Finite Group Databasebeta

← #35204022

Subgroups — #35204022

Order 20288.

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

normal subgroups only

This group has 262 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
1C2×#19099575C211yesmaximal, characteristic, complemented
2#41×D317C211yesmaximal, characteristic, complemented
3#23778812C212yesmaximal, complemented
4#23778811C211yesmaximal, characteristic, complemented
5#23778808C211yesmaximal, characteristic, complemented
6#23778807C212yesmaximal, complemented
7#23778805C211yesmaximal, characteristic, complemented
8#23778803C212yesmaximal, complemented
9#23778801C212yesmaximal, complemented
10M16×D317C211yesmaximal, characteristic, complemented
11#137×C317C211yesmaximal, characteristic, complemented
12#19099575—21no—
13#19099575—21no—
14#19099575C2212yes—
15#19099575C2211yescharacteristic, complemented
16#19099575C2212yescomplemented
17#19099574—21no—
18#19099574C2211yescharacteristic
19#19099573—21no—
20#19099573C2211yescharacteristic, complemented
21Q8×D317C2211yescharacteristic, complemented
22C2×D1268—21no—
23C2×D1268C2211yescharacteristic, complemented
24D4×D317—21no—
25D4×D317C2211yescharacteristic, complemented
26C2×C4×D317—21no—
27C2×C4×D317—21no—
28C2×C4×D317C2211yescharacteristic
29#19099564C2212yescomplemented
30#19099563C2212yescomplemented
31#19099561C2212yescomplemented
32#19099561C2212yescomplemented
33D2536C2212yescomplemented
34#19099559C2212yescomplemented
35#19099558C2211yescharacteristic, complemented
36#19099557C2212yescomplemented
37C8×D317C2212yescomplemented
38#41×C317C2211yescharacteristic
39C2×Q8×C317C2211yescharacteristic, complemented
40Q16×C317C2212yescomplemented
41SD16×C317C2212yescomplemented
42M16×C317C2211yescharacteristic, complemented
43C22×D317—21no—
44C22×D317—21no—
45C22×D317D411yescharacteristic
46Dic634—21no—
47Dic634C2311yescharacteristic
48D1268—21no—
49D1268—22no—
50D1268—22no—
51D1268—21no—
52D1268C2312yes—
53D1268C2311yescharacteristic
54C22⋊D317—21no—
55C22⋊D317—21no—
56C22⋊D317—21no—
57C2×Dic317—21no—
58C2×Dic317D411yescharacteristic
59C4×D317—21no—
60C4×D317—21no—
61C4×D317—21no—
62C4×D317—22no—
63C4×D317—22no—
64C4×D317D411yescharacteristic
65C4×D317D411yescharacteristic
66C4×D317C2311yescharacteristic
67C4×D317C2311yescharacteristic
68#14579261C2312yes—
69Q8×C317—21no—
70Q8×C317C2312yes—
71Q8×C317C2311yescharacteristic
72D4×C317—21no—
73D4×C317C2311yescharacteristic
74C2×C4×C317—21no—
75C2×C4×C317—21no—
76C2×C4×C317C2311yescharacteristic
77C8×C317C2312yes—
78C22×C317—21no—
79C22×C317C2×D411yescharacteristic
80C2×D317—21no—
81C2×D317—22no—
82C2×D317—41no—
83C2×D317—42no—
84C2×D317—42no—
85C2×D317C2×D411yescharacteristic
86C2×D317C2×D411yescharacteristic
87C4×C317—21no—
88C4×C317—22no—
89C4×C317C2×D411yescharacteristic
90C4×C317C2411yescharacteristic
91Dic317—21no—
92Dic317C2×D411yescharacteristic
93Dic317C2×D411yescharacteristic
94C2×C317—21no—
95C2×C317—41no—
96C2×C317C22×D411yescharacteristic
97D317—21no—
98D317—21no—
99D317—41no—
100D317—42no—
101C317#57511yescharacteristic, complemented, Sylow
102#575—3171nomaximal, Sylow
103C2×#41—3171no—
104C2×#41—3171no—
105#137—3171no—
