Finite Group Databasebeta

← #38768767

Subgroups — #38768767

Order 25472.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#15691886C211yesmaximal, characteristic
2C2×#20263869C214yesmaximal, complemented
3#27163881C211yesmaximal, characteristic, complemented
4#381×C199C211yesmaximal, characteristic, complemented
5C2×#15691886—28no—
6C2×#15691886—22no—
7C2×#15691886—22no—
8C22×Dic398C411yescharacteristic, complemented
9C22×D796C411yescharacteristic, complemented
10C22×C22⋊D199C2212yes—
11C22×C4×D199C412yescomplemented
12C2×C22⋊Dic199C2212yes—
13#20263869—28no—
14C199⋊#95—22no—
15C23×C4×C199C2211yescharacteristic
16C2×C22⋊C4×C199C2212yes—
17#95×C199—22no—
18C23×D199C2×C412yes—
19#15691886—216no—
20#15691886—216no—
21C2×Dic398—24no—
22C2×Dic398—22no—
23C2×D796—24no—
24C2×D796—22no—
25C2×C22⋊D199—24no—
26C2×C22⋊D199—22no—
27C2×C22⋊D199—22no—
28C2×C22⋊D199D418yes—
29C22×Dic199—22no—
30C22×Dic199C2×C412yes—
31C2×C4×D199—24no—
32C2×C4×D199—24no—
33C2×C4×D199—24no—
34C22⋊Dic199—24no—
35C24×C199C2311yescharacteristic
36C22×C4×C199—24no—
37C22×C4×C199—22no—
38C22×C4×C199C2×C411yescharacteristic
39C22×C4×C199C2×C411yescharacteristic
40C22×C4×C199D412yes—
41C22×C4×C199D412yes—
42C22⋊C4×C199—24no—
43C22×D199—22no—
44C22×D199—22no—
45C22×D199—44no—
46C22×D199C22⋊C414yes—
47Dic398—28no—
48D796—28no—
49C22⋊D199—28no—
50C22⋊D199—28no—
51C2×Dic199—22no—
52C2×Dic199—22no—
53C2×Dic199—44no—
54C2×Dic199—42no—
55C2×Dic199C22⋊C414yes—
56C4×D199—216no—
57C23×C199—42no—
58C23×C199C22×C411yescharacteristic
59C23×C199C2×D414yes—
60C23×C199C2×D411yescharacteristic
61C23×C199C2×D411yescharacteristic
62C2×C4×C199—22no—
63C2×C4×C199—22no—
64C2×C4×C199—24no—
65C2×C4×C199—24no—
66C2×C4×C199—44no—
67C2×C4×C199—42no—
68C2×C4×C199C22⋊C412yes—
69C2×C4×C199C22⋊C412yes—
70C22×C199—22no—
71C22×C199—22no—
72C22×C199—22no—
73C22×C199—44no—
74C22×C199C2×C22⋊C412yes—
75C22×C199C2×C22⋊C411yescharacteristic
76C22×C199#12012yes—
77C22×C199#12012yes—
78C2×D199—24no—
79C2×D199—44no—
80C2×D199—44no—
81C2×D199—44no—
82C4×C199—22no—
83C4×C199—22no—
84C4×C199—82no—
85Dic199—24no—
86Dic199—82no—
87C2×C199—22no—
88C2×C199—22no—
89C2×C199—22no—
90C2×C199#37911yescharacteristic
91C2×C199#41012yes—
92D199—44no—
93C199#178211yescharacteristic, complemented, Sylow
94#1782—1991nomaximal, Sylow
95C22×#41—1991no—
96C2×#99—1994no—
97#381—1991no—
98#381D19911yescharacteristic, complemented
99C2×#41—3988no—
100C2×#41—3982no—
101C2×#41—3982no—
102C22×Q8—1991no—
103C22×D4—1992no—
104C22×D4—1991no—
105C23×C4—1992no—
106C23×C4C2×D19911yescharacteristic
107C2×C22⋊C4—1992no—
108C2×C22⋊C4C2×D19912yes—
109#99—3988no—
110#95—22no—
111#95—3982no—
112C24—1992no—
113C24C22×D19911yescharacteristic
114#41—39816no—
115#41—39816no—
116C2×Q8—3984no—
117C2×Q8—3982no—
118C2×D4—3984no—
119C2×D4—3984no—
120C2×D4—3982no—
121C2×D4—3982no—
122C2×D4—3982no—
123C2×D4—1998no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—3984no—
127C22×C4—3984no—
128C22×C4—3982no—
129C22×C4—3984no—
130C22×C4—1992no—
131C22×C4C4×D19911yescharacteristic
132C22×C4C4×D19911yescharacteristic
133C22×C4C22⋊D19912yes—
134C22×C4D79612yes—
135C22⋊C4—24no—
136C22⋊C4—3984no—
137C23—42no—
138C23—7964no—
139C23—3982no—
140C23—3982no—
141C23—1994no—
142C23C2×C4×D19911yescharacteristic
143C23D4×D19914yes—
144C23C2×C22⋊D19911yescharacteristic
145C23C2×D79611yescharacteristic
146Q8—3988no—
147D4—3988no—
148D4—3988no—
149D4—3988no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—7964no—
157C2×C4—7962no—
158C2×C4—39816no—
159C2×C4—3982no—
160C2×C4—3982no—
161C2×C4—1994no—
162C2×C4(C2×D199)⋊C412yes—
163C2×C4(C2×D199)⋊C412yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—7964no—
169C22—7964no—
170C22—7964no—
171C22—3984no—
172C22C22⋊C4×D19912yes—
173C22C2×(C2×D199)⋊C411yescharacteristic
174C22#2026392312yes—
175C22#2026392512yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—15922no—
180C4—3984no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—7964no—
185C2#2716387311yescharacteristic
186C2#2716399112yes—
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC23×C199C22×C4
FrattiniC24C22×D199
Fitting#381×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC22×#15691886C2
cosocleC24×C199C23
hypercenter#381D199
hypocenterC199#1782
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20263869C2semidirect
3. #27163881C2semidirect
4. #381×C199C2semidirect
8. C22×Dic398C4semidirect
9. C22×D796C4semidirect
11. C22×C4×D199C4semidirect
93. C199#1782semidirect
98. #381D199semidirect

Maximal subgroups

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

subgroupindex
C22×#156918862
C2×#202638692
#271638812
#381×C1992
#1782199