Finite Group Databasebeta

← #38741288

Subgroups — #38741288

Order 25216.

218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 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#27161129C211yesmaximal, characteristic, complemented
3#27160946C211yesmaximal, 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×C4×Dic197C2211yescharacteristic
18#20263255—21no—
19C22⋊#13096802C2211yescharacteristic
20C8×Dic197—21no—
21#20263233—21no—
22#20263233—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—21no—
38#15691335—21no—
39#15691335—21no—
40#15691335—21no—
41#15691335—21no—
42C4×Dic197—21no—
43C4×Dic197C2×C411yescharacteristic
44C4×Dic197C2×C411yescharacteristic
45C4×Dic197D411yescharacteristic
46C4×Dic197D411yescharacteristic
47C2×Q8×C197C2×C411yescharacteristic
48C22×C4×C197—21no—
49C22×C4×C197C2311yescharacteristic
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197C2×C411yescharacteristic
53C22⋊C4×C197—21no—
54C22×D197—21no—
55D788—41no—
56C22⋊D197—41no—
57C22⋊D197—41no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—41no—
62C2×Dic197—41no—
63C2×Dic197—41no—
64C2×Dic197—41no—
65C2×Dic197C22⋊C411yescharacteristic
66C2×Dic197C22⋊C411yescharacteristic
67C2×Dic197C22⋊C411yescharacteristic
68C2×Dic197C22⋊C411yescharacteristic
69C4×D197—42no—
70#13096802—41no—
71C23×C197C2×D411yescharacteristic
72Q8×C197—41no—
73C2×C4×C197—21no—
74C2×C4×C197—21no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197C22×C411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C8×C197—41no—
82C22×C197—21no—
83C22×C197—21no—
84C22×C197C2×C22⋊C411yescharacteristic
85C22×C197#13511yescharacteristic
86C22×C197#13511yescharacteristic
87C2×D197—41no—
88C2×D197—41no—
89C2×D197—41no—
90C4×C197—21no—
91C4×C197—21no—
92C4×C197—41no—
93C4×C197—41no—
94C4×C197—81no—
95Dic197—21no—
96Dic197—21no—
97Dic197—21no—
98Dic197—21no—
99Dic197—41no—
100Dic197—81no—
101C2×C197—21no—
102C2×C197—21no—
103C2×C197#41011yescharacteristic
104C2×C197#41611yescharacteristic
105C2×C197#42111yescharacteristic
106D197—81no—
107C197#152111yescharacteristic, complemented, Sylow
108#1521—1971nomaximal, Sylow
109#529—1971no—
110#431—1971no—
111#383—1971no—
112#328—1971no—
113#328D19711yescharacteristic, complemented
114#327—1971no—
115#327—1971no—
116#126—3941no—
117#125—1971no—
118#124—1971no—
119#123—3941no—
120C22⋊Q8C2×D19711yescharacteristic
121#121—1971no—
122C4×Q8—1971no—
123C4×D4—3941no—
124C4×D4—1971no—
125#117—3941no—
126C2×C42—1971no—
127C4⋊C8—3941no—
128C22⋊C8—1971no—
129C22⋊C8C2×D19711yescharacteristic
130C4×C8—3941no—
131#95—3941no—
132#95—3941no—
133#95—1971no—
134#95C2×D19711yescharacteristic
135C2×Q8C4×D19711yescharacteristic
136C2×D4—3941no—
137C2×D4—1971no—
138C22×C4—21no—
139C22×C4—3941no—
140C22×C4—3941no—
141C22×C4—1971no—
142C22×C4—1971no—
143C22×C4C22×D19711yescharacteristic
144C2×C8—21no—
145C2×C8—3941no—
146C4⋊C4—21no—
147C4⋊C4—3941no—
148C4⋊C4—3941no—
149C4⋊C4—3941no—
150C4⋊C4—1971no—
151C4⋊C4C4×D19711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3941no—
154C22⋊C4—3941no—
155C22⋊C4—3941no—
156C22⋊C4—3941no—
157C42—3941no—
158C42—1971no—
159C42—1971no—
160C42—1971no—
161C42—1971no—
162C23—3941no—
163C23D4×D19711yescharacteristic
164Q8—41no—
165D4—7881no—
166D4—7881no—
167D4—7881no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—7882no—
175C2×C4—7881no—
176C2×C4—7881no—
177C2×C4—7881no—
178C2×C4—7881no—
179C2×C4—3941no—
180C2×C4—3941no—
181C2×C4—3941no—
182C2×C4—1971no—
183C2×C4—1971no—
184C2×C4—1971no—
185C2×C4—1971no—
186C2×C4C2×C4×D19711yescharacteristic
187C2×C4D4×D19711yescharacteristic
188C8—41no—
189C8—7881no—
190C22—21no—
191C22—21no—
192C22—7881no—
193C22—7881no—
194C22—7881no—
195C22C22⋊C4×D19711yescharacteristic
196C22#2026334711yescharacteristic
197C22#2026334911yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—41no—
202C4—81no—
203C4—7881no—
204C4—15761no—
205C4—3941no—
206C4—3941no—
207C4—3941no—
208C4—3941no—
209C2—21no—
210C2—21no—
211C2—15761no—
212C2#2716105111yescharacteristic
213C2#2716107411yescharacteristic
214C2#2716109411yescharacteristic
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#1521
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. #27161129C2semidirect
3. #27160946C2semidirect
4. #27160816C2semidirect
7. #328×C197C2semidirect
9. #20263317C4semidirect
10. #20263315C4semidirect
13. Q8×Dic197C4semidirect
14. #20263288C4semidirect
107. C197#1521semidirect
113. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271616152
#271611292
#271609462
#271608162
#271608142
#271608122
#328×C1972
#1521197