106#137—3172no—
107#137D31711yescharacteristic, complemented
108AGL(1,ℤ/8)—3171no—
109AGL(1,ℤ/8)—3172no—
110AGL(1,ℤ/8)—3171no—
111#135—3172no—
112#135—3172no—
113C2×M16—3171no—
114#41—3171no—
115#41—3172no—
116#41—3172no—
117#41—3171no—
118#41—3171no—
119#41—6341no—
120#41—6341no—
121#41—6341no—
122#41—6341no—
123#41C2×D31711yescharacteristic
124C2×Q8—3171no—
125C2×Q8C2×D31711yescharacteristic, complemented
126C2×D4—3171no—
127C2×D4—3171no—
128C2×D4—6341no—
129C2×D4—6341no—
130C22×C4—3171no—
131C22×C4—6341no—
132C22×C4—6341no—
133Q16—3172no—
134Q16C2×D31712yescomplemented
135SD16—3172no—
136SD16—3172no—
137SD16—3172no—
138SD16C2×D31712yescomplemented
139D8—3172no—
140D8—3172no—
141M16—3171no—
142M16—3172no—
143M16C2×D31711yescharacteristic, complemented
144C2×C8—3172no—
145C23—3171no—
146C23—6341no—
147C23—6341no—
148Q8—21no—
149Q8—3171no—
150Q8—6341no—
151Q8C22×D31711yescharacteristic
152Q8C22×D31712yes—
153D4—21no—
154D4—3172no—
155D4—3171no—
156D4—6341no—
157D4—6342no—
158D4—6342no—
159D4—6341no—
160D4—6341no—
161D4—6341no—
162D4—6341no—
163D4C22×D31711yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3171no—
167C2×C4—3171no—
168C2×C4—3171no—
169C2×C4—3171no—
170C2×C4—3171no—
171C2×C4—6341no—
172C2×C4—6342no—
173C2×C4—6342no—
174C2×C4—6341no—
175C2×C4—6341no—
176C2×C4—6341no—
177C2×C4C22×D31711yescharacteristic
178C8—3172no—
179C8C22×D31712yes—
180C22—21no—
181C22—3171no—
182C22—3171no—
183C22—6342no—
184C22—6341no—
185C22—12681no—
186C22—12682no—
187C22—12682no—
188C22D4×D31711yescharacteristic
189C4—21no—
190C4—22no—
191C4—3171no—
192C4—3171no—
193C4—6341no—
194C4D4×D31711yescharacteristic
195C4C23×D31711yescharacteristic
196C2—21no—
197C2—41no—
198C2—6341no—
199C2—6341no—
200C2—12682no—
201C2—12681no—
202C2C2×D4×D31711yescharacteristic
subgroupG/H
centerC2C2×D4×D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
Fitting#137×C317C2
socleC2×C317C22×D4
hypercenter#137D317
hypocenterC317#575
LCSG ▷ C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#19099575C2semidirect
2. #41×D317C2semidirect
3. #23778812C2semidirect
4. #23778811C2semidirect
5. #23778808C2semidirect
6. #23778807C2semidirect
7. #23778805C2semidirect
8. #23778803C2semidirect
9. #23778801C2semidirect
10. M16×D317C2semidirect
11. #137×C317C2semidirect
15. #19099575C22semidirect
16. #19099575C22semidirect
20. #19099573C22semidirect
21. Q8×D317C22semidirect
23. C2×D1268C22semidirect
25. D4×D317C22semidirect
29. #19099564C22semidirect
30. #19099563C22semidirect
31. #19099561C22semidirect
32. #19099561C22semidirect
33. D2536C22semidirect
34. #19099559C22semidirect
35. #19099558C22semidirect
36. #19099557C22semidirect
37. C8×D317C22semidirect
39. C2×Q8×C317C22semidirect
40. Q16×C317C22semidirect
41. SD16×C317C22semidirect
42. M16×C317C22semidirect
101. C317#575semidirect
107. #137D317semidirect
125. C2×Q8C2×D317semidirect
134. Q16C2×D317semidirect
138. SD16C2×D317semidirect
143. M16C2×D317semidirect

Maximal subgroups

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

subgroupindex
C2×#190995752
#41×D3172
#237788122
#237788112
#237788082
#237788072
#237788052
#237788032
#237788012
M16×D3172
#137×C3172
#575317