Finite Group Databasebeta

← #38741676

Subgroups — #38741676

Order 25216.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161674C211yesmaximal, characteristic
2#27161237C211yesmaximal, characteristic, complemented
3#27161217C211yesmaximal, characteristic
4#27161000C211yesmaximal, characteristic, complemented
5#27160815C211yesmaximal, characteristic, complemented
6#27160804C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic
8C22×Dic394C2211yescharacteristic
9C2×Dic788—21no—
10C2×#15691346—21no—
11#20263319—21no—
12#20263316—21no—
13#20263307—21no—
14#20263307—21no—
15#20263305C2211yescharacteristic
16#20263303—21no—
17D4×Dic197—21no—
18C2×C4⋊Dic197—21no—
19C2×#15691337C2211yescharacteristic
20#20263251—21no—
21#20263249—21no—
22C22⋊#13096802C2211yescharacteristic
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394—42no—
30C2×Dic394D411yescharacteristic
31C22×Dic197—21no—
32C22×Dic197—21no—
33C22×Dic197D411yescharacteristic
34C22×Dic197D411yescharacteristic
35Dic788—42no—
36#15691346—42no—
37C2×#13096802—21no—
38C4⋊Dic197—21no—
39C4⋊Dic197—42no—
40C4⋊Dic197D411yescharacteristic
41#15691337—21no—
42#15691337—21no—
43#15691337—21no—
44#15691337—21no—
45#15691337—21no—
46C22⋊Dic197—21no—
47C22⋊Dic197—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C4×Dic197—21no—
51C2×D4×C197—21no—
52C2×D4×C197D411yescharacteristic
53C22×C4×C197—21no—
54C22×C4×C197C2311yescharacteristic
55C2×C8×C197—21no—
56C4⋊C4×C197D411yescharacteristic
57C22⋊C4×C197—21no—
58Dic394—22no—
59Dic394—22no—
60Dic394—41no—
61Dic394—41no—
62Dic394—41no—
63C2×Dic197—22no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—21no—
67C2×Dic197—21no—
68C2×Dic197—21no—
69C2×Dic197—21no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197—41no—
73C2×Dic197—81no—
74#13096802—41no—
75C23×C197—21no—
76C23×C197C2×D411yescharacteristic
77D4×C197—41no—
78D4×C197—41no—
79D4×C197—41no—
80C2×C4×C197—22no—
81C2×C4×C197—21no—
82C2×C4×C197—41no—
83C2×C4×C197—41no—
84C2×C4×C197—41no—
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197C2×D411yescharacteristic
87C8×C197—41no—
88C22×C197—21no—
89C22×C197—21no—
90C22×C197—41no—
91C22×C197—41no—
92C22×C197—41no—
93C22×C197#12011yescharacteristic
94C22×C197#13511yescharacteristic
95C22×C197#13711yescharacteristic
96C4×C197—21no—
97C4×C197—21no—
98C4×C197—42no—
99C4×C197—41no—
100Dic197—41no—
101Dic197—41no—
102Dic197—42no—
103Dic197—41no—
104Dic197—41no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197—81no—
108C2×C197#45211yescharacteristic
109C2×C197#45611yescharacteristic
110C2×C197B(4,𝔽2)11yescharacteristic
111C197#160311yescharacteristic, complemented, Sylow
112#1603—1971nomaximal, Sylow
113#539—1971no—
114#451—1971no—
115#448—1971no—
116#395—1971no—
117#328—1971no—
118#327D19711yescharacteristic
119#324—1971no—
120C22×Q8—1971no—
121C2×Q16—3941no—
122C2×SD16—3941no—
123#126—3941no—
124#125—3941no—
125#123—3941no—
126#123—3941no—
127C22⋊Q8—3941no—
128C22⋊Q8—1971no—
129#121C2×D19711yescharacteristic
130C4×D4—3941no—
131C2×C4⋊C4—3941no—
132C2×C4⋊C4—1971no—
133#103—3941no—
134#103—3941no—
135C22⋊C8—1971no—
136C22⋊C8C2×D19711yescharacteristic
137#95—3941no—
138#95C2×D19711yescharacteristic
139C2×Q8—7882no—
140C2×Q8—3941no—
141C2×Q8—3941no—
142C2×Q8—1971no—
143C2×D4—21no—
144C2×D4D78811yescharacteristic
145C22×C4—21no—
146C22×C4—3941no—
147C22×C4—3941no—
148C22×C4—1971no—
149C22×C4—1971no—
150C22×C4C22×D19711yescharacteristic
151Q16—7882no—
152SD16—7882no—
153C2×C8—21no—
154C2×C8—3941no—
155C4⋊C4—7882no—
156C4⋊C4—3941no—
157C4⋊C4—3941no—
158C4⋊C4—3941no—
159C4⋊C4—3941no—
160C4⋊C4—3941no—
161C4⋊C4—3941no—
162C4⋊C4—1971no—
163C4⋊C4D78811yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3941no—
166C22⋊C4—3941no—
167C22⋊C4—3941no—
168C22⋊C4—3941no—
169C42—3941no—
170C23—21no—
171C23D4×D19711yescharacteristic
172Q8—7881no—
173Q8—7881no—
174Q8—7881no—
175Q8—3942no—
176Q8—3942no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—7881no—
186C2×C4—7881no—
187C2×C4—7881no—
188C2×C4—15761no—
189C2×C4—3942no—
190C2×C4—3941no—
191C2×C4—3941no—
192C2×C4—3941no—
193C2×C4—3941no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4D4×D19711yescharacteristic
197C2×C4C2×D78811yescharacteristic
198C8—41no—
199C8—7881no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2026329611yescharacteristic
206C22#2026334511yescharacteristic
207C22#2026335511yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—7881no—
213C4—7881no—
214C4—7882no—
215C4—7881no—
216C4—7881no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2716124211yescharacteristic
221C2#2716126311yescharacteristic
222C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#121×C197C22
hypercenter#327D197
hypocenterC197#1603
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27161237C2semidirect
4. #27161000C2semidirect
5. #27160815C2semidirect
6. #27160804C2semidirect
111. C197#1603semidirect

Maximal subgroups

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

subgroupindex
#271616742
#271612372
#271612172
#271610002
#271608152
#271608042
#327×C1972
#1603197