Finite Group Databasebeta

← #38741697

Subgroups — #38741697

Order 25216.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161641C211yesmaximal, characteristic
2#27161392C212yesmaximal, complemented
3#27160812C212yesmaximal, complemented
4#27160803C211yesmaximal, characteristic, complemented
5#397×C197C211yesmaximal, characteristic, complemented
6C22×C22⋊D197—21no—
7#20263310—22no—
8#20263300—21no—
9#20263300C2212yes—
10#20263296—22no—
11#20263259—22no—
12#20263246—22no—
13C22⋊#13096802C2212yescomplemented
14C2×C4⋊C4×C197C2211yescharacteristic, complemented
15C2×C22⋊C4×C197—21no—
16C2×C22⋊C4×C197C2211yescharacteristic
17#95×C197—21no—
18#95×C197C2211yescharacteristic, complemented
19C23×D197—21no—
20C2×D788—21no—
21C2×D788D412yes—
22C2×C22⋊D197—22no—
23C2×C22⋊D197—22no—
24C2×C22⋊D197—42no—
25C2×C22⋊D197—42no—
26C22×Dic197—21no—
27C2×#13096802—22no—
28C4⋊Dic197—21no—
29C4⋊Dic197D412yes—
30#15691335—22no—
31#15691335—22no—
32C24×C197D411yescharacteristic
33C22×C4×C197—21no—
34C22×C4×C197—21no—
35C22×C4×C197D411yescharacteristic
36C22×C4×C197C2311yescharacteristic
37C4⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—42no—
40C22×D197—22no—
41C22×D197—42no—
42C22×D197—81no—
43C22×D197—81no—
44D788—42no—
45D788—81no—
46C22⋊D197—44no—
47C22⋊D197—42no—
48C22⋊D197—42no—
49C2×Dic197—22no—
50C2×Dic197—42no—
51#13096802—42no—
52C23×C197—21no—
53C23×C197—21no—
54C23×C197—41no—
55C23×C197—41no—
56C23×C197D812yes—
57C23×C197C2×D411yescharacteristic
58C2×C4×C197—21no—
59C2×C4×C197—41no—
60C2×C4×C197—41no—
61C2×C4×C197—41no—
62C2×C4×C197—41no—
63C2×C4×C197—41no—
64C2×C4×C197—41no—
65C2×C4×C197—41no—
66C2×C4×C197—41no—
67C2×C4×C197C2×D412yes—
68C22×C197—21no—
69C22×C197—21no—
70C22×C197—22no—
71C22×C197—42no—
72C22×C197—41no—
73C22×C197—41no—
74C22×C197—81no—
75C22×C197#12011yescharacteristic
76C22×C197C2×D811yescharacteristic
77C22×C197AGL(1,ℤ/8)11yescharacteristic
78C2×D197—42no—
79C2×D197—42no—
80C2×D197—42no—
81C2×D197—84no—
82C4×C197—22no—
83C4×C197—41no—
84C4×C197—81no—
85C4×C197—81no—
86Dic197—42no—
87C2×C197—21no—
88C2×C197—21no—
89C2×C197—42no—
90C2×C197#44711yescharacteristic
91C2×C197#45511yescharacteristic
92C2×C197#45811yescharacteristic
93D197—82no—
94C197#160711yescharacteristic, complemented, Sylow
95#1607—1971nomaximal, Sylow
96#534—1971no—
97#480—1972no—
98#397D19711yescharacteristic, complemented
99#327—1972no—
100#323—1971no—
101C22×D4—3941no—
102#123—3942no—
103#121—3941no—
104#121—1972no—
105#120—3942no—
106C2×C4⋊C4C2×D19711yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D19711yescharacteristic
109#107—3942no—
110#102—3942no—
111C22⋊C8—1972no—
112#95—21no—
113#95C2×D19711yescharacteristic, complemented
114C24—3941no—
115C24C22⋊D19711yescharacteristic
116C2×D4—7882no—
117C2×D4—7882no—
118C2×D4—3941no—
119C2×D4—3942no—
120C2×D4—3942no—
121C2×D4—1972no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—3941no—
125C22×C4C22⋊D19711yescharacteristic
126C22×C4C22×D19711yescharacteristic
127C2×C8—3942no—
128C4⋊C4—22no—
129C4⋊C4—3941no—
130C4⋊C4—1972no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—3942no—
134C22⋊C4—3942no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—7882no—
140C23—15761no—
141C23—15761no—
142C23—3942no—
143C23#1569134312yes—
144C23C2×C22⋊D19711yescharacteristic
145D4—7882no—
146D4—7884no—
147D4—7882no—
148D4—7882no—
149D4—15761no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—7882no—
160C2×C4—3942no—
161C2×C4D4×D19712yes—
162C8—7882no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—7882no—
171C22—7882no—
172C22—7882no—
173C22—15764no—
174C22#2026329411yescharacteristic
175C22C2×#1569134311yescharacteristic
176C22#2026335111yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—7882no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—15762no—
186C2#2716121211yescharacteristic
187C2#2716125611yescharacteristic
188C2#2716126811yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#397×C197C2
socleC22×C197#120
prime radical#27161641C2
hypercenter#397D197
hypocenterC197#1607
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #27161392C2semidirect
3. #27160812C2semidirect
4. #27160803C2semidirect
5. #397×C197C2semidirect
13. C22⋊#13096802C22semidirect
14. C2×C4⋊C4×C197C22semidirect
18. #95×C197C22semidirect
94. C197#1607semidirect
98. #397D197semidirect
106. C2×C4⋊C4C2×D197semidirect
113. #95C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616412
#271613922
#271608122
#271608032
#397×C1972
#1607197