Finite Group Databasebeta

← #38743753

Subgroups — #38743753

Order 25216.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#20263352C212yesmaximal, complemented
2#27161660C211yesmaximal, characteristic, complemented
3C2×#20263254C212yesmaximal, complemented
4#27160848C211yesmaximal, characteristic, complemented
5#390×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2212yes—
7#20263352—24no—
8#20263352—24no—
9C2×#15691346—22no—
10C2×#15691343—22no—
11C2×#15691342C2212yes—
12#20263306—22no—
13#20263301—22no—
14#20263300—22no—
15#20263254—24no—
16C22×D4×C197C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C2×C22⋊C4×C197—22no—
19C2×C42×C197C2211yescharacteristic
20#15691358—22no—
21#15691358—22no—
22#15691358—22no—
23#15691358—22no—
24C2×Dic394D412yes—
25C2×D788—22no—
26C2×D788D412yes—
27C2×C22⋊D197—22no—
28C2×C4×D197—22no—
29#15691346—28no—
30#15691343—28no—
31#15691342—22no—
32#15691342D414yescomplemented
33C2×#13096802—22no—
34C4⋊Dic197—22no—
35#15691335—24no—
36C24×C197—21no—
37C2×D4×C197—22no—
38C2×D4×C197—22no—
39C2×D4×C197—21no—
40C2×D4×C197D412yescomplemented
41C22×C4×C197—21no—
42C22×C4×C197D412yes—
43C22×C4×C197C2311yescharacteristic
44C4⋊C4×C197—22no—
45C22⋊C4×C197—44no—
46C42×C197—22no—
47C22×D197—22no—
48Dic394—22no—
49Dic394—22no—
50D788—22no—
51D788—22no—
52D788—82no—
53C22⋊D197—42no—
54C22⋊D197—42no—
55C2×Dic197—22no—
56C4×D197—42no—
57C4×D197—42no—
58#13096802—24no—
59C23×C197—22no—
60C23×C197—42no—
61C23×C197—42no—
62C23×C197—41no—
63C23×C197—41no—
64C23×C197C2×D411yescharacteristic
65D4×C197—24no—
66D4×C197—42no—
67C2×C4×C197—22no—
68C2×C4×C197—22no—
69C2×C4×C197—41no—
70C2×C4×C197—41no—
71C2×C4×C197—42no—
72C2×C4×C197C2×D412yes—
73C2×C4×C197C2×D412yes—
74C2×C4×C197C2×D411yescharacteristic
75C2×C4×C197#4111yescharacteristic
76C22×C197—21no—
77C22×C197—21no—
78C22×C197—44no—
79C22×C197—42no—
80C22×C197—82no—
81C22×C197—81no—
82C22×C197—81no—
83C22×C197#12011yescharacteristic
84C22×C197#12111yescharacteristic
85C22×C197#12711yescharacteristic
86C2×D197—42no—
87C2×D197—42no—
88C2×D197—42no—
89C4×C197—42no—
90C4×C197—42no—
91C4×C197#12012yes—
92C4×C197#12112yes—
93Dic197—42no—
94C2×C197—21no—
95C2×C197—21no—
96C2×C197—82no—
97C2×C197#39211yescharacteristic
98C2×C197#45311yescharacteristic
99C2×C197#45411yescharacteristic
100D197—82no—
101C197#200611yescharacteristic, complemented, Sylow
102#2006—1971nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1972no—
104#537—1971no—
105C2×#104—1972no—
106#390D19711yescharacteristic, complemented
107#343—1971no—
108C2×#41—1972no—
109C22×D4C2×D19711yescharacteristic
110AGL(1,ℤ/8)—3944no—
111AGL(1,ℤ/8)—3944no—
112C2×SD16—3942no—
113C2×D8—3942no—
114C2×M16—1972no—
115C22⋊Q8—3942no—
116#121—3942no—
117#121—3942no—
118C2×C4⋊C4C2×D19711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D19711yescharacteristic
121#104—3944no—
122C24—21no—
123#41—3942no—
124#41—3942no—
125#41—3942no—
126#41—3942no—
127C2×Q8—1972no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—3942no—
132C2×D4—3942no—
133C2×D4—1972no—
134C2×D4D78812yescomplemented
135C22×C4—21no—
136C22×C4—3942no—
137C22×C4C22⋊D19712yes—
138C22×C4C22×D19711yescharacteristic
139SD16—3948no—
140D8—3948no—
141M16—3942no—
142M16—1974no—
143C2×C8—3942no—
144C4⋊C4—22no—
145C4⋊C4—3942no—
146C22⋊C4—44no—
147C22⋊C4—3944no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—3942no—
155C23C2×C22⋊D19711yescharacteristic
156Q8—3942no—
157Q8—3942no—
158D4—24no—
159D4—42no—
160D4—7882no—
161D4—7882no—
162D4—15762no—
163D4—3942no—
164D4—3942no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—7882no—
171C2×C4—7882no—
172C2×C4—3942no—
173C2×C4D4×D19712yes—
174C2×C4D4×D19712yes—
175C2×C4C2×D78811yescharacteristic
176C2×C4#1569135811yescharacteristic
177C8—3944no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—7882no—
186C22—7882no—
187C22—7882no—
188C22#2026329411yescharacteristic
189C22#2026330011yescharacteristic
190C22#2026332211yescharacteristic
191C4—42no—
192C4—42no—
193C4—7882no—
194C4#2026329612yes—
195C4#2026329912yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—15762no—
200C2#2716098811yescharacteristic
201C2#2716124611yescharacteristic
202C2#2716125111yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#390×C197C2
socleC22×C197#120
hypercenter#390D197
hypocenterC197#2006
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. #27161660C2semidirect
3. C2×#20263254C2semidirect
4. #27160848C2semidirect
5. #390×C197C2semidirect
32. #15691342D4semidirect
40. C2×D4×C197D4semidirect
101. C197#2006semidirect
106. #390D197semidirect
134. C2×D4D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
#271616602
C2×#202632542
#271608482
#390×C1972
#2006197