Finite Group Databasebeta

← #38741840

Subgroups — #38741840

Order 25216.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161749C211yesmaximal, characteristic
2#27161451C211yesmaximal, characteristic, complemented
3#27161375C211yesmaximal, characteristic, complemented
4#27161370C211yesmaximal, characteristic, complemented
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#331×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691346—22no—
10#20263326C2211yescharacteristic
11#20263322—21no—
12#20263298—22no—
13#20263297—21no—
14C4×Dic394C2211yescharacteristic
15C4×D788C2211yescharacteristic
16D4×Dic197—21no—
17#20263260—21no—
18#20263258—21no—
19#20263251—21no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22#127×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691357—42no—
26#15691357—42no—
27C2×Dic394—21no—
28C2×Dic394D411yescharacteristic
29C2×D788D411yescharacteristic
30C2×C22⋊D197—22no—
31C2×C22⋊D197—22no—
32C22×Dic197—22no—
33C2×C4×D197—21no—
34#15691346—44no—
35C2×#13096802—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41C22⋊Dic197—22no—
42#15691335—21no—
43C4×Dic197—21no—
44C2×D4×C197—21no—
45C2×D4×C197—21no—
46C2×D4×C197D412yes—
47C2×C8×C197—21no—
48C2×C8×C197—21no—
49C42×C197C2311yescharacteristic
50C22×D197—21no—
51Dic394—21no—
52Dic394—42no—
53Dic394SD1612yescomplemented
54D788—21no—
55D788—21no—
56C22⋊D197—44no—
57C22⋊D197—82no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—44no—
62C2×Dic197—41no—
63C4×D197—41no—
64C4×D197—41no—
65#13096802—22no—
66C23×C197—22no—
67D4×C197—42no—
68D4×C197—41no—
69D4×C197—41no—
70D4×C197—41no—
71D4×C197—41no—
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C2×C4×C197C2×D411yescharacteristic
75C8×C197—41no—
76C8×C197—41no—
77C22×C197—42no—
78C22×C197—42no—
79C22×C197—42no—
80C22×C197#12011yescharacteristic
81C2×D197—41no—
82C2×D197—41no—
83C2×D197—41no—
84C4×C197—21no—
85C4×C197C2×SD1611yescharacteristic
86C4×C197#13511yescharacteristic
87C4×C197AGL(1,ℤ/8)11yescharacteristic
88C4×C197#13711yescharacteristic
89Dic197—22no—
90Dic197—41no—
91Dic197—41no—
92Dic197—81no—
93C2×C197—82no—
94C2×C197#45011yescharacteristic
95C2×C197#45211yescharacteristic
96C2×C197#45311yescharacteristic
97D197—81no—
98C197#163111yescharacteristic, complemented, Sylow
99#1631—1971nomaximal, Sylow
100#550—1971no—
101#492—1971no—
102#477—1971no—
103#476—1971no—
104#331D19711yescharacteristic, complemented
105Q8⋊C8—1971no—
106D4⋊C8—1971no—
107C2×#41—3941no—
108C2×SD16—3942no—
109C4⋊Q8—1971no—
110#127—3941no—
111#127C2×D19711yescharacteristic
112#121—3942no—
113#121—3941no—
114C4×Q8—1971no—
115C4×D4—3941no—
116C4×D4—1971no—
117#107—3941no—
118#106—3941no—
119C4⋊C8C2×D19711yescharacteristic
120C4⋊C8C2×D19711yescharacteristic
121#103—3941no—
122#102—3941no—
123C4×C8—1971no—
124#41—7882no—
125#41—7882no—
126C2×Q8—3941no—
127C2×Q8—1971no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—3942no—
131C2×D4—3942no—
132C2×D4—1971no—
133C2×D4D78812yes—
134C22×C4—3942no—
135C22×C4—3941no—
136SD16—7884no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—3941no—
140C4⋊C4—3941no—
141C4⋊C4—3941no—
142C4⋊C4—3941no—
143C4⋊C4—1971no—
144C4⋊C4—1971no—
145C22⋊C4—3942no—
146C22⋊C4—3941no—
147C42—3941no—
148C42C22×D19711yescharacteristic
149C23—22no—
150C23—3941no—
151Q8—7882no—
152Q8—3941no—
153Q8—1972no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—7884no—
160D4—15762no—
161D4—3941no—
162D4—3941no—
163C2×C4—7884no—
164C2×C4—7881no—
165C2×C4—7881no—
166C2×C4—7881no—
167C2×C4—3941no—
168C2×C4—3941no—
169C2×C4—3941no—
170C2×C4D4×D19711yescharacteristic
171C2×C4D4×D19711yescharacteristic
172C2×C4C2×D78811yescharacteristic
173C8—41no—
174C8—41no—
175C8—3942no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7881no—
180C22—7881no—
181C22—7881no—
182C22#2026329611yescharacteristic
183C4—21no—
184C4—7881no—
185C4—7881no—
186C4—15761no—
187C4—3942no—
188C4SD16×D19711yescharacteristic
189C4#2026334511yescharacteristic
190C4#2026335011yescharacteristic
191C4#2026335511yescharacteristic
192C2—82no—
193C2—15761no—
194C2#2716123111yescharacteristic
195C2#2716124211yescharacteristic
196C2#2716124911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197#120
prime radical#27161749C2
hypercenter#331D197
hypocenterC197#1631
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27161451C2semidirect
3. #27161375C2semidirect
4. #27161370C2semidirect
5. #27160809C2semidirect
6. #27160806C2semidirect
7. #331×C197C2semidirect
53. Dic394SD16semidirect
98. C197#1631semidirect
104. #331D197semidirect

Maximal subgroups

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

subgroupindex
#271617492
#271614512
#271613752
#271613702
#271608092
#271608062
#331×C1972
#1631197