Finite Group Databasebeta

← #28384387

Subgroups — #28384387

Order 14272.

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#20297351C211yesmaximal, characteristic, complemented
2#20297350C211yesmaximal, characteristic, complemented
3#20297336C212yesmaximal, complemented
4#20297334C211yesmaximal, characteristic, complemented
5#20297329C211yesmaximal, characteristic, complemented
6#20297328C212yesmaximal, complemented
7#20297325C212yesmaximal, complemented
8C2×#16813159C211yesmaximal, characteristic, complemented
9SD16×D223C212yesmaximal, complemented
10#20297310C211yesmaximal, characteristic, complemented
11#135×C223C211yesmaximal, characteristic, complemented
12#16813171C2212yes—
13#16813170—21no—
14#16813170—21no—
15#16813170C2211yescharacteristic
16#16813169—22no—
17#16813169C2212yescomplemented
18C2×Dic446—21no—
19C2×Dic446C2211yescharacteristic, complemented
20Q8×D223C2212yescomplemented
21C2×D892—21no—
22C2×D892C2211yescharacteristic
23D4×D223—22no—
24D4×D223C2212yes—
25#16813160C2212yescomplemented
26#16813159C2212yescomplemented
27#16813159C2212yescomplemented
28#16813158C2212yescomplemented
29#16813157C2212yescomplemented
30#16813155C2212yescomplemented
31#16813154C2211yescharacteristic, complemented
32#16813153C2212yescomplemented
33C8×D223C2212yescomplemented
34#41×C223C2211yescharacteristic
35#41×C223C2211yescharacteristic, complemented
36Q16×C223C2211yescharacteristic, complemented
37SD16×C223C2211yescharacteristic, complemented
38SD16×C223C2211yescharacteristic, complemented
39D8×C223C2211yescharacteristic, complemented
40C2×C8×C223C2211yescharacteristic, complemented
41C22×D223—21no—
42C22×D223—22no—
43Dic446—21no—
44Dic446—22no—
45Dic446D411yescharacteristic
46Dic446C2312yes—
47D892—22no—
48D892—21no—
49D892D411yescharacteristic
50D892C2312yes—
51C22⋊D223—22no—
52C22⋊D223—22no—
53C22⋊D223D412yes—
54C2×Dic223—21no—
55C2×Dic223—22no—
56C4×D223—22no—
57C4×D223—22no—
58C4×D223C2312yes—
59C223⋊C8C2312yes—
60Q8×C223C2311yescharacteristic
61Q8×C223C2311yescharacteristic
62D4×C223—21no—
63D4×C223—21no—
64D4×C223C2311yescharacteristic
65D4×C223C2311yescharacteristic
66C2×C4×C223—21no—
67C2×C4×C223—21no—
68C2×C4×C223C2311yescharacteristic
69C8×C223C2311yescharacteristic
70C8×C223C2311yescharacteristic
71C22×C223—21no—
72C22×C223—21no—
73C22×C223C2×D411yescharacteristic
74C2×D223—22no—
75C2×D223—82no—
76C2×D223—81no—
77C2×D223C2×D412yes—
78C4×C223—21no—
79C4×C223—21no—
80C4×C223C2×D411yescharacteristic
81C4×C223C2411yescharacteristic
82Dic223—22no—
83Dic223C2×D412yes—
84C2×C223—21no—
85C2×C223—41no—
86C2×C223—41no—
87C2×C223C22×D411yescharacteristic
88D223—22no—
89D223—42no—
90C223#57711yescharacteristic, complemented, Sylow
91#577—2231nomaximal, Sylow
922−1+4—2231no—
93H(5,𝔽2)—2231no—
94#137—2231no—
95#137—2232no—
96AGL(1,ℤ/8)—2231no—
97AGL(1,ℤ/8)—2232no—
98#135—2232no—
99#135D22311yescharacteristic, complemented
100C2×SD16—2232no—
101C2×SD16—2231no—
102#131—2231no—
103#41—4461no—
104#41—4461no—
105#41—4462no—
106#41—2232no—
107#41—2232no—
108#41—2231no—
109#41C2×D22311yescharacteristic
110#41C2×D22311yescharacteristic, complemented
111C2×Q8—4461no—
112C2×Q8—2232no—
113C2×Q8—2231no—
114C2×D4—4461no—
115C2×D4—4462no—
116C2×D4—2232no—
117C2×D4—2231no—
118Q16—2232no—
119Q16C2×D22311yescharacteristic, complemented
120SD16—2232no—
121SD16—2232no—
122SD16—2232no—
123SD16—2232no—
124SD16C2×D22311yescharacteristic, complemented
125SD16C2×D22311yescharacteristic, complemented
126D8—2232no—
127D8C2×D22311yescharacteristic, complemented
128M16—2231no—
129M16—2232no—
130C2×C8—2232no—
131C2×C8C2×D22311yescharacteristic, complemented
132C23—4462no—
133C23—4461no—
134Q8—4462no—
135Q8—4461no—
136Q8—2232no—
137Q8—2231no—
138Q8C22×D22311yescharacteristic
139Q8C22×D22311yescharacteristic
140D4—21no—
141D4—21no—
142D4—4462no—
143D4—4462no—
144D4—4462no—
145D4—4461no—
146D4—2232no—
147D4—2231no—
148D4—2232no—
149D4C22×D22311yescharacteristic
150D4C22×D22311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4462no—
154C2×C4—4462no—
155C2×C4—4461no—
156C2×C4—4462no—
157C2×C4—2232no—
158C2×C4C22×D22311yescharacteristic
159C8—2232no—
160C8C22×D22311yescharacteristic
161C8C22×D22311yescharacteristic
162C22—21no—
163C22—21no—
164C22—4462no—
165C22—2232no—
166C22—17841no—
167C22—17842no—
168C22D4×D22311yescharacteristic
169C4—21no—
170C4—21no—
171C4—4462no—
172C4—2232no—
173C4D4×D22311yescharacteristic
174C4C23×D22311yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—8922no—
179C2—4462no—
180C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC2C2×D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
Fitting#135×C223C2
socleC2×C223C22×D4
hypercenter#135D223
hypocenterC223#577
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #20297351C2semidirect
2. #20297350C2semidirect
3. #20297336C2semidirect
4. #20297334C2semidirect
5. #20297329C2semidirect
6. #20297328C2semidirect
7. #20297325C2semidirect
8. C2×#16813159C2semidirect
9. SD16×D223C2semidirect
10. #20297310C2semidirect
11. #135×C223C2semidirect
17. #16813169C22semidirect
19. C2×Dic446C22semidirect
20. Q8×D223C22semidirect
25. #16813160C22semidirect
26. #16813159C22semidirect
27. #16813159C22semidirect
28. #16813158C22semidirect
29. #16813157C22semidirect
30. #16813155C22semidirect
31. #16813154C22semidirect
32. #16813153C22semidirect
33. C8×D223C22semidirect
35. #41×C223C22semidirect
36. Q16×C223C22semidirect
37. SD16×C223C22semidirect
38. SD16×C223C22semidirect
39. D8×C223C22semidirect
40. C2×C8×C223C22semidirect
90. C223#577semidirect
99. #135D223semidirect
110. #41C2×D223semidirect
119. Q16C2×D223semidirect
124. SD16C2×D223semidirect
125. SD16C2×D223semidirect
127. D8C2×D223semidirect
131. C2×C8C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973512
#202973502
#202973362
#202973342
#202973292
#202973282
#202973252
C2×#168131592
SD16×D2232
#202973102
#135×C2232
#577223