Finite Group Databasebeta

← #38769894

Subgroups — #38769894

Order 25472.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263983C212yesmaximal, complemented
2#27164599C211yesmaximal, characteristic, complemented
3C2×#20263881C212yesmaximal, complemented
4#27163919C211yesmaximal, characteristic, complemented
5#343×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2212yes—
7C22×D796C2211yescharacteristic
8#20263983—24no—
9#20263983—24no—
10C2×#15691875—22no—
11C2×D1592—22no—
12#20263935—22no—
13#20263930—22no—
14#20263929—22no—
15C2×(C2×D199)⋊C4—22no—
16C2×C4×Dic199C2211yescharacteristic
17#20263881—24no—
18C2×M16×C199C2212yes—
19C2×C4⋊C4×C199C2211yescharacteristic
20C23×D199—21no—
21#15691886—22no—
22#15691886—22no—
23#15691886—22no—
24#15691886—22no—
25C2×Dic398D412yes—
26C2×D796—22no—
27C2×D796—22no—
28C2×D796—22no—
29C2×D796—21no—
30C2×D796D412yes—
31C2×D796D412yescomplemented
32C2×C22⋊D199—22no—
33C22×Dic199—21no—
34C2×C4×D199—22no—
35#15691875—28no—
36D1592—28no—
37C4⋊Dic199—22no—
38(C2×D199)⋊C4—24no—
39(C2×D199)⋊C4—44no—
40C4×Dic199—22no—
41C22×C4×C199D412yes—
42C22×C4×C199C2311yescharacteristic
43M16×C199—22no—
44M16×C199D414yescomplemented
45C2×C8×C199—22no—
46C4⋊C4×C199—22no—
47C22×D199—22no—
48C22×D199—22no—
49C22×D199—42no—
50C22×D199—42no—
51C22×D199—41no—
52C22×D199—41no—
53Dic398—22no—
54Dic398—22no—
55D796—22no—
56D796—24no—
57D796—22no—
58D796—42no—
59D796—82no—
60C22⋊D199—42no—
61C22⋊D199—42no—
62C2×Dic199—22no—
63C2×Dic199—22no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C4×D199—42no—
67C4×D199—42no—
68C23×C199C2×D411yescharacteristic
69C2×C4×C199—22no—
70C2×C4×C199—42no—
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D412yes—
73C2×C4×C199C2×D412yes—
74C2×C4×C199#4111yescharacteristic
75C8×C199—24no—
76C22×C199—21no—
77C22×C199—21no—
78C22×C199#12011yescharacteristic
79C22×C199#12111yescharacteristic
80C22×C199#12711yescharacteristic
81C2×D199—44no—
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—42no—
85C2×D199—42no—
86C2×D199—82no—
87C2×D199—81no—
88C2×D199—81no—
89C4×C199—42no—
90C4×C199#12012yes—
91C4×C199#12112yes—
92Dic199—42no—
93Dic199—42no—
94C2×C199—21no—
95C2×C199—21no—
96C2×C199#39211yescharacteristic
97C2×C199#45311yescharacteristic
98C2×C199#45411yescharacteristic
99D199—82no—
100D199—82no—
101C199#200611yescharacteristic, complemented, Sylow
102#2006—1991nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1992no—
104#537—1991no—
105C2×#104—1992no—
106#390—1991no—
107#343D19911yescharacteristic, complemented
108C2×#41—1992no—
109C22×D4—1991no—
110AGL(1,ℤ/8)—3984no—
111AGL(1,ℤ/8)—3984no—
112C2×SD16—3982no—
113C2×D8—3982no—
114C2×M16C2×D19912yes—
115C22⋊Q8—3982no—
116#121—3982no—
117#121—3982no—
118C2×C4⋊C4C2×D19911yescharacteristic
119C2×C22⋊C4—3982no—
120C2×C42—1991no—
121#104—3984no—
122C24—3981no—
123#41—3982no—
124#41—3982no—
125#41—3982no—
126#41—3982no—
127C2×Q8—1992no—
128C2×D4—3982no—
129C2×D4—3982no—
130C2×D4—3982no—
131C2×D4—3982no—
132C2×D4—3981no—
133C2×D4—1992no—
134C2×D4—1992no—
135C22×C4—3981no—
136C22×C4—3982no—
137C22×C4D79612yes—
138C22×C4C22×D19911yescharacteristic
139SD16—3988no—
140D8—3988no—
141M16—22no—
142M16D79614yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3982no—
146C22⋊C4—7964no—
147C22⋊C4—3984no—
148C42—3982no—
149C23—7962no—
150C23—7962no—
151C23—7961no—
152C23—7961no—
153C23—3982no—
154C23—3982no—
155C23C2×D79611yescharacteristic
156Q8—3982no—
157Q8—3982no—
158D4—7962no—
159D4—7962no—
160D4—7962no—
161D4—15922no—
162D4—3982no—
163D4—3984no—
164D4—3982no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—7961no—
168C2×C4—7961no—
169C2×C4—7962no—
170C2×C4—7962no—
171C2×C4—3982no—
172C2×C4—3982no—
173C2×C4D4×D19911yescharacteristic
174C2×C4D4×D19912yes—
175C2×C4C2×D79612yes—
176C2×C4#1569188711yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—7964no—
181C22—7962no—
182C22—7962no—
183C22—7962no—
184C22—7962no—
185C22—15922no—
186C22—15921no—
187C22—15921no—
188C22#2026392511yescharacteristic
189C22#2026393011yescharacteristic
190C22#2026395211yescharacteristic
191C4—42no—
192C4—7962no—
193C4—7962no—
194C4#2026392512yes—
195C4#2026393012yes—
196C2—21no—
197C2—21no—
198C2—15922no—
199C2—15922no—
200C2#2716392811yescharacteristic
201C2#2716418811yescharacteristic
202C2#2716419311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#343×C199C2
socleC22×C199#120
hypercenter#343D199
hypocenterC199#2006
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#20263983C2semidirect
2. #27164599C2semidirect
3. C2×#20263881C2semidirect
4. #27163919C2semidirect
5. #343×C199C2semidirect
31. C2×D796D4semidirect
44. M16×C199D4semidirect
101. C199#2006semidirect
107. #343D199semidirect
142. M16D796semidirect

Maximal subgroups

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

subgroupindex
C2×#202639832
#271645992
C2×#202638812
#271639192
#343×C1992
#2006199