Finite Group Databasebeta

← #27133472

Subgroups — #27133472

Order 12224.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#20259370C211yesmaximal, characteristic, complemented
2#20259368C211yesmaximal, characteristic, complemented
3#20259354C212yesmaximal, complemented
4#20259350C211yesmaximal, characteristic, complemented
5#20259349C212yesmaximal, complemented
6#20259345C212yesmaximal, complemented
7#20259342C211yesmaximal, characteristic, complemented
8C2×#15690274C211yesmaximal, characteristic, complemented
9SD16×D191C212yesmaximal, complemented
10#20259330C211yesmaximal, characteristic, complemented
11#137×C191C211yesmaximal, characteristic, complemented
12#15690288—21no—
13#15690288C2212yes—
14#15690288C2212yescomplemented
15#15690287—21no—
16#15690287—22no—
17#15690287C2211yescharacteristic
18#15690287C2211yescharacteristic, complemented
19Q8×D191—21no—
20Q8×D191C2212yescomplemented
21C2×D764—21no—
22C2×D764C2211yescharacteristic
23D4×D191—22no—
24D4×D191C2212yes—
25#15690277C2211yescharacteristic, complemented
26#15690276C2212yescomplemented
27#15690275C2211yescharacteristic, complemented
28#15690274C2212yescomplemented
29#15690274C2212yescomplemented
30#15690274C2211yescharacteristic, complemented
31D1528C2212yescomplemented
32#15690272C2211yescharacteristic, complemented
33#15690270C2212yescomplemented
34C2×C191⋊C8C2211yescharacteristic, complemented
35C8×D191C2212yescomplemented
36#41×C191C2211yescharacteristic
37C2×Q8×C191C2211yescharacteristic, complemented
38Q16×C191C2212yescomplemented
39SD16×C191C2212yescomplemented
40M16×C191C2211yescharacteristic, complemented
41C22×D191—21no—
42C22×D191—22no—
43Dic382—22no—
44Dic382D411yescharacteristic
45Dic382C2311yescharacteristic
46D764—22no—
47D764—22no—
48D764—21no—
49D764D411yescharacteristic
50D764C2311yescharacteristic
51D764C2312yes—
52C22⋊D191—21no—
53C22⋊D191—22no—
54C22⋊D191D412yes—
55C4×D191—22no—
56C4×D191—21no—
57C4×D191—22no—
58C4×D191—22no—
59C4×D191C2312yes—
60C191⋊C8C2311yescharacteristic
61C191⋊C8C2311yescharacteristic
62Q8×C191—21no—
63Q8×C191C2312yes—
64Q8×C191C2311yescharacteristic
65D4×C191—21no—
66D4×C191C2311yescharacteristic
67C2×C4×C191—21no—
68C2×C4×C191—21no—
69C2×C4×C191C2311yescharacteristic
70C8×C191C2312yes—
71C22×C191—21no—
72C22×C191C2×D411yescharacteristic
73C2×D191—21no—
74C2×D191—22no—
75C2×D191—82no—
76C2×D191—81no—
77C2×D191C2×D412yes—
78C4×C191—21no—
79C4×C191—22no—
80C4×C191C2×D411yescharacteristic
81C4×C191C2411yescharacteristic
82Dic191—21no—
83Dic191C2×D412yes—
84C2×C191—21no—
85C2×C191—41no—
86C2×C191C22×D411yescharacteristic
87D191—22no—
88D191—41no—
89D191—42no—
90C191#57711yescharacteristic, complemented, Sylow
91#577—1911nomaximal, Sylow
922−1+4—1911no—
93H(5,𝔽2)—1911no—
94#137—1912no—
95#137D19111yescharacteristic, complemented
96AGL(1,ℤ/8)—1912no—
97AGL(1,ℤ/8)—1911no—
98#135—1911no—
99#135—1912no—
100C2×SD16—1911no—
101C2×SD16—1912no—
102#131—1911no—
103#41—3821no—
104#41—3822no—
105#41—3821no—
106#41—1912no—
107#41—1912no—
108#41—1911no—
109#41—1911no—
110#41C2×D19111yescharacteristic
111C2×Q8—3821no—
112C2×Q8—1912no—
113C2×Q8C2×D19111yescharacteristic, complemented
114C2×D4—3821no—
115C2×D4—3822no—
116C2×D4—1912no—
117C2×D4—1911no—
118Q16—1911no—
119Q16C2×D19112yescomplemented
120SD16—1912no—
121SD16—1912no—
122SD16—1911no—
123SD16—1911no—
124SD16—1912no—
125SD16C2×D19112yescomplemented
126D8—1911no—
127D8—1912no—
128M16—1912no—
129M16C2×D19111yescharacteristic, complemented
130C2×C8—1911no—
131C2×C8—1912no—
132C23—3822no—
133C23—3821no—
134Q8—21no—
135Q8—3822no—
136Q8—1911no—
137Q8—1911no—
138Q8C22×D19112yes—
139Q8C22×D19111yescharacteristic
140D4—21no—
141D4—3822no—
142D4—3822no—
143D4—3821no—
144D4—3822no—
145D4—3821no—
146D4—1912no—
147D4—1911no—
148D4—1911no—
149D4—1912no—
150D4C22×D19111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3822no—
154C2×C4—3822no—
155C2×C4—3822no—
156C2×C4—3821no—
157C2×C4—1912no—
158C2×C4C22×D19111yescharacteristic
159C8—1911no—
160C8—1911no—
161C8C22×D19112yes—
162C22—21no—
163C22—3821no—
164C22—3822no—
165C22—1912no—
166C22—15281no—
167C22—15282no—
168C22D4×D19111yescharacteristic
169C4—21no—
170C4—22no—
171C4—3821no—
172C4—1912no—
173C4D4×D19111yescharacteristic
174C4C23×D19111yescharacteristic
175C2—21no—
176C2—41no—
177C2—3822no—
178C2—7642no—
179C2—7641no—
180C2C2×D4×D19111yescharacteristic
subgroupG/H
centerC2C2×D4×D191
commutator, cosocleC4×C191C24
FrattiniC4C23×D191
Fitting#137×C191C2
socleC2×C191C22×D4
hypercenter#137D191
hypocenterC191#577
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #20259370C2semidirect
2. #20259368C2semidirect
3. #20259354C2semidirect
4. #20259350C2semidirect
5. #20259349C2semidirect
6. #20259345C2semidirect
7. #20259342C2semidirect
8. C2×#15690274C2semidirect
9. SD16×D191C2semidirect
10. #20259330C2semidirect
11. #137×C191C2semidirect
14. #15690288C22semidirect
18. #15690287C22semidirect
20. Q8×D191C22semidirect
25. #15690277C22semidirect
26. #15690276C22semidirect
27. #15690275C22semidirect
28. #15690274C22semidirect
29. #15690274C22semidirect
30. #15690274C22semidirect
31. D1528C22semidirect
32. #15690272C22semidirect
33. #15690270C22semidirect
34. C2×C191⋊C8C22semidirect
35. C8×D191C22semidirect
37. C2×Q8×C191C22semidirect
38. Q16×C191C22semidirect
39. SD16×C191C22semidirect
40. M16×C191C22semidirect
90. C191#577semidirect
95. #137D191semidirect
113. C2×Q8C2×D191semidirect
119. Q16C2×D191semidirect
125. SD16C2×D191semidirect
129. M16C2×D191semidirect

Maximal subgroups

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

subgroupindex
#202593702
#202593682
#202593542
#202593502
#202593492
#202593452
#202593422
C2×#156902742
SD16×D1912
#202593302
#137×C1912
#577191