Finite Group Databasebeta

← #38742902

Subgroups — #38742902

Order 25216.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic197C211yesmaximal, characteristic
2C2×#20263260C212yesmaximal, complemented
3#27160939C211yesmaximal, characteristic
4#27160842C212yesmaximal, complemented
5C2×C22⋊C8×C197C211yesmaximal, characteristic
6C23×Dic197—21no—
7C2×C4⋊Dic197—24no—
8C2×C4⋊Dic197—21no—
9C2×C4⋊Dic197C2211yescharacteristic
10C2×C4⋊Dic197C2211yescharacteristic
11C2×C4⋊Dic197C2212yes—
12C2×C22⋊Dic197—21no—
13#20263260—24no—
14#20263233—21no—
15C23×C4×C197C2211yescharacteristic
16C22×C8×C197C2212yes—
17C22⋊C8×C197C414yescomplemented
18C22×Dic197—24no—
19C22×Dic197—22no—
20C22×Dic197—21no—
21C22×Dic197—21no—
22C22×Dic197—42no—
23C22×Dic197—42no—
24C4⋊Dic197—24no—
25C4⋊Dic197—22no—
26C4⋊Dic197—22no—
27C4⋊Dic197—22no—
28C4⋊Dic197D414yes—
29C22⋊Dic197—42no—
30C24×C197D411yescharacteristic
31C22×C4×C197—22no—
32C22×C4×C197—22no—
33C22×C4×C197C2×C412yes—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197Q812yes—
36C22×C4×C197C2311yescharacteristic
37C2×C8×C197—22no—
38C2×C8×C197—22no—
39C2×C8×C197C2×C414yescomplemented
40C2×Dic197—24no—
41C2×Dic197—48no—
42C2×Dic197—42no—
43C2×Dic197—42no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46C2×Dic197—41no—
47C2×Dic197—41no—
48C2×Dic197—41no—
49C2×Dic197—41no—
50C23×C197—21no—
51C23×C197—21no—
52C23×C197—21no—
53C23×C197—21no—
54C23×C197C4⋊C412yes—
55C23×C197D812yes—
56C23×C197Q1612yes—
57C23×C197C2×D411yescharacteristic
58C2×C4×C197—21no—
59C2×C4×C197—21no—
60C2×C4×C197—24no—
61C2×C4×C197—24no—
62C2×C4×C197C4⋊C412yes—
63C2×C4×C197C22×C411yescharacteristic
64C2×C4×C197C2×D412yes—
65C2×C4×C197C2×Q811yescharacteristic
66C2×C4×C197#4112yes—
67C8×C197—24no—
68C22×C197—22no—
69C22×C197—22no—
70C22×C197—22no—
71C22×C197—22no—
72C22×C197—22no—
73C22×C197—22no—
74C22×C197#10714yescomplemented
75C22×C197C2×C4⋊C411yescharacteristic
76C22×C197#12011yescharacteristic
77C22×C197#12311yescharacteristic
78C22×C197C2×D811yescharacteristic
79C22×C197C2×Q1611yescharacteristic
80C22×C197AGL(1,ℤ/8)11yescharacteristic
81C22×C197#13711yescharacteristic
82C4×C197—22no—
83C4×C197C4×D412yes—
84C4×C197C22⋊Q812yes—
85Dic197—44no—
86Dic197—81no—
87Dic197—81no—
88C2×C197—24no—
89C2×C197#38511yescharacteristic
90C2×C197C2×#10711yescharacteristic
91C2×C197#42811yescharacteristic
92C2×C197#44711yescharacteristic
93C2×C197#45111yescharacteristic
94C2×C197#48011yescharacteristic
95C2×C197#48411yescharacteristic
96C197#185211yescharacteristic, complemented, Sylow
97#1852—1971nomaximal, Sylow
98C22×C4⋊C4—1971no—
99C2×#107—1972no—
100C2×C22⋊C8D19711yescharacteristic
101#380—1971no—
102#340—1972no—
103C23×C4—3941no—
104C23×C4C2×D19711yescharacteristic
105C22×C8C2×D19712yes—
106C2×C4⋊C4—3941no—
107C2×C4⋊C4—3944no—
108C2×C4⋊C4—1971no—
109C2×C4⋊C4—1971no—
110C2×C4⋊C4—1972no—
111C2×C22⋊C4—3941no—
112#107—3944no—
113C22⋊C8Dic19714yescomplemented
114#95—3941no—
115C24D78811yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—7882no—
119C22×C4—7882no—
120C22×C4—3941no—
121C22×C4—3941no—
122C22×C4—3944no—
123C22×C4—3942no—
124C22×C4C2×Dic19712yes—
125C22×C4D78811yescharacteristic
126C22×C4Dic39412yes—
127C22×C4C22×D19711yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic19714yescomplemented
131C4⋊C4—3944no—
132C4⋊C4—3942no—
133C4⋊C4—3942no—
134C4⋊C4—3942no—
135C4⋊C4—1974no—
136C22⋊C4—7882no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic19712yes—
142C23D157612yes—
143C23Dic78812yes—
144C23C2×D78811yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—7888no—
150C2×C4—7882no—
151C2×C4—7882no—
152C2×C4—7881no—
153C2×C4—7881no—
154C2×C4—7881no—
155C2×C4—7881no—
156C2×C4—7881no—
157C2×C4—7881no—
158C2×C4—3944no—
159C2×C4C4⋊Dic19712yes—
160C2×C4C22×Dic19711yescharacteristic
161C2×C4D4×D19712yes—
162C2×C4C2×Dic39411yescharacteristic
163C2×C4#1569135712yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2026326014yescomplemented
172C22C2×C4⋊Dic19711yescharacteristic
173C22#2026329611yescharacteristic
174C22#2026331011yescharacteristic
175C22C2×D157611yescharacteristic
176C22C2×Dic78811yescharacteristic
177C22#2026335411yescharacteristic
178C22#2026335911yescharacteristic
179C4—22no—
180C4—7884no—
181C4—15761no—
182C4—15761no—
183C4D4×Dic19712yes—
184C4#2026330312yes—
185C2—24no—
186C2#2716095411yescharacteristic
187C2C2×#2026326011yescharacteristic
188C2#2716112111yescharacteristic
189C2#2716121511yescharacteristic
190C2#2716123711yescharacteristic
191C2#2716139311yescharacteristic
192C2#2716141511yescharacteristic
subgroupG/H
centerC23C2×D788
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC2×C22⋊C8×C197C2
socleC23×C197C2×D4
prime radicalC24×C197D4
cosocleC22×C4×C197C23
hypercenterC2×C22⋊C8D197
hypocenterC197#1852
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#20263260C2semidirect
4. #27160842C2semidirect
17. C22⋊C8×C197C4semidirect
39. C2×C8×C197C2×C4semidirect
74. C22×C197#107semidirect
96. C197#1852semidirect
113. C22⋊C8Dic197semidirect
130. C2×C8C2×Dic197semidirect
171. C22#20263260semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1972
C2×#202632602
#271609392
#271608422
C2×C22⋊C8×C1972
#1852197