Finite Group Databasebeta

← C2×#496×C199 #38766220

Subgroups — C2×#496×C199 #38766220

Order 25472.

668 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 232 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×SD16×C199C212yesmaximal, complemented
2C22×D8×C199C212yesmaximal, complemented
3C2×#127×C199C211yesmaximal, characteristic, complemented
4C2×#124×C199C211yesmaximal, characteristic, complemented
5#496×C199C218yesmaximal, direct product factor
6C2×#97×C199C211yesmaximal, characteristic, complemented
7C22×Q8×C199C2211yescharacteristic
8C22×D4×C199—22no—
9C22×D4×C199C2211yescharacteristic
10C22×D4×C199C2212yes—
11C2×SD16×C199—28no—
12C2×SD16×C199C2218yescomplemented
13C2×D8×C199—28no—
14C2×D8×C199C2218yescomplemented
15C22×C8×C199C2212yes—
16#127×C199—22no—
17#127×C199C2214yescomplemented
18#124×C199—22no—
19#124×C199C2214yescomplemented
20C2×C22⋊C4×C199—22no—
21C2×C42×C199C2211yescharacteristic
22#97×C199C2214yescomplemented
23C24×C199—21no—
24C24×C199—22no—
25C2×Q8×C199—21no—
26C2×Q8×C199—24no—
27C2×Q8×C199C2312yes—
28C2×D4×C199—24no—
29C2×D4×C199—24no—
30C2×D4×C199—24no—
31C2×D4×C199—21no—
32C2×D4×C199—22no—
33C2×D4×C199—28no—
34C2×D4×C199—48no—
35C2×D4×C199C2312yes—
36C2×D4×C199C2314yes—
37C22×C4×C199—21no—
38C22×C4×C199D412yes—
39C22×C4×C199C2311yescharacteristic
40SD16×C199—216no—
41D8×C199—216no—
42C2×C8×C199D418yescomplemented
43C2×C8×C199C2314yes—
44C22⋊C4×C199—28no—
45C42×C199D412yes—
46C42×C199C2312yes—
47C23×C199—22no—
48C23×C199—24no—
49C23×C199—42no—
50C23×C199—44no—
51C23×C199—48no—
52C23×C199—44no—
53C23×C199C2×D411yescharacteristic
54Q8×C199—24no—
55Q8×C199—42no—
56D4×C199—24no—
57D4×C199—24no—
58D4×C199—24no—
59D4×C199—416no—
60D4×C199—42no—
61D4×C199—44no—
62C2×C4×C199—24no—
63C2×C4×C199—22no—
64C2×C4×C199—42no—
65C2×C4×C199C2×D414yes—
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D414yes—
68C2×C4×C199C2411yescharacteristic
69C8×C199C2×D418yescomplemented
70C22×C199—42no—
71C22×C199—44no—
72C22×C199—48no—
73C22×C199—44no—
74C22×C199—88no—
75C22×C199—84no—
76C22×C199#12712yes—
77C22×C199AGL(1,ℤ/8)14yes—
78C22×C199C22×D411yescharacteristic
79C4×C199—24no—
80C4×C199—42no—
81C4×C199#12712yes—
82C4×C199C22×D412yes—
83C2×C199—84no—
84C2×C199—82no—
85C2×C199#49614yesdirect product factor
86C2×C199C2×#12711yescharacteristic
87C2×C199C2×AGL(1,ℤ/8)12yes—
88C199C2×#49611yescharacteristic, direct product factor, Sylow
89C2×#496C19911yesmaximal, characteristic, direct product factor, Sylow
90C22×SD16C2×C19912yescomplemented
91C22×D8C2×C19912yescomplemented
92C2×#127C2×C19911yescharacteristic, complemented
93C2×#124C2×C19911yescharacteristic, complemented
94#496C2×C19918yesdirect product factor
95C2×#97C2×C19911yescharacteristic, complemented
96C22×Q8C22×C19911yescharacteristic
97C22×D4—22no—
98C22×D4C22×C19911yescharacteristic
