Finite Group Databasebeta

← #38767784

Subgroups — #38767784

Order 25472.

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

normal subgroups only

This group has 310 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#27164599C211yesmaximal, characteristic, complemented
2#27164152C212yesmaximal, complemented
3#27163751C212yesmaximal, complemented
4#27163744C211yesmaximal, characteristic, complemented
5#394×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7C2×#15691871—22no—
8#20263935—22no—
9#20263930—22no—
10#20263929—21no—
11#20263929C2212yes—
12#20263876—22no—
13C22⋊(C199⋊C8)C2212yes—
14C22×D4×C199C2211yescharacteristic
15C2×C4⋊C4×C199C2211yescharacteristic
16C2×C22⋊C4×C199—21no—
17C2×C22⋊C4×C199—21no—
18#95×C199C2211yescharacteristic
19#15691886—42no—
20#15691886—42no—
21C2×Dic398—21no—
22C2×D796—21no—
23C2×D796—21no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C2×C4×D199—22no—
27#15691871—44no—
28C2×C199⋊C8—22no—
29C4⋊Dic199—21no—
30C4⋊Dic199D412yes—
31(C2×D199)⋊C4—22no—
32(C2×D199)⋊C4—22no—
33C24×C199—21no—
34C2×D4×C199—22no—
35C2×D4×C199—42no—
36C22×C4×C199—21no—
37C22×C4×C199D411yescharacteristic
38C22×C4×C199D411yescharacteristic
39C22×C4×C199C2311yescharacteristic
40C4⋊C4×C199—22no—
41C22⋊C4×C199—42no—
42C22⋊C4×C199—42no—
43C22×D199—22no—
44Dic398—42no—
45D796—42no—
46D796—42no—
47D796—81no—
48C22⋊D199—42no—
49C22⋊D199—42no—
50C2×Dic199—22no—
51C4×D199—44no—
52C199⋊C8—42no—
53C23×C199—22no—
54C23×C199—41no—
55C23×C199—41no—
56C23×C199—41no—
57C23×C199—41no—
58C23×C199—41no—
59C23×C199—41no—
60C23×C199C2×D411yescharacteristic
61D4×C199—24no—
62D4×C199—42no—
63C2×C4×C199—21no—
64C2×C4×C199—21no—
65C2×C4×C199—21no—
66C2×C4×C199—41no—
67C2×C4×C199—41no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
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—81no—
79C22×C199—81no—
80C22×C199—81no—
81C22×C199—81no—
82C22×C199#12011yescharacteristic
83C22×C199C2×D811yescharacteristic
84C22×C199AGL(1,ℤ/8)11yescharacteristic
85C2×D199—42no—
86C2×D199—42no—
87C2×D199—42no—
88C4×C199—22no—
89C4×C199—22no—
90C4×C199—41no—
91C4×C199—81no—
92Dic199—42no—
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#537—1991no—
103#447—1992no—
104#394D19911yescharacteristic, complemented
105#327—1992no—
106#324—1991no—
107C2×#41—3981no—
108C22×D4C2×D19911yescharacteristic
109C2×D8—3982no—
110C22⋊Q8—3982no—
111#121—3982no—
112#121—3981no—
113#121—1992no—
114C2×C4⋊C4C2×D19911yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—3982no—
118C22⋊C8—1992no—
119#95C2×D19911yescharacteristic
120C24—21no—
121#41—7962no—
122#41—7962no—
123C2×Q8—3981no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—3981no—
127C2×D4—3981no—
128C2×D4—3982no—
129C2×D4—1992no—
130C22×C4—21no—
131C22×C4—3982no—
132C22×C4C22⋊D19911yescharacteristic
133C22×C4C22⋊D19911yescharacteristic
134C22×C4C22×D19911yescharacteristic
135D8—7964no—
136C2×C8—3982no—
137C4⋊C4—22no—
138C4⋊C4—3981no—
139C4⋊C4—1992no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—3982no—
143C22⋊C4—3982no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—3982no—
152C23C2×C22⋊D19911yescharacteristic
153Q8—7962no—
154D4—24no—
155D4—42no—
156D4—7962no—
157D4—7962no—
158D4—7962no—
159D4—7962no—
160D4—15921no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—7964no—
170C2×C4—3982no—
171C2×C4#1569187112yescomplemented
172C2×C4D4×D19912yes—
173C8—7962no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—7962no—
184C22—7962no—
185C22—7962no—
186C22#2026392311yescharacteristic
187C22C2×#1569187111yescharacteristic
188C22#2026398011yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—7962no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—15922no—
198C2#2716415111yescharacteristic
199C2#2716419011yescharacteristic
200C2#2716420411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#394×C199C2
socleC22×C199#120
hypercenter#394D199
hypocenterC199#1598
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #27164599C2semidirect
2. #27164152C2semidirect
3. #27163751C2semidirect
4. #27163744C2semidirect
5. #394×C199C2semidirect
71. C2×C4×C199D8semidirect
100. C199#1598semidirect
104. #394D199semidirect
171. C2×C4#15691871semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271641522
#271637512
#271637442
#394×C1992
#1598199