Finite Group Databasebeta

← #38752244

Subgroups — #38752244

Order 25216.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15691348C211yesmaximal, characteristic, complemented
2#27161650C211yesmaximal, characteristic
3#27161557C211yesmaximal, characteristic, complemented
4#27161299C214yesmaximal, complemented
5#27161287C214yesmaximal, complemented
6#27161107C211yesmaximal, characteristic, complemented
7#27161089C211yesmaximal, characteristic, complemented
8#27161075C211yesmaximal, characteristic
9C2×#137×C197C211yesmaximal, characteristic
10C22×Dic394C2211yescharacteristic
11C2×#15691348—24no—
12C2×#15691348C2214yes—
13C22×#13096802C2211yescharacteristic
14#20263324C2212yes—
15#20263313C2212yes—
16#20263307—22no—
17#20263305—22no—
18Q8×Dic197—22no—
19Q8×Dic197C2214yes—
20#20263285—22no—
21#20263285C2211yescharacteristic
22#20263285C2211yescharacteristic
23#20263255C2214yes—
24#20263251C2214yes—
25#20263250C2212yes—
26#20263247C2212yes—
27C2×#41×C197C2211yescharacteristic
28C22×Q8×C197C2211yescharacteristic
29#137×C197—24no—
30C2×Q16×C197C2212yes—
31C2×SD16×C197C2212yes—
32C2×M16×C197C2211yescharacteristic
33C2×Dic394—21no—
34C2×Dic394—22no—
35C2×Dic394—22no—
36C2×Dic394C2312yes—
37C22×Dic197—21no—
38#15691348—28no—
39C2×#13096802—22no—
40C2×#13096802C2312yes—
41C4⋊Dic197—24no—
42C4⋊Dic197D412yes—
43C4⋊Dic197C2312yes—
44#15691337—24no—
45C22⋊Dic197—24no—
46C22⋊Dic197—21no—
47C22⋊Dic197D412yes—
48C4×Dic197—24no—
49C4×Dic197—21no—
50C4×Dic197C2312yes—
51#41×C197—22no—
52#41×C197—22no—
53C2×Q8×C197—21no—
54C2×Q8×C197—22no—
55C2×Q8×C197D414yes—
56C2×Q8×C197C2311yescharacteristic
57C2×Q8×C197C2312yes—
58C2×D4×C197—21no—
59C2×D4×C197C2311yescharacteristic
60C22×C4×C197—21no—
61C22×C4×C197—21no—
62C22×C4×C197C2311yescharacteristic
63Q16×C197—24no—
64SD16×C197—24no—
65M16×C197—22no—
66C2×C8×C197C2312yes—
67Dic394—22no—
68Dic394—22no—
69Dic394—42no—
70C2×Dic197—22no—
71C2×Dic197—22no—
72C2×Dic197—42no—
73C2×Dic197C2×D414yes—
74#13096802—22no—
75C23×C197—21no—
76C23×C197C2×D411yescharacteristic
77Q8×C197—24no—
78Q8×C197—22no—
79Q8×C197—21no—
80Q8×C197—21no—
81Q8×C197C2×D414yes—
82D4×C197—21no—
83D4×C197—21no—
84D4×C197—42no—
85C2×C4×C197—24no—
86C2×C4×C197—22no—
87C2×C4×C197—21no—
88C2×C4×C197—42no—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197C2×D412yes—
91C2×C4×C197#4112yes—
92C2×C4×C197C2411yescharacteristic
93C8×C197—22no—
94C22×C197—22no—
95C22×C197—41no—
96C22×C197—42no—
97C22×C197#12112yes—
98C22×C197C22×D411yescharacteristic
99C4×C197—24no—
100C4×C197—41no—
101C4×C197#12112yes—
102C4×C197C22×D411yescharacteristic
103C4×C197C2×#4111yescharacteristic
104Dic197—24no—
105Dic197—42no—
106Dic197—42no—
107C2×C197—22no—
108C2×C197—81no—
109C2×C197C2×#12111yescharacteristic
110C2×C197#57812yes—
111C197#304411yescharacteristic, complemented, Sylow
112#3044—1971nomaximal, Sylow
113C2×#137D19711yescharacteristic
114C22×Q16—1971no—
115#536—1971no—
116#519—1971no—
117#464—1974no—
118#462—1974no—
119#424—1971no—
120#419—1971no—
121#416—1971no—
122C2×#41C2×D19711yescharacteristic
123C22×Q8—1971no—
124C22×Q8C2×D19711yescharacteristic
125#137—24no—
126C2×Q16—3944no—
127C2×Q16—1974no—
128C2×Q16C2×D19712yes—
129C2×SD16C2×D19712yes—
130C2×M16C2×D19711yescharacteristic
131C22×C8—1971no—
132C4⋊Q8—1972no—
133#124—1972no—
134#123—3942no—
135C22⋊Q8—3942no—
136C4×Q8—3942no—
137C4×Q8—1974no—
138#117—3942no—
139#117—1971no—
140#117—1971no—
141C4⋊C8—1974no—
142#103—1972no—
143#103—1974no—
144#102—1972no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3942no—
150C2×Q8—3942no—
151C2×Q8—3941no—
152C2×Q8—1972no—
153C2×Q8C22⋊D19714yes—
154C2×Q8C22×D19711yescharacteristic
155C2×Q8C22×D19712yes—
156C2×D4—21no—
157C2×D4C22×D19711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—3941no—
161C22×C4C22×D19711yescharacteristic
162Q16—24no—
163Q16—3948no—
164SD16—24no—
165M16—22no—
166C2×C8—3942no—
167C2×C8—1972no—
168C2×C8C22×D19712yes—
169C4⋊C4—3944no—
170C4⋊C4—3944no—
171C4⋊C4—1972no—
172C4⋊C4—1972no—
173C22⋊C4—3941no—
174C22⋊C4—3944no—
175C22⋊C4—1972no—
176C42—3944no—
177C42—3941no—
178C42—1972no—
179C23—21no—
180C23D4×D19711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—7882no—
186Q8—3942no—
187Q8—3942no—
188Q8C2×C22⋊D19714yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—7882no—
197C2×C4—3942no—
198C2×C4—3942no—
199C2×C4—1974no—
200C2×C4D4×D19711yescharacteristic
201C2×C4C2×C22⋊D19712yes—
202C2×C4#1569135712yes—
203C2×C4C23×D19711yescharacteristic
204C8—22no—
205C8—3942no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2026329812yes—
210C22C2×D4×D19711yescharacteristic
211C4—24no—
212C4—41no—
213C4—7882no—
214C4—7882no—
215C4—3944no—
216C4#2026329812yes—
217C4C22×C22⋊D19711yescharacteristic
218C4C2×#1569135711yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2026329811yescharacteristic
222C2#2716189512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197C22×D4
prime radicalC2×#41×C197C22
hypercenterC2×#137D197
hypocenterC197#3044
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#15691348C2semidirect
3. #27161557C2semidirect
4. #27161299C2semidirect
5. #27161287C2semidirect
6. #27161107C2semidirect
7. #27161089C2semidirect
111. C197#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#156913482
#271616502
#271615572
#271612992
#271612872
#271611072
#271610892
#271610752
C2×#137×C1972
#3044197