Finite Group Databasebeta

← #38741937

Subgroups — #38741937

Order 25216.

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

normal subgroups only

This group has 212 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#27161795C211yesmaximal, characteristic
2#27161356C211yesmaximal, characteristic, complemented
3#27161281C211yesmaximal, characteristic, complemented
4#27160828C211yesmaximal, characteristic, complemented
5#27160808C211yesmaximal, characteristic, complemented
6#27160808C211yesmaximal, characteristic, complemented
7#487×C197C211yesmaximal, characteristic, complemented
8C2×#15691345—21no—
9#20263315—22no—
10#20263301—21no—
11#20263301—21no—
12C4×D788C2211yescharacteristic
13C4×D788C2211yescharacteristic
14#20263284—21no—
15#20263284—21no—
16#20263259—21no—
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263250—21no—
20#20263246—21no—
21C4⋊Q8×C197C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23#103×C197—22no—
24C4×C8×C197C2211yescharacteristic
25C2×D788—22no—
26C2×D788D411yescharacteristic
27C2×D788D411yescharacteristic
28C2×C4×D197—21no—
29C2×C4×D197—21no—
30#15691345—42no—
31C2×#13096802—21no—
32C2×#13096802—21no—
33C4⋊Dic197D411yescharacteristic
34C4⋊Dic197D411yescharacteristic
35#15691335—22no—
36#15691335—22no—
37#15691335—21no—
38#15691335—21no—
39C4×Dic197—22no—
40C2×Q8×C197—21no—
41C2×Q8×C197D412yes—
42C2×C8×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49C22×D197—21no—
50D788—21no—
51D788—21no—
52D788—21no—
53D788—21no—
54D788—82no—
55C2×Dic197—21no—
56C2×Dic197—21no—
57C4×D197—41no—
58C4×D197—41no—
59C4×D197—41no—
60C4×D197—41no—
61#13096802—41no—
62#13096802—41no—
63Q8×C197—42no—
64Q8×C197—41no—
65Q8×C197—41no—
66C2×C4×C197—22no—
67C2×C4×C197—41no—
68C2×C4×C197C2×D411yescharacteristic
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C8×C197—41no—
72C22×C197#12011yescharacteristic
73C2×D197—41no—
74C2×D197—41no—
75C2×D197—41no—
76C2×D197—41no—
77C2×D197—41no—
78C2×D197—41no—
79C4×C197—21no—
80C4×C197—42no—
81C4×C197—81no—
82C4×C197#13511yescharacteristic
83C4×C197#13511yescharacteristic
84C4×C197AGL(1,ℤ/8)11yescharacteristic
85C4×C197#13711yescharacteristic
86Dic197—41no—
87Dic197—41no—
88C2×C197#44911yescharacteristic
89C2×C197#45211yescharacteristic
90C2×C197#45511yescharacteristic
91D197—81no—
92D197—81no—
93C197#164611yescharacteristic, complemented, Sylow
94#1646—1971nomaximal, Sylow
95#555—1971no—
96#487D19711yescharacteristic, complemented
97#474—1971no—
98#461—1971no—
99#333—1971no—
100D4⋊C8—1971no—
101D4⋊C8—1971no—
102C2×SD16—3941no—
103C4⋊Q8C2×D19711yescharacteristic
104C4⋊Q8C2×D19711yescharacteristic
105#124—3942no—
106#121—3941no—
107#121—3941no—
108C4×D4—1971no—
109C4×D4—1971no—
110#117—3941no—
111#117—3941no—
112#107—3941no—
113C4⋊C8—1971no—
114C4⋊C8—1971no—
115#103—22no—
116#103—3941no—
117#102—3941no—
118C4×C8C2×D19711yescharacteristic
119C2×Q8—21no—
120C2×Q8C22⋊D19712yes—
121C2×D4—3942no—
122C2×D4—1971no—
123C2×D4—1971no—
124C22×C4—3941no—
125C22×C4—3941no—
126SD16—7882no—
127C2×C8—21no—
128C2×C8—3941no—
129C2×C8—3941no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1971no—
135C4⋊C4—1971no—
136C22⋊C4—3942no—
137C22⋊C4—3942no—
138C22⋊C4—3941no—
139C22⋊C4—3941no—
140C42—3942no—
141C42C22×D19711yescharacteristic
142C23—3941no—
143C23—3941no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—15762no—
148D4—3941no—
149D4—3941no—
150D4—3941no—
151D4—3941no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—7881no—
155C2×C4—7881no—
156C2×C4—7881no—
157C2×C4—7881no—
158C2×C4—3941no—
159C2×C4—3941no—
160C2×C4D4×D19711yescharacteristic
161C2×C4D4×D19711yescharacteristic
162C2×C4C2×C22⋊D19711yescharacteristic
163C8—41no—
164C8—7881no—
165C8—7881no—
166C22—7881no—
167C22—7881no—
168C22—7881no—
169C22—7881no—
170C22—7881no—
171C22—7881no—
172C22#2026329411yescharacteristic
173C4—21no—
174C4—42no—
175C4—81no—
176C4—7881no—
177C4—7881no—
178C4#2026334711yescharacteristic
179C4#2026334911yescharacteristic
180C4#2026335311yescharacteristic
181C4#2026335811yescharacteristic
182C2—15761no—
183C2—15761no—
184C2#2716122311yescharacteristic
185C2#2716124011yescharacteristic
186C2#2716125611yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#487×C197C2
socleC22×C197#120
prime radical#27161795C2
hypercenter#487D197
hypocenterC197#1646
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #27161356C2semidirect
3. #27161281C2semidirect
4. #27160828C2semidirect
5. #27160808C2semidirect
6. #27160808C2semidirect
7. #487×C197C2semidirect
93. C197#1646semidirect
96. #487D197semidirect

Maximal subgroups

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

subgroupindex
#271617952
#271613562
#271612812
#271608282
#271608082
#487×C1972
#1646197