99C22×D4C22×C19912yes—
100C2×SD16—28no—
101C2×SD16C22×C19918yescomplemented
102C2×D8—28no—
103C2×D8C22×C19918yescomplemented
104C22×C8C22×C19912yes—
105#127—22no—
106#127C22×C19914yescomplemented
107#124—22no—
108#124C22×C19914yescomplemented
109C2×C22⋊C4—22no—
110C2×C42C22×C19911yescharacteristic
111#97C22×C19914yescomplemented
112C24—21no—
113C24—22no—
114C2×Q8—21no—
115C2×Q8—24no—
116C2×Q8C23×C19912yes—
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—24no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—28no—
123C2×D4—48no—
124C2×D4C23×C19912yes—
125C2×D4C23×C19914yes—
126C22×C4—21no—
127C22×C4D4×C19912yes—
128C22×C4C23×C19911yescharacteristic
129SD16—216no—
130D8—216no—
131C2×C8D4×C19918yescomplemented
132C2×C8C23×C19914yes—
133C22⋊C4—28no—
134C42D4×C19912yes—
135C42C23×C19912yes—
136C23—22no—
137C23—24no—
138C23—42no—
139C23—44no—
140C23—48no—
141C23—44no—
142C23C2×D4×C19911yescharacteristic
143Q8—24no—
144Q8—42no—
145D4—24no—
146D4—24no—
147D4—24no—
148D4—416no—
149D4—42no—
150D4—44no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4C2×D4×C19914yes—
155C2×C4C2×D4×C19911yescharacteristic
156C2×C4C2×D4×C19914yes—
157C2×C4C24×C19911yescharacteristic
158C8C2×D4×C19918yescomplemented
159C22—42no—
160C22—44no—
161C22—48no—
162C22—44no—
163C22—88no—
164C22—84no—
165C22#127×C19912yes—
166C22AGL(1,ℤ/8)×C19914yes—
167C22C22×D4×C19911yescharacteristic
168C4—24no—
169C4—42no—
170C4#127×C19912yes—
171C4C22×D4×C19912yes—
172C2—84no—
173C2—82no—
174C2#496×C19914yesdirect product factor
175C2C2×#127×C19911yescharacteristic
176C2C2×AGL(1,ℤ/8)×C19912yes—
subgroupG/H
center, socleC23×C199C2×D4
commutator, Frattini, cosocleC2×C4C24×C199
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C199 ◁ C2×C42×C199 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×SD16×C199C2semidirect
2. C22×D8×C199C2semidirect
3. C2×#127×C199C2semidirect
4. C2×#124×C199C2semidirect
5. #496×C199C2direct (N × H)
6. C2×#97×C199C2semidirect
12. C2×SD16×C199C22semidirect
14. C2×D8×C199C22semidirect
17. #127×C199C22semidirect
19. #124×C199C22semidirect
22. #97×C199C22semidirect
42. C2×C8×C199D4semidirect
69. C8×C199C2×D4semidirect
85. C2×C199#496direct (N × H)
88. C199C2×#496direct (N × H)
89. C2×#496C199direct (N × H)
90. C22×SD16C2×C199semidirect
91. C22×D8C2×C199semidirect
92. C2×#127C2×C199semidirect
93. C2×#124C2×C199semidirect
94. #496C2×C199direct (N × H)
95. C2×#97C2×C199semidirect
101. C2×SD16C22×C199semidirect
103. C2×D8C22×C199semidirect
106. #127C22×C199semidirect
108. #124C22×C199semidirect
111. #97C22×C199semidirect
131. C2×C8D4×C199semidirect
158. C8C2×D4×C199semidirect
174. C2#496×C199direct (N × H)

Maximal subgroups

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

subgroupindex
C22×SD16×C1992
C22×D8×C1992
C2×#127×C1992
C2×#124×C1992
#496×C1992
C2×#97×C1992
C2×#496199