Finite Group Databasebeta

← #38741653

Subgroups — #38741653

Order 25216.

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

normal subgroups only

This group has 266 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#27161708C211yesmaximal, characteristic, complemented
2#27161232C211yesmaximal, characteristic
3#27161229C211yesmaximal, characteristic
4#27160997C211yesmaximal, characteristic
5#27160815C211yesmaximal, characteristic, complemented
6#27160804C211yesmaximal, characteristic, complemented
7#328×C197C211yesmaximal, characteristic, complemented
8C22×D788C2211yescharacteristic
9C2×#15691347—21no—
10C2×#15691345—21no—
11#20263325—21no—
12#20263324—21no—
13#20263316—21no—
14#20263316—21no—
15#20263305C2211yescharacteristic
16#20263303—21no—
17#20263282—21no—
18C2×#15691337C2211yescharacteristic
19C2×#15691335—21no—
20C2×#15691335—21no—
21#20263248—21no—
22#20263246—21no—
23C22⋊#13096802C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197—21no—
28C2×Dic394—21no—
29C2×Dic394D411yescharacteristic
30C2×D788—21no—
31C2×D788—21no—
32C2×D788—42no—
33C22×Dic197D411yescharacteristic
34C22×Dic197D411yescharacteristic
35#15691347—42no—
36#15691345—42no—
37C2×#13096802—21no—
38C4⋊Dic197—21no—
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43#15691337—21no—
44#15691337—21no—
45#15691337—21no—
46#15691337—21no—
47C22⋊Dic197—21no—
48#15691335—42no—
49#15691335—42no—
50C4×Dic197—21no—
51C4×Dic197—21no—
52C2×Q8×C197D411yescharacteristic
53C22×C4×C197—21no—
54C22×C4×C197C2311yescharacteristic
55C2×C8×C197—21no—
56C4⋊C4×C197—21no—
57C4⋊C4×C197D411yescharacteristic
58C22⋊C4×C197—21no—
59C22×D197—22no—
60C22×D197—41no—
61C22×D197—41no—
62C22×D197—41no—
63C22×D197—41no—
64C22×D197—41no—
65C22×D197—41no—
66Dic394—41no—
67Dic394—81no—
68D788—22no—
69D788—22no—
70D788—41no—
71D788—41no—
72C2×Dic197—21no—
73C2×Dic197—21no—
74C2×Dic197—21no—
75C2×Dic197—21no—
76C2×Dic197—21no—
77C2×Dic197—41no—
78C2×Dic197SD1612yescomplemented
79#13096802—41no—
80C23×C197C2×D411yescharacteristic
81Q8×C197—41no—
82C2×C4×C197—21no—
83C2×C4×C197—21no—
84C2×C4×C197—41no—
85C2×C4×C197—41no—
86C2×C4×C197—41no—
87C2×C4×C197—41no—
88C2×C4×C197C2×D411yescharacteristic
89C2×C4×C197C2×D411yescharacteristic
90C8×C197—41no—
91C22×C197—21no—
92C22×C197—21no—
93C22×C197#12011yescharacteristic
94C22×C197C2×SD1611yescharacteristic
95C22×C197#13711yescharacteristic
96C2×D197—42no—
97C2×D197—42no—
98C2×D197—42no—
99C2×D197—81no—
100C2×D197—81no—
101C2×D197—81no—
102C2×D197—81no—
103C4×C197—21no—
104C4×C197—21no—
105C4×C197—41no—
106C4×C197—41no—
107C4×C197—81no—
108Dic197—22no—
109Dic197—41no—
110Dic197—41no—
111Dic197—41no—
112C2×C197—21no—
113C2×C197—21no—
114C2×C197#44811yescharacteristic
115C2×C197#45411yescharacteristic
116C2×C197B(4,𝔽2)11yescharacteristic
117D197—82no—
118C197#159911yescharacteristic, complemented, Sylow
119#1599—1971nomaximal, Sylow
120#544—1971no—
121#450—1971no—
122#450—1971no—
123#394—1971no—
124#328—1971no—
125#328D19711yescharacteristic, complemented
126#324—1971no—
127C22×D4—1971no—
128C2×SD16—3941no—
129C2×SD16—3941no—
130C4⋊Q8—3941no—
131C4⋊Q8—3941no—
132#125—3941no—
133#125—3941no—
134C22⋊Q8—3941no—
135C22⋊Q8—1971no—
136C22⋊Q8C2×D19711yescharacteristic
137#117—3941no—
138C2×C4⋊C4—1971no—
139C2×C22⋊C4—3941no—
140C2×C22⋊C4—3941no—
141#102—3941no—
142#102—3941no—
143C22⋊C8—1971no—
144C22⋊C8C2×D19711yescharacteristic
145#95C2×D19711yescharacteristic
146C24—3941no—
147C2×Q8—3941no—
148C2×Q8—1971no—
149C2×Q8D78811yescharacteristic
150C2×D4—7882no—
151C2×D4—3941no—
152C2×D4—3941no—
153C22×C4—21no—
154C22×C4—1971no—
155C22×C4—1971no—
156C22×C4C22×D19711yescharacteristic
157SD16—7882no—
158SD16—7882no—
159C2×C8—21no—
160C2×C8—3941no—
161C4⋊C4—21no—
162C4⋊C4—3941no—
163C4⋊C4—3941no—
164C4⋊C4—3941no—
165C4⋊C4—3941no—
166C4⋊C4—3941no—
167C4⋊C4—3941no—
168C4⋊C4—3941no—
169C4⋊C4—3941no—
170C4⋊C4—1971no—
171C4⋊C4D78811yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—7882no—
174C22⋊C4—7882no—
175C22⋊C4—3941no—
176C42—3941no—
177C42—3941no—
178C23—7881no—
179C23—7881no—
180C23—7881no—
181C23—7881no—
182C23—7881no—
183C23—7881no—
184C23—3942no—
185C23D4×D19711yescharacteristic
186Q8—41no—
187Q8—7881no—
188Q8—15761no—
189D4—7881no—
190D4—7881no—
191D4—3942no—
192D4—3942no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—7881no—
200C2×C4—3941no—
201C2×C4—3941no—
202C2×C4—3941no—
203C2×C4—3941no—
204C2×C4—3941no—
205C2×C4—1972no—
206C2×C4D4×D19711yescharacteristic
207C2×C4C2×D78811yescharacteristic
208C8—41no—
209C8—7881no—
210C22—21no—
211C22—21no—
212C22—7882no—
213C22—7882no—
214C22—7882no—
215C22—15761no—
216C22—15761no—
217C22—15761no—
218C22—15761no—
219C22#2026329611yescharacteristic
220C22SD16×D19711yescharacteristic
221C22#2026335811yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—7881no—
228C4—7881no—
229C4—7881no—
230C4—3942no—
231C2—21no—
232C2—21no—
233C2—15762no—
234C2#2716121911yescharacteristic
235C2#2716125311yescharacteristic
236C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radicalC22×D788C22
hypercenter#328D197
hypocenterC197#1599
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27161708C2semidirect
5. #27160815C2semidirect
6. #27160804C2semidirect
7. #328×C197C2semidirect
78. C2×Dic197SD16semidirect
118. C197#1599semidirect
125. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271617082
#271612322
#271612292
#271609972
#271608152
#271608042
#328×C1972
#1599197