Finite Group Databasebeta

← #38742003

Subgroups — #38742003

Order 25216.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161823C211yesmaximal, characteristic, complemented
2#27161466C211yesmaximal, characteristic, complemented
3#27161443C211yesmaximal, characteristic, complemented
4#27160820C211yesmaximal, characteristic, complemented
5#27160819C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic, complemented
7#496×C197C211yesmaximal, characteristic, complemented
8C2×#15691346—21no—
9C2×#15691343—21no—
10#20263318C2211yescharacteristic
11#20263314C2211yescharacteristic
12#20263310—21no—
13#20263308—21no—
14#20263303—21no—
15#20263302—21no—
16#20263299—21no—
17#20263298—21no—
18#20263247—21no—
19#20263247—21no—
20#20263237C2211yescharacteristic
21#20263237C2211yescharacteristic
22C2×SD16×C197—21no—
23C2×D8×C197—21no—
24#127×C197C2211yescharacteristic
25#124×C197C2211yescharacteristic
26#97×C197C2211yescharacteristic
27C2×Dic394D411yescharacteristic
28C2×D788D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C22×Dic197—21no—
32C2×C4×D197—21no—
33#15691346—42no—
34#15691343—42no—
35C2×#13096802—21no—
36C2×#13096802—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43C22⋊Dic197—21no—
44#15691335—21no—
45#15691335—21no—
46#15691335—21no—
47C2×Q8×C197D411yescharacteristic
48C2×D4×C197—21no—
49C2×D4×C197—21no—
50C2×D4×C197—21no—
51C2×D4×C197D411yescharacteristic
52SD16×C197—42no—
53D8×C197—42no—
54C2×C8×C197—21no—
55C22⋊C4×C197—21no—
56C22⋊C4×C197—21no—
57C42×C197C2311yescharacteristic
58C22×D197—21no—
59Dic394—41no—
60D788—41no—
61C22⋊D197—81no—
62C22⋊D197—81no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—81no—
67C4×D197—81no—
68#13096802—22no—
69#13096802—41no—
70C23×C197—21no—
71C23×C197—41no—
72Q8×C197—41no—
73D4×C197—41no—
74D4×C197—41no—
75D4×C197—41no—
76D4×C197—42no—
77D4×C197—41no—
78D4×C197—41no—
79C2×C4×C197—21no—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2×D411yescharacteristic
83C8×C197—22no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197—41no—
87C22×C197—81no—
88C22×C197—81no—
89C22×C197—81no—
90C22×C197#12011yescharacteristic
91C2×D197—41no—
92C2×D197—41no—
93C2×D197—41no—
94C4×C197—21no—
95C4×C197—21no—
96C4×C197—21no—
97C4×C197—41no—
98Dic197—41no—
99Dic197—41no—
100Dic197—41no—
101C2×C197—81no—
102C2×C197—161no—
103C2×C197#45311yescharacteristic
104C2×C197#45311yescharacteristic
105C2×C197#45411yescharacteristic
106D197—81no—
107C197#165611yescharacteristic, complemented, Sylow
108#1656—1971nomaximal, Sylow
109#562—1971no—
110#496—1971no—
111#496D19711yescharacteristic, complemented
112#490—1971no—
113#330—1971no—
114#329—1971no—
115#329—1971no—
116C2×SD16—21no—
117C2×SD16—3941no—
118C2×D8—21no—
119C2×D8—3941no—
120#127C2×D19711yescharacteristic
121#125—1971no—
122#124—1971no—
123#124C2×D19711yescharacteristic
124#123—3941no—
125#123—3941no—
126C22⋊Q8—3941no—
127C22⋊Q8—3941no—
128#121—3941no—
129#121—3941no—
130#102—3941no—
131#102—3941no—
132#97—1971no—
133#97—1971no—
134#97C2×D19711yescharacteristic
135C2×Q8—1971no—
136C2×Q8C22⋊D19711yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3941no—
141C2×D4—3941no—
142C2×D4—1971no—
143C2×D4C22⋊D19711yescharacteristic
144C22×C4—3941no—
145C22×C4—3941no—
146SD16—42no—
147SD16—7882no—
148D8—42no—
149D8—7882no—
150C2×C8—21no—
151C2×C8—3941no—
152C2×C8—3941no—
153C4⋊C4—3941no—
154C4⋊C4—3941no—
155C4⋊C4—3941no—
156C4⋊C4—3941no—
157C4⋊C4—1971no—
158C4⋊C4—1971no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—3941no—
162C22⋊C4—3941no—
163C22⋊C4—3941no—
164C22⋊C4—3941no—
165C42C22×D19711yescharacteristic
166C23—21no—
167C23—41no—
168C23—3941no—
169Q8—41no—
170Q8—7881no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—7881no—
178D4—15761no—
179D4—15761no—
180C2×C4—21no—
181C2×C4—15761no—
182C2×C4—15761no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4—3941no—
186C2×C4D4×D19711yescharacteristic
187C2×C4D4×D19711yescharacteristic
188C2×C4C2×C22⋊D19711yescharacteristic
189C8—22no—
190C8—7881no—
191C8—3942no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—7881no—
199C22—7881no—
200C22—7881no—
201C22#2026329411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—7881no—
207C4—7881no—
208C4—7881no—
209C2—81no—
210C2—161no—
211C2—15761no—
212C2#2716124511yescharacteristic
213C2#2716124611yescharacteristic
214C2#2716125011yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#496×C197C2
socleC22×C197#120
hypercenter#496D197
hypocenterC197#1656
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #27161823C2semidirect
2. #27161466C2semidirect
3. #27161443C2semidirect
4. #27160820C2semidirect
5. #27160819C2semidirect
6. #27160818C2semidirect
7. #496×C197C2semidirect
107. C197#1656semidirect
111. #496D197semidirect

Maximal subgroups

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

subgroupindex
#271618232
#271614662
#271614432
#271608202
#271608192
#271608182
#496×C1972
#1656197