Finite Group Databasebeta

← #38743843

Subgroups — #38743843

Order 25216.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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
2C2×#20263309C212yesmaximal, complemented
3C2×#20263240C212yesmaximal, complemented
4#27160846C211yesmaximal, characteristic, complemented
5#381×C197C211yesmaximal, characteristic, complemented
6C22×C22⋊D197C2212yes—
7#20263310—22no—
8#20263309—24no—
9#20263309—24no—
10#20263300—22no—
11#20263296—22no—
12C2×#15691337—22no—
13C2×C22⋊Dic197C2212yes—
14C2×#15691335—22no—
15#20263240—24no—
16C23×C4×C197C2211yescharacteristic
17C2×C22⋊C4×C197C2211yescharacteristic
18C2×C22⋊C4×C197C2211yescharacteristic
19#95×C197—22no—
20C23×D197D412yes—
21C2×D788—22no—
22C2×C22⋊D197—22no—
23C2×C22⋊D197—22no—
24C2×C22⋊D197—22no—
25C2×C22⋊D197—22no—
26C2×C22⋊D197—22no—
27C2×C22⋊D197—22no—
28C22×Dic197—22no—
29C22×Dic197D412yes—
30C4⋊Dic197—22no—
31#15691337—28no—
32C22⋊Dic197—22no—
33C22⋊Dic197D414yescomplemented
34#15691335—28no—
35#15691335—24no—
36C24×C197C2311yescharacteristic
37C22×C4×C197—21no—
38C22×C4×C197—22no—
39C22×C4×C197—22no—
40C22×C4×C197—21no—
41C22×C4×C197—21no—
42C22×C4×C197D412yes—
43C22×C4×C197D412yescomplemented
44C22⋊C4×C197—22no—
45C22⋊C4×C197—22no—
46C22×D197—22no—
47C22×D197—22no—
48C22×D197—22no—
49C22×D197—44no—
50D788—82no—
51C22⋊D197—42no—
52C22⋊D197—42no—
53C22⋊D197—42no—
54C22⋊D197—42no—
55C2×Dic197—24no—
56C2×Dic197—22no—
57C2×Dic197—22no—
58C2×Dic197—22no—
59C2×Dic197—44no—
60C23×C197—42no—
61C23×C197C2×D412yes—
62C23×C197C2×D412yes—
63C23×C197C2×D411yescharacteristic
64C23×C197C2×D411yescharacteristic
65C23×C197#4111yescharacteristic
66C2×C4×C197—22no—
67C2×C4×C197—24no—
68C2×C4×C197—22no—
69C2×C4×C197—41no—
70C2×C4×C197—42no—
71C2×C4×C197—41no—
72C2×C4×C197—42no—
73C2×C4×C197—42no—
74C2×C4×C197—41no—
75C22×C197—22no—
76C22×C197—24no—
77C22×C197—42no—
78C22×C197—41no—
79C22×C197—41no—
80C22×C197#12011yescharacteristic
81C22×C197#12012yes—
82C22×C197#12112yes—
83C22×C197#12111yescharacteristic
84C22×C197#12711yescharacteristic
85C2×D197—44no—
86C2×D197—42no—
87C2×D197—44no—
88C2×D197—82no—
89C4×C197—42no—
90C4×C197—42no—
91C4×C197—81no—
92Dic197—44no—
93Dic197—42no—
94C2×C197—22no—
95C2×C197—21no—
96C2×C197—22no—
97C2×C197—21no—
98C2×C197#39211yescharacteristic
99C2×C197#45812yes—
100D197—82no—
101C197#202211yescharacteristic, complemented, Sylow
102#2022—1971nomaximal, Sylow
103#534—1971no—
104C2×#123—1972no—
105C2×#99—1972no—
106#381D19711yescharacteristic, complemented
107#342—1971no—
108C22×D4—1972no—
109C23×C4C2×D19711yescharacteristic
110#123—3942no—
111#123—3944no—
112#123—3944no—
113#121—3942no—
114#120—3942no—
115C2×C4⋊C4—3942no—
116C2×C22⋊C4—3942no—
117C2×C22⋊C4—1972no—
118C2×C22⋊C4C2×D19711yescharacteristic
119C2×C22⋊C4C2×D19711yescharacteristic
120#99—3944no—
121#95—22no—
122C24—1972no—
123C24C22×D19711yescharacteristic
124C2×D4—3942no—
125C2×D4—3942no—
126C2×D4—3942no—
127C2×D4—3942no—
128C2×D4—3942no—
129C2×D4—3942no—
130C2×D4—3942no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—3942no—
137C22×C4—1972no—
138C22×C4C22⋊D19712yes—
139C22×C4D78812yescomplemented
140C4⋊C4—3948no—
141C4⋊C4—3942no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3948no—
145C22⋊C4—3944no—
146C22⋊C4—3942no—
147C22⋊C4—1974no—
148C23—42no—
149C23—7884no—
150C23—3942no—
151C23—3942no—
152C23—3942no—
153C23D4×D19712yes—
154C23D4×D19712yes—
155C23C2×C22⋊D19711yescharacteristic
156C23C2×D78811yescharacteristic
157C23#1569135811yescharacteristic
158D4—7882no—
159D4—7882no—
160D4—7882no—
161D4—7882no—
162D4—15762no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—7884no—
173C2×C4—3942no—
174C2×C4—3944no—
175C2×C4—3942no—
176C2×C4—3942no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—7884no—
183C22—7882no—
184C22—7884no—
185C22—15762no—
186C22#2026329411yescharacteristic
187C22#2026329612yes—
188C22#2026329912yes—
189C22#2026330011yescharacteristic
190C22#2026332211yescharacteristic
191C4—42no—
192C4—42no—
193C4—81no—
194C4—7884no—
195C4—7882no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—15762no—
201C2#2716098811yescharacteristic
202C2#2716126812yes—
subgroupG/H
centerC22#20263294
commutator, cosocleC24×C197C23
FrattiniC24C22×D197
Fitting#381×C197C2
socleC22×C197#120
prime radical#27161641C2
hypercenter#381D197
hypocenterC197#2022
LCSG ▷ C24×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20263309C2semidirect
3. C2×#20263240C2semidirect
4. #27160846C2semidirect
5. #381×C197C2semidirect
33. C22⋊Dic197D4semidirect
43. C22×C4×C197D4semidirect
101. C197#2022semidirect
106. #381D197semidirect
139. C22×C4D788semidirect

Maximal subgroups

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

subgroupindex
#271616412
C2×#202633092
C2×#202632402
#271608462
#381×C1972
#2022197