Finite Group Databasebeta

← #38769899

Subgroups — #38769899

Order 25472.

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

normal subgroups only

This group has 370 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×#20263988C212yesmaximal, complemented
2#27164599C211yesmaximal, characteristic
3C2×#20263881C212yesmaximal, complemented
4#27163923C211yesmaximal, characteristic, complemented
5#343×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2212yes—
7C22×Dic398C2211yescharacteristic
8#20263988—24no—
9#20263988—24no—
10C2×Dic796—22no—
11C2×#15691875—22no—
12#20263935—22no—
13#20263930—22no—
14#20263929—22no—
15C2×C4×Dic199C2211yescharacteristic
16#20263881—24no—
17C199⋊#95—22no—
18C2×M16×C199C2212yes—
19C2×C4⋊C4×C199C2211yescharacteristic
20#15691886—22no—
21#15691886—22no—
22#15691886—22no—
23#15691886—22no—
24C2×Dic398—22no—
25C2×Dic398—22no—
26C2×Dic398—21no—
27C2×Dic398D412yes—
28C2×Dic398D412yescomplemented
29C2×D796—22no—
30C2×D796D412yes—
31C2×C22⋊D199—22no—
32C22×Dic199—21no—
33C22×Dic199—21no—
34C2×C4×D199—22no—
35Dic796—28no—
36#15691875—28no—
37C4⋊Dic199—22no—
38(C2×D199)⋊C4—24no—
39C4×Dic199—22no—
40C22×C4×C199D412yes—
41C22×C4×C199C2311yescharacteristic
42M16×C199—22no—
43M16×C199D414yescomplemented
44C2×C8×C199—22no—
45C4⋊C4×C199—22no—
46C22×D199—22no—
47Dic398—24no—
48Dic398—22no—
49Dic398—22no—
50Dic398—42no—
51D796—22no—
52D796—22no—
53D796—82no—
54C22⋊D199—42no—
55C22⋊D199—42no—
56C2×Dic199—22no—
57C2×Dic199—22no—
58C2×Dic199—22no—
59C2×Dic199—41no—
60C2×Dic199—41no—
61C2×Dic199—41no—
62C2×Dic199—41no—
63C4×D199—42no—
64C4×D199—42no—
65C23×C199C2×D411yescharacteristic
66C2×C4×C199—22no—
67C2×C4×C199—42no—
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D412yes—
70C2×C4×C199C2×D412yes—
71C2×C4×C199#4111yescharacteristic
72C8×C199—24no—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199#12011yescharacteristic
76C22×C199#12111yescharacteristic
77C22×C199#12711yescharacteristic
78C2×D199—42no—
79C2×D199—42no—
80C2×D199—42no—
81C4×C199—42no—
82C4×C199#12012yes—
83C4×C199#12112yes—
84Dic199—42no—
85Dic199—42no—
86Dic199—42no—
87C2×C199—21no—
88C2×C199—21no—
89C2×C199#39211yescharacteristic
90C2×C199#45511yescharacteristic
91C2×C199#45611yescharacteristic
92D199—82no—
93C199#200711yescharacteristic, complemented, Sylow
94#2007—1991nomaximal, Sylow
95C2×#137—1992no—
96#537—1991no—
97C2×#104—1992no—
98#391—1991no—
99#343D19911yescharacteristic, complemented
100C2×#41—1992no—
101C22×Q8—1991no—
102#137—3984no—
103#137—3984no—
104C2×Q16—3982no—
105C2×SD16—3982no—
106C2×M16C2×D19912yes—
107C22⋊Q8—3982no—
108#121—3982no—
109#121—3982no—
110C2×C4⋊C4C2×D19911yescharacteristic
111C2×C42—1991no—
112#104—3984no—
113#95—3982no—
114#41—3982no—
115#41—3982no—
116#41—3982no—
117#41—3982no—
118C2×Q8—3982no—
119C2×Q8—3982no—
120C2×Q8—3981no—
121C2×Q8—1992no—
122C2×Q8—1992no—
123C2×D4—3982no—
124C2×D4—3982no—
125C2×D4—1992no—
126C22×C4—3981no—
127C22×C4—3981no—
128C22×C4—3982no—
129C22×C4D79612yes—
130C22×C4C22×D19911yescharacteristic
131Q16—3988no—
132SD16—3988no—
133M16—22no—
134M16D79614yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—3982no—
138C22⋊C4—3984no—
139C42—3982no—
140C23—3982no—
141C23C2×D79611yescharacteristic
142Q8—7962no—
143Q8—3984no—
144Q8—3982no—
145Q8—3982no—
146D4—7962no—
147D4—7962no—
148D4—15922no—
149D4—3982no—
150D4—3982no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—7961no—
154C2×C4—7961no—
155C2×C4—7961no—
156C2×C4—7961no—
157C2×C4—7962no—
158C2×C4—7962no—
159C2×C4—3982no—
160C2×C4—3982no—
161C2×C4—3982no—
162C2×C4D4×D19911yescharacteristic
163C2×C4D4×D19912yes—
164C2×C4C2×D79612yes—
165C2×C4#1569188711yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—7962no—
170C22—7962no—
171C22—7962no—
172C22#2026392511yescharacteristic
173C22#2026393011yescharacteristic
174C22#2026395211yescharacteristic
175C4—42no—
176C4—7962no—
177C4—7962no—
178C4—7962no—
179C4#2026392512yes—
180C4#2026393012yes—
181C2—21no—
182C2—21no—
183C2—15922no—
184C2#2716392811yescharacteristic
185C2#2716419811yescharacteristic
186C2#2716420311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#343×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#343D199
hypocenterC199#2007
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×#20263988C2semidirect
3. C2×#20263881C2semidirect
4. #27163923C2semidirect
5. #343×C199C2semidirect
28. C2×Dic398D4semidirect
43. M16×C199D4semidirect
93. C199#2007semidirect
99. #343D199semidirect
134. M16D796semidirect

Maximal subgroups

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

subgroupindex
C2×#202639882
#271645992
C2×#202638812
#271639232
#343×C1992
#2007199