Finite Group Databasebeta

← #38741342

Subgroups — #38741342

Order 25216.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161614C211yesmaximal, characteristic
2#27161134C211yesmaximal, characteristic, complemented
3#27161134C211yesmaximal, characteristic, complemented
4#27160821C211yesmaximal, characteristic, complemented
5#27160818C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic, complemented
7#330×C197C211yesmaximal, characteristic, complemented
8#20263321—21no—
9#20263314C2211yescharacteristic
10#20263311—21no—
11#20263306C411yescharacteristic
12#20263301C411yescharacteristic
13#20263291C411yescharacteristic
14#20263288—21no—
15#20263288C411yescharacteristic
16#20263284—21no—
17C42×D197C2211yescharacteristic
18#20263255—21no—
19#20263255—21no—
20#20263238—21no—
21#20263238—21no—
22#20263237C2211yescharacteristic
23#20263237C2211yescharacteristic
24#125×C197C2211yescharacteristic
25#97×C197C2211yescharacteristic
26#97×C197C2211yescharacteristic
27C2×Dic394C2×C411yescharacteristic
28C2×D788—21no—
29C2×D788C2×C411yescharacteristic
30C2×C4×D197—21no—
31C2×C4×D197C2×C411yescharacteristic
32C2×C4×D197D411yescharacteristic
33C2×C4×D197D411yescharacteristic
34C2×#13096802—21no—
35C2×#13096802—21no—
36C4⋊Dic197—21no—
37#15691337—21no—
38#15691335—21no—
39#15691335—21no—
40#15691335—21no—
41#15691335—21no—
42#15691335—21no—
43C4×Dic197—21no—
44C4×Dic197C2×C411yescharacteristic
45C4×Dic197D411yescharacteristic
46C4×Dic197D411yescharacteristic
47C2×C8×C197—21no—
48C2×C8×C197—21no—
49C4⋊C4×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197C2×C411yescharacteristic
52C4⋊C4×C197C2×C411yescharacteristic
53C42×C197C2311yescharacteristic
54C22×D197—21no—
55C22×D197C22⋊C411yescharacteristic
56Dic394—41no—
57D788—41no—
58D788—42no—
59C2×Dic197—21no—
60C2×Dic197C22⋊C411yescharacteristic
61C2×Dic197C22⋊C411yescharacteristic
62C2×Dic197C22⋊C411yescharacteristic
63C4×D197—41no—
64C4×D197—41no—
65C4×D197—41no—
66C4×D197—42no—
67#13096802—41no—
68#13096802—41no—
69C2×C4×C197—21no—
70C2×C4×C197—21no—
71C2×C4×C197C22×C411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C8×C197—41no—
75C8×C197—41no—
76C22×C197C2×C22⋊C411yescharacteristic
77C2×D197—21no—
78C2×D197—21no—
79C2×D197—21no—
80C2×D197—41no—
81C2×D197—41no—
82C2×D197—41no—
83C4×C197—21no—
84C4×C197—21no—
85C4×C197—21no—
86C4×C197—41no—
87C4×C197—41no—
88Dic197—21no—
89Dic197—21no—
90Dic197—21no—
91Dic197—41no—
92C2×C197#41311yescharacteristic
93C2×C197#42111yescharacteristic
94C2×C197#42111yescharacteristic
95D197—41no—
96D197—81no—
97C197#153511yescharacteristic, complemented, Sylow
98#1535—1971nomaximal, Sylow
99#529—1971no—
100#432—1971no—
101#432—1971no—
102#330—1971no—
103#330D19711yescharacteristic, complemented
104#329—1971no—
105#329—1971no—
106#126—3941no—
107#125C2×D19711yescharacteristic
108#124—1971no—
109#123—3941no—
110C22⋊Q8—1971no—
111#121—1971no—
112C4×Q8—1971no—
113C4×D4—3941no—
114C4×D4—1971no—
115#117—3941no—
116C2×C42—1971no—
117C4⋊C8—3941no—
118C4⋊C8—3941no—
119C22⋊C8—3941no—
120C22⋊C8—3941no—
121#97—1971no—
122#97—1971no—
123#97C2×D19711yescharacteristic
124#97C2×D19711yescharacteristic
125C2×Q8—1971no—
126C2×D4—3941no—
127C2×D4—1971no—
128C22×C4—3941no—
129C22×C4—1971no—
130C22×C4—1971no—
131C22×C4—1971no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—3941no—
135C2×C8—3941no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3941no—
139C4⋊C4—3941no—
140C4⋊C4C4×D19711yescharacteristic
141C4⋊C4C4×D19711yescharacteristic
142C22⋊C4—3941no—
143C22⋊C4—3941no—
144C22⋊C4—3941no—
145C22⋊C4—3941no—
146C22⋊C4—3941no—
147C42—3941no—
148C42—1971no—
149C42—1971no—
150C42—1971no—
151C42C22×D19711yescharacteristic
152C23—3941no—
153C23—1971no—
154Q8—7881no—
155D4—7881no—
156D4—7882no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—7881no—
160C2×C4—7881no—
161C2×C4—7881no—
162C2×C4—7882no—
163C2×C4—3941no—
164C2×C4—1971no—
165C2×C4—1971no—
166C2×C4—1971no—
167C2×C4C2×C4×D19711yescharacteristic
168C2×C4D4×D19711yescharacteristic
169C2×C4D4×D19711yescharacteristic
170C8—41no—
171C8—41no—
172C8—7881no—
173C8—7881no—
174C22—7881no—
175C22—7881no—
176C22—7881no—
177C22—3941no—
178C22—3941no—
179C22—3941no—
180C22C22⋊C4×D19711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—7881no—
187C4—3941no—
188C4—3941no—
189C4—3941no—
190C2—7881no—
191C2—15761no—
192C2#2716106211yescharacteristic
193C2#2716109411yescharacteristic
194C2#2716109411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#330×C197C2
socleC22×C197C2×C22⋊C4
prime radical#20263301C4
cosocleC42×C197C23
hypercenter#330D197
hypocenterC197#1535
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #27161134C2semidirect
3. #27161134C2semidirect
4. #27160821C2semidirect
5. #27160818C2semidirect
6. #27160818C2semidirect
7. #330×C197C2semidirect
97. C197#1535semidirect
103. #330D197semidirect

Maximal subgroups

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

subgroupindex
#271616142
#271611342
#271608212
#271608182
#330×C1972
#1535197