Finite Group Databasebeta

← #38742673

Subgroups — #38742673

Order 25216.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20263374C211yesmaximal, characteristic
2C2×#20263252C212yesmaximal, complemented
3#27161081C212yesmaximal, complemented
4#27160986C211yesmaximal, characteristic, complemented
5C2×#101×C197C211yesmaximal, characteristic, complemented
6#20263374—24no—
7#20263374—22no—
8#20263374C414yescomplemented
9C2×#15691359—22no—
10C2×#15691358—22no—
11C2×#15691358C2212yes—
12C2×Q8×D197—22no—
13C2×C4⋊Dic197C2211yescharacteristic
14C2×C4×Dic197C2211yescharacteristic
15#20263252—24no—
16#20263251—22no—
17#20263248—22no—
18#20263233—22no—
19C22×Q8×C197C2211yescharacteristic
20C2×M16×C197C2212yes—
21#101×C197—22no—
22#15691359—28no—
23#15691359—28no—
24#15691358—28no—
25#15691358—28no—
26#15691358—22no—
27#15691358—22no—
28#15691358C2×C414yes—
29#15691358D414yes—
30C2×Dic394—22no—
31C2×Dic394D412yes—
32Q8×D197—28no—
33Q8×D197—28no—
34C2×D788—22no—
35C2×D788D412yes—
36C2×C22⋊D197—22no—
37C22×Dic197—21no—
38C22×Dic197—21no—
39C2×C4×D197—24no—
40C2×C4×D197—22no—
41C4⋊Dic197—22no—
42C4×Dic197—22no—
43C2×Q8×C197—22no—
44C2×Q8×C197—22no—
45C2×Q8×C197C2×C412yes—
46C2×Q8×C197D412yes—
47C22×C4×C197D412yes—
48C22×C4×C197C2311yescharacteristic
49M16×C197—22no—
50M16×C197—22no—
51C2×C8×C197—22no—
52C22×D197—22no—
53Dic394—28no—
54Dic394—22no—
55Dic394C22⋊C414yes—
56D788—28no—
57D788—22no—
58D788C22⋊C414yes—
59C22⋊D197—24no—
60C22⋊D197—24no—
61C2×Dic197—22no—
62C2×Dic197—22no—
63C2×Dic197—41no—
64C2×Dic197—41no—
65C2×Dic197—42no—
66C2×Dic197—41no—
67C4×D197—216no—
68C4×D197—24no—
69C4×D197—24no—
70C23×C197C2×D411yescharacteristic
71Q8×C197—24no—
72Q8×C197—24no—
73C2×C4×C197—22no—
74C2×C4×C197—22no—
75C2×C4×C197C22⋊C414yes—
76C2×C4×C197C22×C411yescharacteristic
77C2×C4×C197C2×D412yes—
78C2×C4×C197C2×D412yes—
79C2×C4×C197C2×D411yescharacteristic
80C8×C197—42no—
81C22×C197—21no—
82C22×C197—21no—
83C22×C197C2×C22⋊C411yescharacteristic
84C22×C197#12011yescharacteristic
85C22×C197#12011yescharacteristic
86C2×D197—24no—
87C2×D197—42no—
88C2×D197—42no—
89C4×C197—24no—
90C4×C197C2×C22⋊C412yes—
91C4×C197#12012yes—
92Dic197—24no—
93Dic197—42no—
94Dic197—81no—
95C2×C197—21no—
96C2×C197—21no—
97C2×C197#37911yescharacteristic
98C2×C197#45411yescharacteristic
99C2×C197#45611yescharacteristic
100D197—44no—
101C197#179311yescharacteristic, complemented, Sylow
102#1793—1971nomaximal, Sylow
103C2×2−1+4—1971no—
104C2×#104—1972no—
105#417—1972no—
106C2×#101D19711yescharacteristic, complemented
107#391—1971no—
1082−1+4—3944no—
1092−1+4—3942no—
1102−1+4—1974no—
111C2×#41—3942no—
112C2×#41—3942no—
113C2×#41—1972no—
114C22×Q8—3942no—
115C22×Q8C2×D19711yescharacteristic
116C2×M16C2×D19712yes—
117C2×C4⋊C4—1971no—
118C2×C42—1971no—
119#104—3944no—
120#103—3942no—
121#102—3942no—
122#101—22no—
123#95—3942no—
124#41—3948no—
125#41—3948no—
126#41—3948no—
127#41—3948no—
128#41—3942no—
129#41—3942no—
130#41—1974no—
131#41—1974no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—3942no—
135C2×Q8—3948no—
136C2×Q8—3948no—
137C2×Q8—1972no—
138C2×Q8C4×D19712yes—
139C2×Q8C22⋊D19712yes—
140C2×D4—3942no—
141C2×D4—3942no—
142C2×D4—1972no—
143C22×C4—3944no—
144C22×C4—3941no—
145C22×C4—3941no—
146C22×C4—3942no—
147C22×C4D78812yes—
148C22×C4C22×D19711yescharacteristic
149M16—22no—
150M16—22no—
151C2×C8—22no—
152C4⋊C4—3942no—
153C42—3942no—
154C23—3942no—
155C23C2×D78811yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—3948no—
159Q8—3942no—
160Q8—1974no—
161D4—3948no—
162D4—3944no—
163D4—3944no—
164D4—3942no—
165D4—1974no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—7881no—
169C2×C4—7881no—
170C2×C4—7882no—
171C2×C4—7881no—
172C2×C4—39416no—
173C2×C4—3942no—
174C2×C4—3942no—
175C2×C4—3944no—
176C2×C4—3944no—
177C2×C4#1569133514yes—
178C2×C4C2×C4×D19711yescharacteristic
179C2×C4D4×D19712yes—
180C2×C4D4×D19712yes—
181C2×C4C2×C22⋊D19711yescharacteristic
182C8—42no—
183C22—21no—
184C22—21no—
185C22—7882no—
186C22—7882no—
187C22—3944no—
188C22C2×#1569133511yescharacteristic
189C22#2026329611yescharacteristic
190C22#2026329611yescharacteristic
191C4—24no—
192C4—7882no—
193C4—15761no—
194C4—3944no—
195C4C22⋊C4×D19712yes—
196C4#2026329412yes—
197C2—21no—
198C2—21no—
199C2—7884no—
200C2#2716093411yescharacteristic
201C2#2716125411yescharacteristic
202C2#2716126411yescharacteristic
subgroupG/H
centerC22#20263296
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC2×#101×C197C2
socleC22×C197#120
prime radicalC2×#20263374C2
cosocleC22×C4×C197C23
hypercenterC2×#101D197
hypocenterC197#1793
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#20263252C2semidirect
3. #27161081C2semidirect
4. #27160986C2semidirect
5. C2×#101×C197C2semidirect
8. #20263374C4semidirect
101. C197#1793semidirect
106. C2×#101D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
C2×#202632522
#271610812
#271609862
C2×#101×C1972
#1793197