Finite Group Databasebeta

← #38741948

Subgroups — #38741948

Order 25216.

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

normal subgroups only

This group has 240 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#27161732C211yesmaximal, characteristic
2#27161482C211yesmaximal, characteristic, complemented
3#27161360C211yesmaximal, characteristic, complemented
4#27161285C211yesmaximal, characteristic, complemented
5#27160806C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#332×C197C211yesmaximal, characteristic, complemented
8C2×#15691359—21no—
9C2×#15691347—21no—
10#20263326C2211yescharacteristic
11#20263311—22no—
12#20263302—21no—
13#20263301—21no—
14C4×D788C2211yescharacteristic
15C4×D788C2211yescharacteristic
16C4⋊C4×D197—21no—
17#20263260—21no—
18#20263259—22no—
19#20263251—21no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691359—42no—
26#15691359—42no—
27C2×Dic394—21no—
28C2×D788—22no—
29C2×D788D411yescharacteristic
30C2×D788D411yescharacteristic
31C2×C4×D197—22no—
32C2×C4×D197—21no—
33C2×C4×D197—21no—
34#15691347—42no—
35C2×#13096802—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39C4⋊Dic197D411yescharacteristic
40#15691337—22no—
41#15691335—22no—
42#15691335—21no—
43#15691335—21no—
44C2×Q8×C197—21no—
45C2×C8×C197—21no—
46C2×C8×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197D412yes—
49C42×C197C2311yescharacteristic
50C22×D197—21no—
51C22×D197—21no—
52Dic394—41no—
53Dic394—41no—
54D788—21no—
55D788—21no—
56D788—21no—
57D788—44no—
58D788D812yes—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—41no—
62C4×D197—44no—
63C4×D197—41no—
64C4×D197—41no—
65C4×D197—41no—
66C4×D197—41no—
67#13096802—22no—
68Q8×C197—41no—
69Q8×C197—41no—
70C2×C4×C197—22no—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C8×C197—41no—
75C8×C197—41no—
76C22×C197#12011yescharacteristic
77C2×D197—22no—
78C2×D197—41no—
79C2×D197—41no—
80C2×D197—41no—
81C2×D197—41no—
82C2×D197—41no—
83C4×C197—21no—
84C4×C197—42no—
85C4×C197C2×D811yescharacteristic
86C4×C197#13511yescharacteristic
87C4×C197AGL(1,ℤ/8)11yescharacteristic
88C4×C197#13711yescharacteristic
89Dic197—41no—
90Dic197—41no—
91Dic197—81no—
92C2×C197#44711yescharacteristic
93C2×C197#45211yescharacteristic
94C2×C197#45611yescharacteristic
95D197—42no—
96D197—81no—
97C197#164711yescharacteristic, complemented, Sylow
98#1647—1971nomaximal, Sylow
99#547—1971no—
100#500—1971no—
101#474—1971no—
102#461—1971no—
103#332D19711yescharacteristic, complemented
104D4⋊C8—1971no—
105D4⋊C8—1971no—
106C2×#41—3941no—
107C2×SD16—3941no—
108C4⋊Q8—1971no—
109C4⋊Q8C2×D19711yescharacteristic
110#123—3942no—
111C22⋊Q8—3941no—
112#121—3941no—
113C4×D4—1971no—
114C4×D4—1971no—
115C2×C4⋊C4—3941no—
116#107—3941no—
117#107—3942no—
118C4⋊C8C2×D19711yescharacteristic
119C4⋊C8C2×D19711yescharacteristic
120#103—3941no—
121#102—3941no—
122C4×C8—1971no—
123#41—7882no—
124#41—7882no—
125C2×Q8—21no—
126C2×Q8—3941no—
127C2×D4—3942no—
128C2×D4—1971no—
129C2×D4—1971no—
130C22×C4—3942no—
131C22×C4—3941no—
132C22×C4—3941no—
133SD16—7882no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—3941no—
137C4⋊C4—21no—
138C4⋊C4—3942no—
139C4⋊C4—3941no—
140C4⋊C4—3941no—
141C4⋊C4—1971no—
142C4⋊C4—1971no—
143C4⋊C4D78812yes—
144C22⋊C4—3942no—
145C22⋊C4—3941no—
146C22⋊C4—3941no—
147C42C22×D19711yescharacteristic
148C23—3941no—
149C23—3941no—
150Q8—41no—
151Q8—41no—
152Q8—7881no—
153Q8—7881no—
154D4—7884no—
155D4—3941no—
156D4—3941no—
157D4—3941no—
158D4—1972no—
159C2×C4—22no—
160C2×C4—7884no—
161C2×C4—7881no—
162C2×C4—7881no—
163C2×C4—7881no—
164C2×C4—7881no—
165C2×C4—7881no—
166C2×C4—3941no—
167C2×C4—3941no—
168C2×C4D4×D19711yescharacteristic
169C2×C4D4×D19711yescharacteristic
170C2×C4C2×D78811yescharacteristic
171C8—41no—
172C8—41no—
173C8—3942no—
174C22—7881no—
175C22—7881no—
176C22—7881no—
177C22—7881no—
178C22—7881no—
179C22—3942no—
180C22#2026329611yescharacteristic
181C4—21no—
182C4—42no—
183C4—7881no—
184C4—7881no—
185C4—15761no—
186C4D8×D19711yescharacteristic
187C4#2026334711yescharacteristic
188C4#2026335311yescharacteristic
189C4#2026335811yescharacteristic
190C2—7882no—
191C2—15761no—
192C2#2716121411yescharacteristic
193C2#2716124411yescharacteristic
194C2#2716126411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
prime radical#27161732C2
hypercenter#332D197
hypocenterC197#1647
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27161482C2semidirect
3. #27161360C2semidirect
4. #27161285C2semidirect
5. #27160806C2semidirect
6. #27160806C2semidirect
7. #332×C197C2semidirect
97. C197#1647semidirect
103. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271617322
#271614822
#271613602
#271612852
#271608062
#332×C1972
#1647197