Finite Group Databasebeta

← #38741674

Subgroups — #38741674

Order 25216.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161678C211yesmaximal, characteristic
2#27161218C211yesmaximal, characteristic
3#27160998C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160814C211yesmaximal, characteristic
6#27160804C211yesmaximal, characteristic, complemented
7#451×C197C211yesmaximal, characteristic, complemented
8C2×#15691345—21no—
9#20263321—21no—
10#20263317—21no—
11#20263311—21no—
12#20263310—21no—
13#20263302—21no—
14#20263300C2211yescharacteristic
15#20263286—21no—
16C2×#15691337—21no—
17C2×#15691337C2211yescharacteristic
18#20263250—21no—
19C22⋊#13096802C2211yescharacteristic
20#20263233—21no—
21#20263233C2211yescharacteristic
22C22×Q8×C197C2211yescharacteristic
23C2×Q16×C197—21no—
24C22⋊Q8×C197C2211yescharacteristic
25#103×C197—21no—
26C22⋊C8×C197C2211yescharacteristic
27C2×D788D411yescharacteristic
28C2×C22⋊D197—21no—
29C22×Dic197—21no—
30C22×Dic197D411yescharacteristic
31C22×Dic197D411yescharacteristic
32C2×C4×D197—21no—
33#15691345—42no—
34C2×#13096802—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197D411yescharacteristic
38#15691337—21no—
39#15691337—21no—
40#15691337—21no—
41#15691337—42no—
42#15691335—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46C4×Dic197—21no—
47C2×Q8×C197—21no—
48C2×Q8×C197—21no—
49C2×Q8×C197—42no—
50C2×Q8×C197D411yescharacteristic
51C22×C4×C197—21no—
52C22×C4×C197C2311yescharacteristic
53Q16×C197—42no—
54C2×C8×C197—21no—
55C4⋊C4×C197—21no—
56C4⋊C4×C197D411yescharacteristic
57C22⋊C4×C197—21no—
58C22×D197—21no—
59D788—41no—
60C22⋊D197—41no—
61C22⋊D197—41no—
62C2×Dic197—22no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—21no—
68C2×Dic197—41no—
69C2×Dic197—41no—
70C2×Dic197—41no—
71C4×D197—81no—
72#13096802—41no—
73C23×C197C2×D411yescharacteristic
74Q8×C197—22no—
75Q8×C197—22no—
76Q8×C197—41no—
77Q8×C197—41no—
78Q8×C197—41no—
79C2×C4×C197—22no—
80C2×C4×C197—21no—
81C2×C4×C197—21no—
82C2×C4×C197—41no—
83C2×C4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197C2×D411yescharacteristic
87C8×C197—41no—
88C22×C197—21no—
89C22×C197—21no—
90C22×C197#12011yescharacteristic
91C22×C197#13511yescharacteristic
92C22×C197#13711yescharacteristic
93C2×D197—41no—
94C2×D197—41no—
95C2×D197—41no—
96C4×C197—21no—
97C4×C197—21no—
98C4×C197—41no—
99C4×C197—42no—
100C4×C197—41no—
101Dic197—42no—
102Dic197—41no—
103Dic197—41no—
104Dic197—41no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197#45211yescharacteristic
108C2×C197#45611yescharacteristic
109C2×C197B(4,𝔽2)11yescharacteristic
110D197—81no—
111C197#160311yescharacteristic, complemented, Sylow
112#1603—1971nomaximal, Sylow
113#539—1971no—
114#451D19711yescharacteristic, complemented
115#448—1971no—
116#395—1971no—
117#328—1971no—
118#327—1971no—
119#324—1971no—
120C22×Q8C2×D19711yescharacteristic
121C2×Q16—21no—
122C2×SD16—3941no—
123#126—3941no—
124#125—3941no—
125#123—3941no—
126#123—3941no—
127C22⋊Q8—3941no—
128C22⋊Q8C2×D19711yescharacteristic
129#121—1971no—
130C4×D4—3941no—
131C2×C4⋊C4—3941no—
132C2×C4⋊C4—1971no—
133#103—21no—
134#103—3941no—
135C22⋊C8—1971no—
136C22⋊C8C2×D19711yescharacteristic
137#95—3941no—
138#95—1971no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D19711yescharacteristic
143C2×D4—3941no—
144C2×D4—1971no—
145C22×C4—21no—
146C22×C4—3941no—
147C22×C4—3941no—
148C22×C4—1971no—
149C22×C4—1971no—
150C22×C4C22×D19711yescharacteristic
151Q16—42no—
152SD16—7882no—
153C2×C8—21no—
154C2×C8—3941no—
155C4⋊C4—21no—
156C4⋊C4—7882no—
157C4⋊C4—3941no—
158C4⋊C4—3941no—
159C4⋊C4—3941no—
160C4⋊C4—3941no—
161C4⋊C4—3941no—
162C4⋊C4—1971no—
163C4⋊C4C22⋊D19711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3941no—
166C22⋊C4—3941no—
167C22⋊C4—3941no—
168C22⋊C4—3941no—
169C42—3941no—
170C23—3941no—
171C23D4×D19711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—7881no—
178D4—7881no—
179D4—7881no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—7881no—
187C2×C4—7881no—
188C2×C4—7881no—
189C2×C4—15761no—
190C2×C4—3942no—
191C2×C4—3941no—
192C2×C4—3941no—
193C2×C4—3941no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4D4×D19711yescharacteristic
197C2×C4C2×C22⋊D19711yescharacteristic
198C8—41no—
199C8—7881no—
200C22—21no—
201C22—21no—
202C22—7881no—
203C22—7881no—
204C22—7881no—
205C22#2026329411yescharacteristic
206C22#2026334911yescharacteristic
207C22#2026335811yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—7882no—
214C4—7881no—
215C4—7881no—
216C4—7881no—
217C2—21no—
218C2—21no—
219C2—15761no—
220C2#2716124011yescharacteristic
221C2#2716126011yescharacteristic
222C2#2716126511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#451×C197C2
socleC22×C197#120
prime radical#20263300C22
hypercenter#451D197
hypocenterC197#1603
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #27160998C2semidirect
4. #27160816C2semidirect
6. #27160804C2semidirect
7. #451×C197C2semidirect
111. C197#1603semidirect
114. #451D197semidirect

Maximal subgroups

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

subgroupindex
#271616782
#271612182
#271609982
#271608162
#271608142
#271608042
#451×C1972
#1603197