—41no—
64C4×D197—41no—
65#13096802—41no—
66Q8×C197—41no—
67Q8×C197—41no—
68C2×C4×C197—22no—
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C8×C197—41no—
73C8×C197—41no—
74C22×C197#12011yescharacteristic
75C2×D197—41no—
76C2×D197—41no—
77C2×D197—41no—
78C2×D197—81no—
79C2×D197—81no—
80C2×D197—81no—
81C4×C197—21no—
82C4×C197—42no—
83C4×C197#13511yescharacteristic
84C4×C197#13511yescharacteristic
85C4×C197AGL(1,ℤ/8)11yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87Dic197—41no—
88Dic197—41no—
89Dic197—41no—
90C2×C197#44911yescharacteristic
91C2×C197#44911yescharacteristic
92C2×C197#45411yescharacteristic
93D197—81no—
94D197—161no—
95C197#162811yescharacteristic, complemented, Sylow
96#1628—1971nomaximal, Sylow
97#556—1971no—
98#486—1971no—
99#460—1971no—
100#459—1971no—
101#332D19711yescharacteristic, complemented
102Q8⋊C8—1971no—
103D4⋊C8—1971no—
104C2×SD16—3941no—
105C2×D8—3941no—
106C4⋊Q8C2×D19711yescharacteristic
107#127—1971no—
108#126—3942no—
109#124—3941no—
110C22⋊Q8—3941no—
111C4×Q8—3941no—
112C4×Q8—1971no—
113C4×D4—1971no—
114#117—3941no—
115C4⋊C8C2×D19711yescharacteristic
116C4⋊C8C2×D19711yescharacteristic
117#102—3941no—
118#102—3941no—
119#102—3942no—
120C4×C8—1971no—
121C2×Q8—21no—
122C2×Q8—1971no—
123C2×D4—3941no—
124C2×D4—3941no—
125C2×D4—3941no—
126C2×D4—1971no—
127C22×C4—3941no—
128SD16—7882no—
129D8—7882no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—3941no—
133C4⋊C4—21no—
134C4⋊C4—3942no—
135C4⋊C4—3941no—
136C4⋊C4—1971no—
137C4⋊C4—1971no—
138C4⋊C4D78812yes—
139C22⋊C4—3942no—
140C22⋊C4—3942no—
141C22⋊C4—3941no—
142C42—3942no—
143C42—3941no—
144C42C22×D19711yescharacteristic
145C23—7881no—
146C23—3941no—
147Q8—41no—
148Q8—41no—
149Q8—3941no—
150Q8—3941no—
151D4—7882no—
152D4—7881no—
153D4—7881no—
154D4—7881no—
155D4—7881no—
156D4—3941no—
157D4—3941no—
158C2×C4—22no—
159C2×C4—7881no—
160C2×C4—7881no—
161C2×C4—3941no—
162C2×C4—3941no—
163C2×C4—3941no—
164C2×C4D4×D19711yescharacteristic
165C2×C4D4×D19711yescharacteristic
166C2×C4C2×D78811yescharacteristic
167C8—41no—
168C8—41no—
169C8—7881no—
170C22—7881no—
171C22—7881no—
172C22—7881no—
173C22—15761no—
174C22—15761no—
175C22—15761no—
176C22#2026329611yescharacteristic
177C4—21no—
178C4—42no—
179C4—7881no—
180C4—7881no—
181C4—7881no—
182C4#2026334711yescharacteristic
183C4#2026334911yescharacteristic
184C4#2026335011yescharacteristic
185C4#2026335311yescharacteristic
186C2—15761no—
187C2—31521no—
188C2#2716122511yescharacteristic
189C2#2716122711yescharacteristic
190C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
prime radical#27161275C2
hypercenter#332D197
hypocenterC197#1628
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27161803C2semidirect
2. #27161427C2semidirect
3. #27161280C2semidirect
5. #27160809C2semidirect
6. #27160806C2semidirect
7. #332×C197C2semidirect
95. C197#1628semidirect
101. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271618032
#271614272
#271612802
#271612752
#271608092
#271608062
#332×C1972
#1628197