Finite Group Databasebeta

← #38744361

Subgroups — #38744361

Order 25216.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#27161643C211yesmaximal, characteristic
2#27161063C211yesmaximal, characteristic, complemented
3#27160865C214yesmaximal, complemented
4#410×C197C211yesmaximal, characteristic, complemented
5C22×D788C411yescharacteristic, complemented
6C2×#15691342—21no—
7#20263322—21no—
8#20263315—21no—
9#20263300—21no—
10#20263300—21no—
11#20263294C412yescomplemented
12#20263294C2212yes—
13#20263285C411yescharacteristic, complemented
14#20263245—21no—
15#20263243C2212yes—
16C2×#41×C197C2211yescharacteristic
17#117×C197—21no—
18#99×C197—21no—
19#99×C197C2212yes—
20C23×D197C2×C412yes—
21C2×D788—21no—
22C2×D788—21no—
23C2×D788—41no—
24C2×D788—41no—
25C2×C22⋊D197—22no—
26C2×C22⋊D197—22no—
27#15691342—22no—
28#15691342—41no—
29C2×#13096802—41no—
30C4⋊Dic197—21no—
31C22⋊Dic197C2×C412yes—
32#15691335—22no—
33#15691335—22no—
34C4×Dic197—21no—
35#41×C197—41no—
36#41×C197—41no—
37C2×Q8×C197D411yescharacteristic
38C2×D4×C197C2×C411yescharacteristic
39C2×D4×C197D411yescharacteristic
40C2×D4×C197C2311yescharacteristic
41C22×C4×C197C2×C411yescharacteristic
42C22×C4×C197D411yescharacteristic
43C22×C4×C197D411yescharacteristic
44C4⋊C4×C197—41no—
45C22⋊C4×C197—22no—
46C42×C197—41no—
47C22×D197—22no—
48C22×D197—42no—
49C22×D197—82no—
50D788—42no—
51D788—41no—
52D788—41no—
53C22⋊D197—82no—
54C22⋊D197—82no—
55C2×Dic197—22no—
56#13096802—42no—
57C23×C197C22⋊C411yescharacteristic
58C23×C197C22×C411yescharacteristic
59C23×C197C2×D411yescharacteristic
60Q8×C197—41no—
61D4×C197—41no—
62D4×C197—41no—
63D4×C197—41no—
64C2×C4×C197—21no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—21no—
69C2×C4×C197—21no—
70C2×C4×C197—42no—
71C2×C4×C197C22⋊C411yescharacteristic
72C2×C4×C197C22⋊C411yescharacteristic
73C2×C4×C197C22⋊C411yescharacteristic
74C2×C4×C197C2×D411yescharacteristic
75C22×C197—21no—
76C22×C197—21no—
77C22×C197—21no—
78C22×C197—41no—
79C22×C197—41no—
80C22×C197—41no—
81C22×C197C2×C22⋊C411yescharacteristic
82C2×D197—42no—
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—84no—
86C4×C197—21no—
87C4×C197—21no—
88C4×C197—21no—
89C4×C197—82no—
90C4×C197#9911yescharacteristic
91C4×C197#9911yescharacteristic
92Dic197—42no—
93C2×C197—21no—
94C2×C197—41no—
95C2×C197—41no—
96C2×C197—41no—
97C2×C197C2×#9911yescharacteristic
98D197—82no—
99C197#212211yescharacteristic, complemented, Sylow
100#2122—1971nomaximal, Sylow
101#535—1971no—
102#413—1971no—
103#410D19711yescharacteristic, complemented
104AΣL(1,𝔽16)—1974no—
105C2×#41C2×D19711yescharacteristic
106C22×D4—1971no—
107C2×M16—3941no—
108#127—3941no—
109#124—3941no—
110#121—3941no—
111#121—3941no—
112#120—1972no—
113#120—1972no—
114#117—21no—
115#117—1971no—
116#101—3941no—
117#100—1972no—
118#99—21no—
119#99C2×D19712yes—
120C24—1972no—
121#41—41no—
122#41—41no—
123C2×Q8D78811yescharacteristic
124C2×D4—7881no—
125C2×D4—7881no—
126C2×D4—3942no—
127C2×D4—3942no—
128C2×D4—3941no—
129C2×D4—3941no—
130C2×D4C4×D19711yescharacteristic
131C2×D4C22⋊D19711yescharacteristic
132C2×D4C22×D19711yescharacteristic
133C22×C4C4×D19711yescharacteristic
134C22×C4C22⋊D19711yescharacteristic
135C22×C4D78811yescharacteristic
136M16—7881no—
137M16—3942no—
138C2×C8—7881no—
139C4⋊C4—41no—
140C4⋊C4—3941no—
141C22⋊C4—22no—
142C22⋊C4—3942no—
143C22⋊C4—3942no—
144C22⋊C4—1972no—
145C42—41no—
146C42—3941no—
147C23—7882no—
148C23—15762no—
149C23—3942no—
150C23#1569133511yescharacteristic
151C23C2×C4×D19711yescharacteristic
152C23C2×C22⋊D19711yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—7882no—
158D4—7881no—
159D4—7881no—
160D4—15762no—
161D4—15762no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—3942no—
170C2×C4#1569133511yescharacteristic
171C2×C4#1569133511yescharacteristic
172C2×C4#1569133511yescharacteristic
173C2×C4C2×D78811yescharacteristic
174C8—7882no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—7882no—
182C22—7882no—
183C22—7882no—
184C22—15764no—
185C22C2×#1569133511yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—82no—
190C4—7882no—
191C4#2026324011yescharacteristic
192C4#2026324011yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—15762no—
198C2C2×#2026324011yescharacteristic
subgroupG/H
centerC2C2×#20263240
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#410×C197C2
socleC2×C197C2×#99
prime radical#27161643C2
cosocleC2×D4×C197C23
hypercenter#410D197
hypocenterC197#2122
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161063C2semidirect
3. #27160865C2semidirect
4. #410×C197C2semidirect
5. C22×D788C4semidirect
11. #20263294C4semidirect
13. #20263285C4semidirect
99. C197#2122semidirect
103. #410D197semidirect

Maximal subgroups

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

subgroupindex
#271616432
#271610632
#271608652
#410×C1972
#2122197