Finite Group Databasebeta

← #38741431

Subgroups — #38741431

Order 25216.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#27161627C211yesmaximal, characteristic
2#27161135C211yesmaximal, characteristic, complemented
3#27161101C211yesmaximal, characteristic, complemented
4#27160826C211yesmaximal, characteristic, complemented
5#27160824C211yesmaximal, characteristic, complemented
6#27160823C211yesmaximal, characteristic, complemented
7#331×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C2×#15691341—21no—
10#20263326C2211yescharacteristic
11#20263313—21no—
12#20263297—21no—
13#20263297C412yes—
14D4×Dic197—21no—
15D4×Dic197C412yes—
16C42×D197C2211yescharacteristic
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263255—21no—
20#20263238—21no—
21#20263236—21no—
22#127×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691357—42no—
26#15691357—42no—
27C2×Dic394—21no—
28C2×C22⋊D197—22no—
29C22×Dic197—22no—
30C2×C4×D197C2×C411yescharacteristic
31C2×C4×D197D411yescharacteristic
32C2×C4×D197D411yescharacteristic
33#15691341—42no—
34C2×#13096802—21no—
35C2×#13096802—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197C2×C412yes—
38C22⋊Dic197—22no—
39C22⋊Dic197—22no—
40C4×Dic197C2×C411yescharacteristic
41C4×Dic197D411yescharacteristic
42C4×Dic197D411yescharacteristic
43C2×D4×C197—21no—
44C2×D4×C197—21no—
45C2×D4×C197C2×C412yes—
46C2×C8×C197—21no—
47C2×C8×C197—21no—
48C42×C197C2311yescharacteristic
49C22×D197C22⋊C411yescharacteristic
50Dic394—41no—
51Dic394—41no—
52C22⋊D197—44no—
53C2×Dic197—22no—
54C2×Dic197—44no—
55C2×Dic197C22⋊C411yescharacteristic
56C2×Dic197C22⋊C411yescharacteristic
57C2×Dic197C22⋊C411yescharacteristic
58C4×D197—21no—
59C4×D197—21no—
60C4×D197—21no—
61C4×D197—21no—
62C4×D197—41no—
63#13096802—41no—
64#13096802—41no—
65C23×C197—22no—
66D4×C197—42no—
67D4×C197—41no—
68D4×C197—41no—
69D4×C197—41no—
70D4×C197—41no—
71C2×C4×C197C22×C411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C8×C197—41no—
75C8×C197—41no—
76C22×C197—42no—
77C22×C197—42no—
78C22×C197—42no—
79C22×C197C2×C22⋊C411yescharacteristic
80C2×D197—21no—
81C2×D197—21no—
82C2×D197—21no—
83C4×C197—21no—
84C4×C197AGL(1,ℤ/8)11yescharacteristic
85C4×C197AGL(1,ℤ/8)11yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87C4×C197#13711yescharacteristic
88Dic197—21no—
89Dic197—21no—
90Dic197—21no—
91Dic197—42no—
92C2×C197—82no—
93C2×C197#41311yescharacteristic
94C2×C197#41711yescharacteristic
95C2×C197#41811yescharacteristic
96D197—41no—
97C197#155711yescharacteristic, complemented, Sylow
98#1557—1971nomaximal, Sylow
99#532—1971no—
100#432—1971no—
101#423—1971no—
102#332—1971no—
103#332—1971no—
104#331—1971no—
105#331D19711yescharacteristic, complemented
106C2×#41—3941no—
107C2×M16—3941no—
108C4⋊Q8—1971no—
109#127C2×D19711yescharacteristic
110#124—3941no—
111#121—3941no—
112#121—1972no—
113C4×D4—3941no—
114C4×D4—1972no—
115C2×C42—1971no—
116C4⋊C8—3941no—
117C4⋊C8—1971no—
118C4⋊C8—1971no—
119C4⋊C8C2×D19711yescharacteristic
120C4⋊C8C2×D19711yescharacteristic
121C22⋊C8—3941no—
122#97—3941no—
123#41—7882no—
124#41—7882no—
125C2×Q8—3941no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—3942no—
129C2×D4C4×D19712yes—
130C22×C4—3942no—
131C22×C4—1971no—
132C22×C4—1971no—
133C22×C4—1971no—
134M16—7882no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—3941no—
138C2×C8—3941no—
139C4⋊C4—3941no—
140C4⋊C4—1972no—
141C22⋊C4—3942no—
142C22⋊C4—3942no—
143C42—1971no—
144C42—1971no—
145C42—1971no—
146C42C22×D19711yescharacteristic
147C23—22no—
148C23—1971no—
149Q8—7881no—
150Q8—7881no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—7884no—
157C2×C4—7881no—
158C2×C4—7884no—
159C2×C4—3942no—
160C2×C4—3941no—
161C2×C4—3941no—
162C2×C4—3941no—
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—42no—
175C22—42no—
176C22—42no—
177C22—3941no—
178C22—3941no—
179C22—3941no—
180C22C22⋊C4×D19711yescharacteristic
181C4—21no—
182C4—7882no—
183C4—3941no—
184C4—3941no—
185C4—3941no—
186C4#2026335011yescharacteristic
187C4#2026335011yescharacteristic
188C4#2026335311yescharacteristic
189C4#2026335511yescharacteristic
190C2—82no—
191C2—7881no—
192C2#2716106011yescharacteristic
193C2#2716107711yescharacteristic
194C2#2716108211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161627C2
cosocleC42×C197C23
hypercenter#331D197
hypocenterC197#1557
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27161135C2semidirect
3. #27161101C2semidirect
4. #27160826C2semidirect
5. #27160824C2semidirect
6. #27160823C2semidirect
7. #331×C197C2semidirect
97. C197#1557semidirect
105. #331D197semidirect

Maximal subgroups

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

subgroupindex
#271616272
#271611352
#271611012
#271608262
#271608242
#271608232
#331×C1972
#1557197