Finite Group Databasebeta

← #38742028

Subgroups — #38742028

Order 25216.

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

normal subgroups only

This group has 208 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
2#27161485C211yesmaximal, characteristic, complemented
3#27161471C211yesmaximal, characteristic, complemented
4#27160820C211yesmaximal, characteristic, complemented
5#27160819C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic, complemented
7#489×C197C211yesmaximal, characteristic, complemented
8C2×#15691348—21no—
9C2×#15691345—21no—
10#20263318C2211yescharacteristic
11#20263314C2211yescharacteristic
12#20263310—21no—
13#20263308—21no—
14#20263303—21no—
15#20263302—21no—
16#20263299—21no—
17#20263298—21no—
18#20263259—21no—
19#20263257—21no—
20#20263237C2211yescharacteristic
21#20263237C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23#124×C197C2211yescharacteristic
24#103×C197—21no—
25#102×C197—21no—
26#97×C197C2211yescharacteristic
27C2×Dic394D411yescharacteristic
28C2×D788D411yescharacteristic
29C2×C22⋊D197—21no—
30C2×C22⋊D197—21no—
31C22×Dic197—21no—
32C2×C4×D197—21no—
33#15691348—42no—
34#15691345—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×C197—21no—
48C2×Q8×C197D411yescharacteristic
49C2×D4×C197D411yescharacteristic
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C22⋊C4×C197—21no—
54C22⋊C4×C197—21no—
55C42×C197C2311yescharacteristic
56C22×D197—21no—
57Dic394—41no—
58D788—41no—
59C22⋊D197—81no—
60C22⋊D197—81no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—81no—
65C4×D197—81no—
66#13096802—22no—
67#13096802—22no—
68C23×C197—21no—
69Q8×C197—41no—
70Q8×C197—41no—
71Q8×C197—41no—
72D4×C197—41no—
73C2×C4×C197—21no—
74C2×C4×C197—41no—
75C2×C4×C197C2×D411yescharacteristic
76C2×C4×C197C2×D411yescharacteristic
77C2×C4×C197C2×D411yescharacteristic
78C8×C197—41no—
79C22×C197—41no—
80C22×C197—41no—
81C22×C197—41no—
82C22×C197#12011yescharacteristic
83C2×D197—41no—
84C2×D197—41no—
85C2×D197—41no—
86C4×C197—21no—
87C4×C197—21no—
88C4×C197—21no—
89C4×C197—41no—
90C4×C197—81no—
91Dic197—41no—
92Dic197—41no—
93Dic197—41no—
94C2×C197—81no—
95C2×C197#45311yescharacteristic
96C2×C197#45511yescharacteristic
97C2×C197#45611yescharacteristic
98D197—81no—
99C197#166111yescharacteristic, complemented, Sylow
100#1661—1971nomaximal, Sylow
101#562—1971no—
102#501—1971no—
103#497—1971no—
104#489D19711yescharacteristic, complemented
105#330—1971no—
106#329—1971no—
107#329—1971no—
108C2×Q16—3941no—
109C2×SD16—3941no—
110C4⋊Q8C2×D19711yescharacteristic
111#125—1971no—
112#124—1971no—
113#124C2×D19711yescharacteristic
114#123—3941no—
115#123—3941no—
116C22⋊Q8—3941no—
117C22⋊Q8—3941no—
118#121—3941no—
119#121—3941no—
120#107—3941no—
121#106—3941no—
122#103—21no—
123#102—21no—
124#97—1971no—
125#97—1971no—
126#97C2×D19711yescharacteristic
127C2×Q8—21no—
128C2×Q8—1971no—
129C2×Q8C22⋊D19711yescharacteristic
130C2×D4—3941no—
131C2×D4—3941no—
132C2×D4—1971no—
133C2×D4C22⋊D19711yescharacteristic
134C22×C4—3941no—
135C22×C4—3941no—
136Q16—7882no—
137SD16—7882no—
138C2×C8—21no—
139C2×C8—3941no—
140C2×C8—3941no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—3941no—
144C4⋊C4—3941no—
145C4⋊C4—3941no—
146C4⋊C4—3941no—
147C4⋊C4—1971no—
148C4⋊C4—1971no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—3941no—
152C22⋊C4—3941no—
153C22⋊C4—3941no—
154C22⋊C4—3941no—
155C42C22×D19711yescharacteristic
156C23—21no—
157C23—3941no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—7881no—
162D4—41no—
163D4—7881no—
164D4—15761no—
165D4—15761no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—15761no—
169C2×C4—15761no—
170C2×C4—3941no—
171C2×C4—3941no—
172C2×C4—3941no—
173C2×C4D4×D19711yescharacteristic
174C2×C4D4×D19711yescharacteristic
175C2×C4C2×C22⋊D19711yescharacteristic
176C8—41no—
177C8—3942no—
178C8—3942no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7881no—
183C22—7881no—
184C22—7881no—
185C22#2026329411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—7881no—
192C4—7881no—
193C4—7881no—
194C2—81no—
195C2—15761no—
196C2#2716124511yescharacteristic
197C2#2716125611yescharacteristic
198C2#2716126011yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#489×C197C2
socleC22×C197#120
prime radical#27161823C2
hypercenter#489D197
hypocenterC197#1661
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #27161485C2semidirect
3. #27161471C2semidirect
4. #27160820C2semidirect
5. #27160819C2semidirect
6. #27160818C2semidirect
7. #489×C197C2semidirect
99. C197#1661semidirect
104. #489D197semidirect

Maximal subgroups

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

subgroupindex
#271618232
#271614852
#271614712
#271608202
#271608192
#271608182
#489×C1972
#1661197