Finite Group Databasebeta

← #38743233

Subgroups — #38743233

Order 25216.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#27161845C211yesmaximal, characteristic
2#27161655C211yesmaximal, characteristic
3#27161089C211yesmaximal, characteristic, complemented
4#27161087C211yesmaximal, characteristic, complemented
5#27161052C211yesmaximal, characteristic
6#27160835C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic
9C22×Dic394C2211yescharacteristic
10#20263334—22no—
11#20263332—22no—
12C2×#15691342C2211yescharacteristic
13M16×D197—22no—
14#20263326C412yescomplemented
15#20263314C412yescomplemented
16#20263310—22no—
17#20263306—22no—
18#20263285C2211yescharacteristic
19#20263251—22no—
20#20263249—22no—
21#20263240—22no—
22C2×M16×C197C2211yescharacteristic
23#117×C197C2211yescharacteristic
24#117×C197C2211yescharacteristic
25#15691358—21no—
26#15691358—21no—
27#15691358—21no—
28#15691358—21no—
29C2×Dic394—21no—
30C2×Dic394—21no—
31C2×Dic394—21no—
32C2×Dic394—21no—
33C2×Dic394—21no—
34C2×Dic394C2×C412yes—
35C2×Dic394C2×C411yescharacteristic
36C2×D788C2×C411yescharacteristic
37C2×C22⋊D197D412yes—
38C22×Dic197—21no—
39C2×C4×D197D412yes—
40#15691342—21no—
41#15691342—21no—
42#15691341—24no—
43C2×#13096802D412yes—
44C8×D197—24no—
45C4⋊Dic197—24no—
46C4⋊Dic197—21no—
47C22⋊Dic197—21no—
48#15691335—24no—
49C4×Dic197—21no—
50C22×C4×C197C2311yescharacteristic
51M16×C197—21no—
52M16×C197—21no—
53C2×C8×C197D412yes—
54C4⋊C4×C197—21no—
55C4⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—21no—
58C42×C197—21no—
59C42×C197C2×C412yes—
60C22×D197C22⋊C412yes—
61Dic394—21no—
62Dic394—21no—
63Dic394—22no—
64Dic394—22no—
65Dic394—41no—
66Dic394—41no—
67D788—21no—
68D788—21no—
69C22⋊D197—22no—
70C22⋊D197—22no—
71C2×Dic197—22no—
72C2×Dic197—41no—
73C2×Dic197—41no—
74C2×Dic197—41no—
75C2×Dic197C22⋊C412yes—
76C4×D197—22no—
77C4×D197—22no—
78#13096802—22no—
79C23×C197C2×D411yescharacteristic
80C2×C4×C197—22no—
81C2×C4×C197—41no—
82C2×C4×C197C22×C411yescharacteristic
83C2×C4×C197C2×D411yescharacteristic
84C2×C4×C197C2×D411yescharacteristic
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197#4111yescharacteristic
87C2×C4×C197#4111yescharacteristic
88C8×C197—22no—
89C22×C197—41no—
90C22×C197C2×C22⋊C411yescharacteristic
91C22×C197#12111yescharacteristic
92C22×C197#12111yescharacteristic
93C2×D197—22no—
94C2×D197—42no—
95C4×C197—42no—
96C4×C197—81no—
97C4×C197C4×D411yescharacteristic
98C4×C197C4×D411yescharacteristic
99C4×C197#12011yescharacteristic
100C4×C197#12311yescharacteristic
101Dic197—22no—
102Dic197—42no—
103Dic197—81no—
104C2×C197—21no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197#38611yescharacteristic
108D197—42no—
109C197#190911yescharacteristic, complemented, Sylow
110#1909—1971nomaximal, Sylow
111#568—1971no—
112#536—1971no—
113#419—1971no—
114#419—1971no—
115#410—1971no—
116H(3,ℤ/4)—1971no—
117H(3,ℤ/4)D19711yescharacteristic, complemented
118C2×#41—1971no—
119C22×Q8—1971no—
120#131—3942no—
121#131—3942no—
122C2×M16—3942no—
123C2×M16—1971no—
124C2×M16C2×D19711yescharacteristic
125C4⋊Q8—1972no—
126#124—1972no—
127#123—3942no—
128C22⋊Q8—3942no—
129#117—1971no—
130#117C2×D19711yescharacteristic
131#117C2×D19711yescharacteristic
132#103—3942no—
133#103—3942no—
134#99—3942no—
135#41—3941no—
136#41—3941no—
137#41—3941no—
138#41—3941no—
139C2×Q8—3941no—
140C2×Q8—3941no—
141C2×Q8—3941no—
142C2×Q8—3941no—
143C2×Q8—3941no—
144C2×Q8—1972no—
145C2×Q8—1971no—
146C2×D4—1972no—
147C2×D4—1971no—
148C22×C4—3941no—
149C22×C4—1972no—
150C22×C4C22×D19711yescharacteristic
151M16—21no—
152M16—21no—
153M16—3944no—
154M16—3941no—
155M16—3941no—
156C2×C8—3944no—
157C2×C8—1972no—
158C2×C8D78812yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3944no—
162C4⋊C4—3941no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—3944no—
166C22⋊C4—3941no—
167C42—21no—
168C42—3941no—
169C42C4×D19712yes—
170C23—1972no—
171C23D4×D19711yescharacteristic
172Q8—7881no—
173Q8—7881no—
174Q8—3942no—
175Q8—3942no—
176Q8—3941no—
177Q8—3941no—
178D4—3942no—
179D4—3942no—
180D4—3941no—
181D4—3941no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—7881no—
185C2×C4—7881no—
186C2×C4—7881no—
187C2×C4—3942no—
188C2×C4—3942no—
189C2×C4—3942no—
190C2×C4—1972no—
191C2×C4C2×C4×D19711yescharacteristic
192C2×C4D4×D19711yescharacteristic
193C2×C4D4×D19711yescharacteristic
194C2×C4C2×D78811yescharacteristic
195C2×C4#1569135811yescharacteristic
196C2×C4#1569135911yescharacteristic
197C8—22no—
198C8—3942no—
199C22—41no—
200C22—7882no—
201C22—3942no—
202C22C22⋊C4×D19711yescharacteristic
203C22#2026329911yescharacteristic
204C22#2026330111yescharacteristic
205C4—42no—
206C4—81no—
207C4—7882no—
208C4—15761no—
209C4—3942no—
210C4#2026328811yescharacteristic
211C4C4×D78811yescharacteristic
212C4#2026329611yescharacteristic
213C4#2026331111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—7882no—
218C2#2716095711yescharacteristic
subgroupG/H
centerC2#27160957
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingH(3,ℤ/4)×C197C2
socleC2×C197#386
prime radicalC2×#15691358C22
cosocleC22×C4×C197C23
hypercenterH(3,ℤ/4)D197
hypocenterC197#1909
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #27161089C2semidirect
4. #27161087C2semidirect
6. #27160835C2semidirect
7. H(3,ℤ/4)×C197C2semidirect
14. #20263326C4semidirect
15. #20263314C4semidirect
109. C197#1909semidirect
117. H(3,ℤ/4)D197semidirect

Maximal subgroups

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

subgroupindex
#271618452
#271616552
#271610892
#271610872
#271610522
#271608352
H(3,ℤ/4)×C1972
#1909197