Finite Group Databasebeta

← #38741701

Subgroups — #38741701

Order 25216.

228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161704C211yesmaximal, characteristic
2#27161409C211yesmaximal, characteristic, complemented
3#27161005C211yesmaximal, characteristic
4#27160817C211yesmaximal, characteristic, complemented
5#27160816C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic
7#483×C197C211yesmaximal, characteristic, complemented
8#20263306—21no—
9#20263305C2211yescharacteristic
10#20263304—21no—
11#20263304—21no—
12#20263303—21no—
13#20263302—21no—
14C2×#15691335—21no—
15C2×#15691335C2211yescharacteristic
16C22⋊C4×D197—21no—
17#20263257—21no—
18#20263249—21no—
19C22⋊#13096802C2211yescharacteristic
20#20263233—21no—
21#20263233C2211yescharacteristic
22C22⋊Q8×C197C2211yescharacteristic
23C2×C4⋊C4×C197C2211yescharacteristic
24#106×C197—21no—
25#103×C197—21no—
26C22⋊C8×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394—21no—
29C2×Dic394D411yescharacteristic
30C22×Dic197—21no—
31C22×Dic197D411yescharacteristic
32C2×C4×D197—21no—
33C2×C4×D197—21no—
34C2×#13096802—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197D411yescharacteristic
37#15691337—21no—
38#15691337—21no—
39#15691337—21no—
40C22⋊Dic197—21no—
41#15691335—21no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—42no—
46C2×Q8×C197D411yescharacteristic
47C22×C4×C197—21no—
48C22×C4×C197C2311yescharacteristic
49C2×C8×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197D411yescharacteristic
54C22⋊C4×C197—21no—
55C22×D197—21no—
56C22×D197—21no—
57C22×D197—41no—
58C22×D197—41no—
59C22×D197SD1612yes—
60Dic394—41no—
61Dic394—81no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C4×D197—81no—
70C4×D197—81no—
71#13096802—41no—
72C23×C197C2×D411yescharacteristic
73Q8×C197—41no—
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C8×C197—41no—
83C22×C197—21no—
84C22×C197—21no—
85C22×C197#12011yescharacteristic
86C22×C197C2×SD1611yescharacteristic
87C22×C197#13711yescharacteristic
88C2×D197—22no—
89C2×D197—41no—
90C2×D197—41no—
91C2×D197—42no—
92C2×D197—81no—
93C4×C197—21no—
94C4×C197—21no—
95C4×C197—41no—
96C4×C197—41no—
97C4×C197—81no—
98Dic197—41no—
99Dic197—41no—
100Dic197—41no—
101Dic197—81no—
102C2×C197—21no—
103C2×C197—21no—
104C2×C197#44811yescharacteristic
105C2×C197#45511yescharacteristic
106C2×C197#45811yescharacteristic
107D197—42no—
108C197#160811yescharacteristic, complemented, Sylow
109#1608—1971nomaximal, Sylow
110#543—1971no—
111#483—1971no—
112#483D19711yescharacteristic, complemented
113#397—1971no—
114#328—1971no—
115#328—1971no—
116#323—1971no—
117C22⋊Q8—3941no—
118C22⋊Q8—3941no—
119C22⋊Q8—3941no—
120C22⋊Q8—3941no—
121C22⋊Q8—3941no—
122C22⋊Q8—1971no—
123C22⋊Q8C2×D19711yescharacteristic
124C2×C4⋊C4C2×D19711yescharacteristic
125C2×C22⋊C4—3941no—
126C2×C22⋊C4—3941no—
127C2×C22⋊C4—1971no—
128#106—21no—
129#106—3941no—
130#103—21no—
131#103—3941no—
132C22⋊C8—1971no—
133C22⋊C8C2×D19711yescharacteristic
134#95—3941no—
135#95—1971no—
136C24—1971no—
137C2×Q8—3941no—
138C2×Q8—1971no—
139C2×Q8C22⋊D19711yescharacteristic
140C22×C4—21no—
141C22×C4—3941no—
142C22×C4—3941no—
143C22×C4—3941no—
144C22×C4—1971no—
145C22×C4C22×D19711yescharacteristic
146C2×C8—21no—
147C2×C8—3941no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—3941no—
152C4⋊C4—3941no—
153C4⋊C4—3941no—
154C4⋊C4—3941no—
155C4⋊C4—1971no—
156C4⋊C4C22⋊D19711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—7882no—
159C22⋊C4—3941no—
160C22⋊C4—3941no—
161C22⋊C4—3941no—
162C22⋊C4—3941no—
163C22⋊C4—3941no—
164C23—7881no—
165C23—7881no—
166C23—3941no—
167C23—3941no—
168C23—1972no—
169C23D4×D19711yescharacteristic
170Q8—41no—
171Q8—7881no—
172Q8—15761no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—7881no—
180C2×C4—7881no—
181C2×C4—7881no—
182C2×C4—7881no—
183C2×C4—15761no—
184C2×C4—15761no—
185C2×C4—3941no—
186C2×C4—3941no—
187C2×C4—3941no—
188C2×C4D4×D19711yescharacteristic
189C2×C4C2×C22⋊D19711yescharacteristic
190C8—41no—
191C8—7881no—
192C22—21no—
193C22—21no—
194C22—7882no—
195C22—7881no—
196C22—7881no—
197C22—15761no—
198C22—3942no—
199C22#2026329411yescharacteristic
200C22SD16×D19711yescharacteristic
201C22#2026335511yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—7881no—
208C4—7881no—
209C4—7881no—
210C4—15761no—
211C2—21no—
212C2—21no—
213C2—7882no—
214C2#2716121611yescharacteristic
215C2#2716125511yescharacteristic
216C2#2716126811yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#483×C197C2
socleC22×C197#120
prime radicalC23×D197D4
hypercenter#483D197
hypocenterC197#1608
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. #27161409C2semidirect
4. #27160817C2semidirect
5. #27160816C2semidirect
7. #483×C197C2semidirect
108. C197#1608semidirect
112. #483D197semidirect

Maximal subgroups

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

subgroupindex
#271617042
#271614092
#271610052
#271608172
#271608162
#271608022
#483×C1972
#1608197