Finite Group Databasebeta

← #38741433

Subgroups — #38741433

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#27161630C211yesmaximal, characteristic
2#27161135C211yesmaximal, characteristic, complemented
3#27161101C211yesmaximal, characteristic, complemented
4#27160825C211yesmaximal, characteristic, complemented
5#27160822C211yesmaximal, characteristic, complemented
6#27160822C211yesmaximal, characteristic, complemented
7#332×C197C211yesmaximal, characteristic, complemented
8C2×#15691359—21no—
9C2×#15691341—21no—
10#20263323C2211yescharacteristic
11#20263315—21no—
12#20263301—21no—
13#20263301C412yes—
14#20263288—21no—
15#20263288C412yes—
16C42×D197C2211yescharacteristic
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263255—21no—
20#20263238—21no—
21#20263236—21no—
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691359—42no—
26#15691359—42no—
27C2×D788—22no—
28C2×D788—21no—
29C2×D788—21no—
30C2×D788C2×C412yes—
31C2×C4×D197—22no—
32C2×C4×D197C2×C411yescharacteristic
33C2×C4×D197D411yescharacteristic
34C2×C4×D197D411yescharacteristic
35#15691341—42no—
36C2×#13096802—21no—
37C2×#13096802—21no—
38#15691335—22no—
39#15691335—22no—
40C4×Dic197C2×C411yescharacteristic
41C4×Dic197D411yescharacteristic
42C4×Dic197D411yescharacteristic
43C2×Q8×C197—21no—
44C2×C8×C197—21no—
45C2×C8×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197C2×C412yes—
48C42×C197C2311yescharacteristic
49C22×D197—22no—
50C22×D197C22⋊C411yescharacteristic
51D788—42no—
52D788—44no—
53D788—41no—
54D788—41no—
55D788—41no—
56D788—41no—
57C2×Dic197C22⋊C411yescharacteristic
58C2×Dic197C22⋊C411yescharacteristic
59C2×Dic197C22⋊C411yescharacteristic
60C4×D197—21no—
61C4×D197—21no—
62C4×D197—21no—
63C4×D197—21no—
64C4×D197—41no—
65C4×D197—44no—
66#13096802—41no—
67#13096802—41no—
68Q8×C197—41no—
69Q8×C197—41no—
70C2×C4×C197—22no—
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—42no—
81C2×D197—42no—
82C2×D197—42no—
83C4×C197—21no—
84C4×C197—42no—
85C4×C197AGL(1,ℤ/8)11yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87C4×C197AGL(1,ℤ/8)11yescharacteristic
88C4×C197#13711yescharacteristic
89Dic197—21no—
90Dic197—21no—
91Dic197—21no—
92C2×C197#41311yescharacteristic
93C2×C197#41711yescharacteristic
94C2×C197#41811yescharacteristic
95D197—41no—
96D197—82no—
97C197#155711yescharacteristic, complemented, Sylow
98#1557—1971nomaximal, Sylow
99#532—1971no—
100#432—1971no—
101#423—1971no—
102#332—1971no—
103#332D19711yescharacteristic, complemented
104#331—1971no—
105#331—1971no—
106C2×#41—3941no—
107C2×M16—3941no—
108C4⋊Q8C2×D19711yescharacteristic
109#127—1971no—
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—21no—
126C2×D4—3942no—
127C2×D4—3941no—
128C2×D4—3941no—
129C2×D4—1972no—
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—21no—
140C4⋊C4C4×D19712yes—
141C22⋊C4—3942no—
142C22⋊C4—3942no—
143C42—1971no—
144C42—1971no—
145C42—1971no—
146C42C22×D19711yescharacteristic
147C23—3942no—
148C23—1971no—
149Q8—41no—
150Q8—41no—
151D4—7882no—
152D4—7884no—
153D4—7881no—
154D4—7881no—
155D4—7881no—
156D4—7881no—
157C2×C4—22no—
158C2×C4—7881no—
159C2×C4—7884no—
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—7882no—
175C22—7882no—
176C22—7882no—
177C22—3941no—
178C22—3941no—
179C22—3941no—
180C22C22⋊C4×D19711yescharacteristic
181C4—21no—
182C4—42no—
183C4—3941no—
184C4—3941no—
185C4—3941no—
186C4#2026335011yescharacteristic
187C4#2026335311yescharacteristic
188C4#2026335311yescharacteristic
189C4#2026335811yescharacteristic
190C2—7881no—
191C2—15762no—
192C2#2716106211yescharacteristic
193C2#2716107911yescharacteristic
194C2#2716108211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161630C2
cosocleC42×C197C23
hypercenter#332D197
hypocenterC197#1557
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27161135C2semidirect
3. #27161101C2semidirect
4. #27160825C2semidirect
5. #27160822C2semidirect
6. #27160822C2semidirect
7. #332×C197C2semidirect
97. C197#1557semidirect
103. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271616302
#271611352
#271611012
#271608252
#271608222
#332×C1972
#1557197