Finite Group Databasebeta

← #38741709

Subgroups — #38741709

Order 25216.

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

normal subgroups only

This group has 216 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#27161407C211yesmaximal, characteristic, complemented
2#27161390C211yesmaximal, characteristic
3#27161009C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic
6#27160805C211yesmaximal, characteristic, complemented
7#539×C197C211yesmaximal, characteristic
8C2×C4⋊Dic197C2211yescharacteristic
9C2×#15691337—21no—
10#20263259—21no—
11#20263257—21no—
12#20263250—21no—
13#20263247—21no—
14C22⋊#13096802C2211yescharacteristic
15C22⋊#13096802C2211yescharacteristic
16#20263233—21no—
17#20263233C2211yescharacteristic
18#126×C197—21no—
19#125×C197—21no—
20#123×C197—21no—
21#123×C197—21no—
22C22⋊Q8×C197—21no—
23C22⋊Q8×C197C2211yescharacteristic
24#121×C197C2211yescharacteristic
25C4×D4×C197—21no—
26C2×C4⋊C4×C197C2211yescharacteristic
27C22×Dic197—21no—
28C22×Dic197—21no—
29C2×#13096802—21no—
30C2×#13096802—21no—
31C4⋊Dic197—21no—
32C4⋊Dic197—21no—
33#15691337—42no—
34C2×Q8×C197D411yescharacteristic
35C2×D4×C197—21no—
36C2×D4×C197D411yescharacteristic
37C22×C4×C197—21no—
38C22×C4×C197D411yescharacteristic
39C22×C4×C197D411yescharacteristic
40C22×C4×C197C2311yescharacteristic
41C4⋊C4×C197—21no—
42C4⋊C4×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197D411yescharacteristic
48C4⋊C4×C197D411yescharacteristic
49C22⋊C4×C197—21no—
50C22⋊C4×C197—21no—
51C22⋊C4×C197—21no—
52C22⋊C4×C197—21no—
53C22⋊C4×C197—21no—
54C42×C197—21no—
55C2×Dic197—22no—
56C2×Dic197—41no—
57C2×Dic197—41no—
58C2×Dic197—41no—
59C2×Dic197—41no—
60C2×Dic197—41no—
61#13096802—41no—
62#13096802—41no—
63C23×C197—21no—
64C23×C197C2×D411yescharacteristic
65Q8×C197—41no—
66D4×C197—41no—
67D4×C197—41no—
68D4×C197—41no—
69C2×C4×C197—21no—
70C2×C4×C197—21no—
71C2×C4×C197—21no—
72C2×C4×C197—21no—
73C2×C4×C197—21no—
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—81no—
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C22×C197—21no—
82C22×C197—21no—
83C22×C197—41no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197#12011yescharacteristic
87C22×C197#13511yescharacteristic
88C22×C197AGL(1,ℤ/8)11yescharacteristic
89C4×C197—21no—
90C4×C197—21no—
91C4×C197—41no—
92C4×C197—41no—
93C4×C197—41no—
94C4×C197—41no—
95C4×C197—41no—
96Dic197—42no—
97Dic197—81no—
98C2×C197—21no—
99C2×C197—21no—
100C2×C197—81no—
101C2×C197#44911yescharacteristic
102C2×C197#45611yescharacteristic
103C2×C197#45811yescharacteristic
104C197#160911yescharacteristic, complemented, Sylow
105#1609—1971nomaximal, Sylow
106#539D19711yescharacteristic
107#483—1971no—
108#480—1971no—
109#398—1971no—
110#328—1971no—
111#327—1971no—
112#324—1971no—
113#126—21no—
114#125—21no—
115#123—21no—
116#123—21no—
117C22⋊Q8—21no—
118C22⋊Q8C2×D19711yescharacteristic
119#121C2×D19711yescharacteristic
120C4×D4—21no—
121C2×C4⋊C4—3941no—
122C2×C4⋊C4—1971no—
123C2×C4⋊C4C2×D19711yescharacteristic
124#107—3941no—
125#106—3941no—
126#103—3941no—
127#102—3941no—
128C22⋊C8—1971no—
129C22⋊C8—1971no—
130#95—3941no—
131#95—1971no—
132C2×Q8C22⋊D19711yescharacteristic
133C2×D4—21no—
134C2×D4C22⋊D19711yescharacteristic
135C22×C4—21no—
136C22×C4—3941no—
137C22×C4—3941no—
138C22×C4C22⋊D19711yescharacteristic
139C22×C4C22⋊D19711yescharacteristic
140C22×C4C22×D19711yescharacteristic
141C2×C8—3941no—
142C2×C8—3941no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—7882no—
150C4⋊C4—3941no—
151C4⋊C4—3941no—
152C4⋊C4C22⋊D19711yescharacteristic
153C4⋊C4C22⋊D19711yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C42—21no—
160C23—21no—
161C23C2×C22⋊D19711yescharacteristic
162Q8—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—7881no—
177C2×C4—7881no—
178C2×C4—7881no—
179C2×C4—7881no—
180C2×C4—7881no—
181C2×C4—3942no—
182C2×C4C2×C22⋊D19711yescharacteristic
183C2×C4C2×C22⋊D19711yescharacteristic
184C8—7881no—
185C8—7881no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#2026329511yescharacteristic
192C22#2026334611yescharacteristic
193C22#2026335111yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—7882no—
202C4—15761no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#2716122411yescharacteristic
207C2#2716126211yescharacteristic
208C2#2716126911yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#539×C197C2
socleC22×C197#120
prime radical#121×C197C22
hypercenter#539D197
hypocenterC197#1609
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #27161407C2semidirect
3. #27161009C2semidirect
4. #27160816C2semidirect
6. #27160805C2semidirect
104. C197#1609semidirect

Maximal subgroups

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

subgroupindex
#271614072
#271613902
#271610092
#271608162
#271608132
#271608052
#539×C1972
#1609197