Finite Group Databasebeta

← #38741913

Subgroups — #38741913

Order 25216.

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

normal subgroups only

This group has 250 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#27161425C211yesmaximal, characteristic, complemented
3#27161365C211yesmaximal, characteristic, complemented
4#27161285C211yesmaximal, characteristic, complemented
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#331×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691347—21no—
10#20263326C2211yescharacteristic
11#20263322—21no—
12#20263298—22no—
13#20263297—21no—
14C4×Dic394C2211yescharacteristic
15C4×D788C2211yescharacteristic
16D4×Dic197—21no—
17#20263258—21no—
18#20263251—21no—
19#20263251—21no—
20#20263247—22no—
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#15691347—42no—
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—41no—
53Dic394—41no—
54Dic394SD1612yescomplemented
55D788—21no—
56D788—21no—
57C22⋊D197—44no—
58C22⋊D197—82no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—44no—
63C2×Dic197—41no—
64C4×D197—41no—
65C4×D197—41no—
66#13096802—41no—
67C23×C197—22no—
68D4×C197—42no—
69D4×C197—41no—
70D4×C197—41no—
71D4×C197—41no—
72D4×C197—41no—
73C2×C4×C197C2×D411yescharacteristic
74C2×C4×C197C2×D411yescharacteristic
75C2×C4×C197C2×D411yescharacteristic
76C8×C197—41no—
77C8×C197—41no—
78C22×C197—42no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197#12011yescharacteristic
82C2×D197—41no—
83C2×D197—41no—
84C2×D197—41no—
85C4×C197—21no—
86C4×C197C2×SD1611yescharacteristic
87C4×C197#13511yescharacteristic
88C4×C197AGL(1,ℤ/8)11yescharacteristic
89C4×C197AGL(1,ℤ/8)11yescharacteristic
90Dic197—22no—
91Dic197—41no—
92Dic197—41no—
93Dic197—81no—
94C2×C197—82no—
95C2×C197#44911yescharacteristic
96C2×C197#45011yescharacteristic
97C2×C197#45511yescharacteristic
98D197—81no—
99C197#164211yescharacteristic, complemented, Sylow
100#1642—1971nomaximal, Sylow
101#550—1971no—
102#486—1971no—
103#475—1971no—
104#461—1971no—
105#331D19711yescharacteristic, complemented
106Q8⋊C8—1971no—
107D4⋊C8—1971no—
108C2×#41—3941no—
109C2×SD16—3941no—
110C4⋊Q8—1971no—
111#127—3941no—
112#127C2×D19711yescharacteristic
113#121—3942no—
114#121—3941no—
115C4×Q8—1971no—
116C4×D4—3941no—
117C4×D4—1971no—
118#106—3941no—
119C4⋊C8C2×D19711yescharacteristic
120C4⋊C8C2×D19711yescharacteristic
121#103—3941no—
122#103—3941no—
123#102—3942no—
124C4×C8—1971no—
125#41—7882no—
126#41—7882no—
127C2×Q8—3941no—
128C2×Q8—1971no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3942no—
132C2×D4—3942no—
133C2×D4—1971no—
134C2×D4D78812yes—
135C22×C4—3942no—
136C22×C4—3941no—
137SD16—7882no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3941no—
141C4⋊C4—3941no—
142C4⋊C4—3941no—
143C4⋊C4—3941no—
144C4⋊C4—1971no—
145C4⋊C4—1971no—
146C22⋊C4—3942no—
147C22⋊C4—3941no—
148C42—3941no—
149C42C22×D19711yescharacteristic
150C23—22no—
151C23—3941no—
152Q8—7881no—
153Q8—7881no—
154Q8—3941no—
155Q8—1972no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—7884no—
162D4—15762no—
163D4—3941no—
164D4—3941no—
165C2×C4—7884no—
166C2×C4—7881no—
167C2×C4—7881no—
168C2×C4—7881no—
169C2×C4—3941no—
170C2×C4—3941no—
171C2×C4—3941no—
172C2×C4D4×D19711yescharacteristic
173C2×C4D4×D19711yescharacteristic
174C2×C4C2×D78811yescharacteristic
175C8—41no—
176C8—41no—
177C8—7881no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7881no—
182C22—7881no—
183C22—7881no—
184C22#2026329611yescharacteristic
185C4—21no—
186C4—7881no—
187C4—7881no—
188C4—15761no—
189C4—3942no—
190C4SD16×D19711yescharacteristic
191C4#2026334711yescharacteristic
192C4#2026335011yescharacteristic
193C4#2026335011yescharacteristic
194C2—82no—
195C2—15761no—
196C2#2716122511yescharacteristic
197C2#2716123111yescharacteristic
198C2#2716125911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197#120
prime radical#27161749C2
hypercenter#331D197
hypocenterC197#1642
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. #27161425C2semidirect
3. #27161365C2semidirect
4. #27161285C2semidirect
5. #27160809C2semidirect
6. #27160806C2semidirect
7. #331×C197C2semidirect
54. Dic394SD16semidirect
99. C197#1642semidirect
105. #331D197semidirect

Maximal subgroups

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

subgroupindex
#271617492
#271614252
#271613652
#271612852
#271608092
#271608062
#331×C1972
#1642197