Finite Group Databasebeta

← #27133469

Subgroups — #27133469

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#20259367C211yesmaximal, characteristic, complemented
3#20259353C211yesmaximal, characteristic, complemented
4#20259351C212yesmaximal, complemented
5#20259349C212yesmaximal, complemented
6#20259348C211yesmaximal, characteristic, complemented
7#20259344C211yesmaximal, characteristic, complemented
8#20259343C212yesmaximal, complemented
9SD16×D191C212yesmaximal, complemented
10#20259328C211yesmaximal, characteristic, complemented
11C2×SD16×C191C211yesmaximal, characteristic, complemented
12#15690288—21no—
13#15690288C2212yescomplemented
14#15690287—22no—
15#15690287C2211yescharacteristic
16#15690287C2211yescharacteristic
17#15690287C2211yescharacteristic, complemented
18#15690286—21no—
19#15690286C2212yes—
20Q8×D191—21no—
21Q8×D191C2212yescomplemented
22C2×C22⋊D191—22no—
23D4×D191—21no—
24D4×D191C2212yes—
25Dic764C2211yescharacteristic, complemented
26#15690277C2212yescomplemented
27#15690276C2211yescharacteristic, complemented
28#15690276C2211yescharacteristic, complemented
29#15690275C2212yescomplemented
30#15690274C2212yescomplemented
31D1528C2211yescharacteristic, complemented
32#15690272C2212yescomplemented
33#15690271C2211yescharacteristic, complemented
34#15690270C2212yescomplemented
35C8×D191C2212yescomplemented
36C2×Q8×C191C2211yescharacteristic, complemented
37C2×D4×C191C2211yescharacteristic
38SD16×C191C2212yescomplemented
39SD16×C191C2212yescomplemented
40C2×C8×C191C2211yescharacteristic, complemented
41C22×D191—22no—
42Dic382—22no—
43Dic382D411yescharacteristic
44Dic382C2311yescharacteristic
45Dic382C2311yescharacteristic
46D764—22no—
47D764D411yescharacteristic
48D764C2311yescharacteristic
49D764C2311yescharacteristic
50C22⋊D191—21no—
51C22⋊D191—21no—
52C22⋊D191—22no—
53C22⋊D191—22no—
54C22⋊D191D412yes—
55C2×Dic191—22no—
56C4×D191—22no—
57C4×D191—21no—
58C4×D191—21no—
59C4×D191—22no—
60C4×D191C2312yes—
61C191⋊C8C2312yes—
62C23×C191—21no—
63Q8×C191—21no—
64Q8×C191C2312yes—
65D4×C191—21no—
66D4×C191C2312yes—
67C2×C4×C191—21no—
68C2×C4×C191C2311yescharacteristic
69C8×C191C2311yescharacteristic
70C8×C191C2311yescharacteristic
71C22×C191—22no—
72C22×C191—81no—
73C22×C191C2×D411yescharacteristic
74C2×D191—21no—
75C2×D191—21no—
76C2×D191—82no—
77C2×D191C2×D412yes—
78C4×C191—22no—
79C4×C191C2×D411yescharacteristic
80C4×C191C2411yescharacteristic
81Dic191—21no—
82Dic191—21no—
83Dic191C2×D412yes—
84C2×C191—21no—
85C2×C191—42no—
86C2×C191C22×D411yescharacteristic
87D191—22no—
88D191—41no—
89D191—41no—
90C191#57711yescharacteristic, complemented, Sylow
91#577—1911nomaximal, Sylow
922−1+4—1911no—
93H(5,𝔽2)—1911no—
94#137—1911no—
95#137—1912no—
96AGL(1,ℤ/8)—1911no—
97AGL(1,ℤ/8)—1912no—
98#135—1912no—
99#135—1911no—
100C2×SD16—1912no—
101C2×SD16D19111yescharacteristic, complemented
102#131—1911no—
103#41—3822no—
104#41—3821no—
105#41—3821no—
106#41—1912no—
107#41—1912no—
108#41—1911no—
109#41—1911no—
110#41—1911no—
111C2×Q8—3821no—
112C2×Q8—1912no—
113C2×Q8C2×D19111yescharacteristic, complemented
114C2×D4—3821no—
115C2×D4—3822no—
116C2×D4—1912no—
117C2×D4C2×D19111yescharacteristic
118Q16—1912no—
119Q16—1911no—
120SD16—1912no—
121SD16—1912no—
122SD16—1911no—
123SD16—1911no—
124SD16C2×D19112yescomplemented
125SD16C2×D19112yescomplemented
126D8—1912no—
127D8—1911no—
128M16—1911no—
129M16—1912no—
130C2×C8—1912no—
131C2×C8C2×D19111yescharacteristic, complemented
132C23—21no—
133C23—3822no—
134Q8—21no—
135Q8—3822no—
136Q8—1911no—
137Q8—1911no—
138Q8—1911no—
139Q8C22×D19112yes—
140D4—21no—
141D4—3822no—
142D4—3822no—
143D4—3821no—
144D4—3821no—
145D4—3822no—
146D4—1912no—
147D4—1911no—
148D4—1911no—
149D4—1911no—
150D4C22×D19112yes—
151C2×C4—21no—
152C2×C4—3822no—
153C2×C4—3822no—
154C2×C4—3822no—
155C2×C4—3821no—
156C2×C4—3821no—
157C2×C4—1912no—
158C2×C4C22×D19111yescharacteristic
159C8—1912no—
160C8C22×D19111yescharacteristic
161C8C22×D19111yescharacteristic
162C22—22no—
163C22—81no—
164C22—3821no—
165C22—3821no—
166C22—1912no—
167C22—15282no—
168C22D4×D19111yescharacteristic
169C4—22no—
170C4—3821no—
171C4—3821no—
172C4—1912no—
173C4D4×D19111yescharacteristic
174C4C23×D19111yescharacteristic
175C2—21no—
176C2—42no—
177C2—3822no—
178C2—7641no—
179C2—7641no—
180C2C2×D4×D19111yescharacteristic
subgroupG/H
centerC2C2×D4×D191
commutator, cosocleC4×C191C24
FrattiniC4C23×D191
FittingC2×SD16×C191C2
socleC2×C191C22×D4
hypercenterC2×SD16D191
hypocenterC191#577
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #20259370C2semidirect
2. #20259367C2semidirect
3. #20259353C2semidirect
4. #20259351C2semidirect
5. #20259349C2semidirect
6. #20259348C2semidirect
7. #20259344C2semidirect
8. #20259343C2semidirect
9. SD16×D191C2semidirect
10. #20259328C2semidirect
11. C2×SD16×C191C2semidirect
13. #15690288C22semidirect
17. #15690287C22semidirect
21. Q8×D191C22semidirect
25. Dic764C22semidirect
26. #15690277C22semidirect
27. #15690276C22semidirect
28. #15690276C22semidirect
29. #15690275C22semidirect
30. #15690274C22semidirect
31. D1528C22semidirect
32. #15690272C22semidirect
33. #15690271C22semidirect
34. #15690270C22semidirect
35. C8×D191C22semidirect
36. C2×Q8×C191C22semidirect
38. SD16×C191C22semidirect
39. SD16×C191C22semidirect
40. C2×C8×C191C22semidirect
90. C191#577semidirect
101. C2×SD16D191semidirect
113. C2×Q8C2×D191semidirect
124. SD16C2×D191semidirect
125. SD16C2×D191semidirect
131. C2×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#202593702
#202593672
#202593532
#202593512
#202593492
#202593482
#202593442
#202593432
SD16×D1912
#202593282
C2×SD16×C1912
#577191