Finite Group Databasebeta

← #38767785

Subgroups — #38767785

Order 25472.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164151C212yesmaximal, complemented
2#27163935C211yesmaximal, characteristic, complemented
3#27163752C212yesmaximal, complemented
4#27163744C211yesmaximal, characteristic, complemented
5#537×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C2×#15691871—22no—
8C2×(C2×D199)⋊C4—21no—
9C2×(C2×D199)⋊C4—21no—
10#20263875—22no—
11C22⋊(C199⋊C8)C2212yes—
12C199⋊#95C2211yescharacteristic
13C2×#41×C199—21no—
14C22⋊Q8×C199—22no—
15#121×C199—22no—
16#121×C199—21no—
17#121×C199C2212yes—
18C2×C4⋊C4×C199C2211yescharacteristic
19C23×D199—21no—
20C2×D796—22no—
21C2×D796—42no—
22C22×Dic199—21no—
23#15691871—44no—
24C2×C199⋊C8—22no—
25(C2×D199)⋊C4—42no—
26(C2×D199)⋊C4—42no—
27#41×C199—42no—
28#41×C199—42no—
29C2×Q8×C199—21no—
30C2×D4×C199—21no—
31C2×D4×C199—22no—
32C2×D4×C199—21no—
33C2×D4×C199D412yes—
34C22×C4×C199—22no—
35C22×C4×C199D411yescharacteristic
36C22×C4×C199D411yescharacteristic
37C22×C4×C199C2311yescharacteristic
38C4⋊C4×C199—21no—
39C4⋊C4×C199—22no—
40C4⋊C4×C199D412yes—
41C22⋊C4×C199—22no—
42C22⋊C4×C199—22no—
43C22×D199—22no—
44C22×D199—41no—
45C22×D199—41no—
46C22×D199—41no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—41no—
50D796—24no—
51D796—42no—
52C2×Dic199—41no—
53C2×Dic199—41no—
54C2×Dic199—41no—
55C199⋊C8—42no—
56C23×C199—22no—
57C23×C199C2×D411yescharacteristic
58Q8×C199—42no—
59D4×C199—42no—
60D4×C199—42no—
61D4×C199—42no—
62D4×C199—42no—
63D4×C199—81no—
64C2×C4×C199—21no—
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—22no—
68C2×C4×C199—41no—
69C2×C4×C199—44no—
70C2×C4×C199—41no—
71C2×C4×C199D812yescomplemented
72C2×C4×C199C2×D412yes—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199—42no—
76C22×C199—42no—
77C22×C199—42no—
78C22×C199#12011yescharacteristic
79C22×C199C2×D811yescharacteristic
80C22×C199AGL(1,ℤ/8)11yescharacteristic
81C2×D199—42no—
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—81no—
85C2×D199—81no—
86C2×D199—81no—
87C2×D199—81no—
88C4×C199—22no—
89C4×C199—22no—
90C4×C199—42no—
91C4×C199—41no—
92Dic199—81no—
93C2×C199—21no—
94C2×C199—21no—
95C2×C199—82no—
96C2×C199#44711yescharacteristic
97C2×C199#45411yescharacteristic
98C2×C199B(4,𝔽2)11yescharacteristic
99D199—82no—
100C199#159811yescharacteristic, complemented, Sylow
101#1598—1991nomaximal, Sylow
102#537D19911yescharacteristic, complemented
103#447—1992no—
104#394—1991no—
105#327—1992no—
106#324—1991no—
107C2×#41—21no—
108C22×D4—1991no—
109C2×D8—3982no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D19912yes—
114C2×C4⋊C4C2×D19911yescharacteristic
115C2×C22⋊C4—3981no—
116C2×C22⋊C4—3981no—
117#102—3982no—
118C22⋊C8—1992no—
119#95—1991no—
120C24—3981no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—7962no—
128C2×D4—3982no—
129C2×D4C22⋊D19912yes—
130C22×C4—22no—
131C22×C4—3981no—
132C22×C4C22⋊D19911yescharacteristic
133C22×C4C22⋊D19911yescharacteristic
134C22×C4C22×D19911yescharacteristic
135D8—7964no—
136C2×C8—3982no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D19912yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—7962no—
143C22⋊C4—7962no—
144C23—22no—
145C23—7961no—
146C23—7961no—
147C23—7961no—
148C23—7961no—
149C23—7961no—
150C23—7961no—
151C23—3982no—
152C23C2×C22⋊D19911yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—7962no—
160D4—3984no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—7961no—
169C2×C4—7961no—
170C2×C4—7961no—
171C2×C4#1569187112yescomplemented
172C2×C4C2×C22⋊D19912yes—
173C8—7962no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7962no—
180C22—7962no—
181C22—7962no—
182C22—15921no—
183C22—15921no—
184C22—15921no—
185C22—15921no—
186C22#2026392411yescharacteristic
187C22C2×#1569187111yescharacteristic
188C22#2026398111yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—42no—
193C4—15921no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—15922no—
198C2#2716415211yescharacteristic
199C2#2716419111yescharacteristic
200C2#2716420511yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#537×C199C2
socleC22×C199#120
hypercenter#537D199
hypocenterC199#1598
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27164151C2semidirect
2. #27163935C2semidirect
3. #27163752C2semidirect
4. #27163744C2semidirect
5. #537×C199C2semidirect
71. C2×C4×C199D8semidirect
100. C199#1598semidirect
102. #537D199semidirect
171. C2×C4#15691871semidirect

Maximal subgroups

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

subgroupindex
#271641512
#271639352
#271637522
#271637442
#537×C1992
#1598199