Finite Group Databasebeta

← #38741236

Subgroups — #38741236

Order 25216.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#27161615C211yesmaximal, characteristic
2#27161133C211yesmaximal, characteristic, complemented
3#27160930C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160814C211yesmaximal, characteristic
6#27160812C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic, complemented
8#20263321—21no—
9#20263317C411yescharacteristic, complemented
10#20263315C411yescharacteristic, complemented
11#20263310—21no—
12#20263300C2211yescharacteristic
13Q8×Dic197C411yescharacteristic, complemented
14#20263288C411yescharacteristic, complemented
15#20263286—21no—
16#20263284—21no—
17C2×#15691337—21no—
18C2×C4×Dic197—21no—
19C2×C4×Dic197C2211yescharacteristic
20#20263255—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263236—21no—
23#20263233C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×D788C2×C411yescharacteristic
28C2×C22⋊D197—21no—
29C22×Dic197—21no—
30C22×Dic197D411yescharacteristic
31C22×Dic197D411yescharacteristic
32C2×C4×D197—21no—
33C2×#13096802—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197C2×C411yescharacteristic
37#15691337—24no—
38#15691337—21no—
39#15691335—21no—
40#15691335—21no—
41#15691335—21no—
42#15691335—21no—
43C4×Dic197—21no—
44C4×Dic197—42no—
45C4×Dic197C2×C411yescharacteristic
46C4×Dic197C2×C411yescharacteristic
47C4×Dic197D411yescharacteristic
48C4×Dic197D411yescharacteristic
49C2×Q8×C197C2×C411yescharacteristic
50C22×C4×C197—21no—
51C22×C4×C197C2311yescharacteristic
52C2×C8×C197—21no—
53C4⋊C4×C197—21no—
54C4⋊C4×C197C2×C411yescharacteristic
55C22⋊C4×C197—21no—
56C22×D197—21no—
57D788—41no—
58C22⋊D197—41no—
59C22⋊D197—41no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—22no—
63C2×Dic197—22no—
64C2×Dic197—21no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197C22⋊C411yescharacteristic
68C2×Dic197C22⋊C411yescharacteristic
69C2×Dic197C22⋊C411yescharacteristic
70C2×Dic197C22⋊C411yescharacteristic
71C4×D197—42no—
72#13096802—41no—
73C23×C197C2×D411yescharacteristic
74Q8×C197—41no—
75C2×C4×C197—22no—
76C2×C4×C197—21no—
77C2×C4×C197—21no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197—41no—
81C2×C4×C197C22×C411yescharacteristic
82C2×C4×C197C2×D411yescharacteristic
83C8×C197—41no—
84C22×C197—21no—
85C22×C197—21no—
86C22×C197C2×C22⋊C411yescharacteristic
87C22×C197AGL(1,ℤ/8)11yescharacteristic
88C22×C197#13711yescharacteristic
89C2×D197—41no—
90C2×D197—41no—
91C2×D197—41no—
92C4×C197—21no—
93C4×C197—21no—
94C4×C197—42no—
95C4×C197—41no—
96C4×C197—41no—
97Dic197—21no—
98Dic197—21no—
99Dic197—21no—
100Dic197—42no—
101Dic197—41no—
102Dic197#10412yes—
103C2×C197—21no—
104C2×C197—21no—
105C2×C197#41011yescharacteristic
106C2×C197#41711yescharacteristic
107C2×C197C2×#10411yescharacteristic
108D197—81no—
109C197#150911yescharacteristic, complemented, Sylow
110#1509—1971nomaximal, Sylow
111#529—1971no—
112#432—1971no—
113C4×C4⋊C4—1971no—
114#328—1971no—
115#328D19711yescharacteristic, complemented
116#327—1971no—
117#327—1971no—
118#126—3941no—
119#125—1971no—
120#124—1971no—
121#123—3941no—
122C22⋊Q8C2×D19711yescharacteristic
123#121—1971no—
124C4×Q8—1971no—
125C4×D4—3941no—
126C4×D4—1971no—
127#117—3941no—
128C2×C4⋊C4—3941no—
129C2×C42—3941no—
130C2×C42—1971no—
131C4⋊C8—3941no—
132C22⋊C8—1971no—
133C22⋊C8C2×D19711yescharacteristic
134#97—3941no—
135#95—1971no—
136#95C2×D19711yescharacteristic
137C2×Q8C4×D19711yescharacteristic
138C2×D4—3941no—
139C2×D4—1971no—
140C22×C4—21no—
141C22×C4—3941no—
142C22×C4—3941no—
143C22×C4—1971no—
144C22×C4—1971no—
145C22×C4C22×D19711yescharacteristic
146C2×C8—21no—
147C2×C8—3941no—
148C4⋊C4—21no—
149C4⋊C4—3944no—
150C4⋊C4—3941no—
151C4⋊C4—3941no—
152C4⋊C4—3941no—
153C4⋊C4—1971no—
154C4⋊C4C4×D19711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3941no—
157C22⋊C4—3941no—
158C22⋊C4—3941no—
159C22⋊C4—3941no—
160C42—7882no—
161C42—3941no—
162C42—1971no—
163C42—1971no—
164C42—1971no—
165C42—1971no—
166C23—3941no—
167C23D4×D19711yescharacteristic
168Q8—41no—
169D4—7881no—
170D4—7881no—
171D4—7881no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—7882no—
179C2×C4—7881no—
180C2×C4—7881no—
181C2×C4—3942no—
182C2×C4—3941no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4—3942no—
186C2×C4—1971no—
187C2×C4—1971no—
188C2×C4—1971no—
189C2×C4—1971no—
190C2×C4C2×C4×D19711yescharacteristic
191C2×C4D4×D19711yescharacteristic
192C8—41no—
193C8—7881no—
194C22—21no—
195C22—21no—
196C22—7881no—
197C22—7881no—
198C22—7881no—
199C22C22⋊C4×D19711yescharacteristic
200C22#2026335311yescharacteristic
201C22#2026335811yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—7882no—
208C4—7881no—
209C4—3941no—
210C4—3941no—
211C4—3941no—
212C4—1972no—
213C2—21no—
214C2—21no—
215C2—15761no—
216C2#2716105111yescharacteristic
217C2#2716107911yescharacteristic
218C2#104×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197C2×C22⋊C4
prime radical#20263300C22
cosocleC22×C4×C197C23
hypercenter#328D197
hypocenterC197#1509
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27161133C2semidirect
3. #27160930C2semidirect
4. #27160816C2semidirect
7. #328×C197C2semidirect
9. #20263317C4semidirect
10. #20263315C4semidirect
13. Q8×Dic197C4semidirect
14. #20263288C4semidirect
109. C197#1509semidirect
115. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271616152
#271611332
#271609302
#271608162
#271608142
#271608122
#328×C1972
#1509197