Finite Group Databasebeta

← #35204019

Subgroups — #35204019

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×#19099573C211yesmaximal, characteristic, complemented
2#41×D317C211yesmaximal, characteristic, complemented
3#23778813C211yesmaximal, characteristic, complemented
4#23778810C211yesmaximal, characteristic, complemented
5#23778809C212yesmaximal, complemented
6#23778806C211yesmaximal, characteristic, complemented
7#23778804C212yesmaximal, complemented
8#23778801C212yesmaximal, complemented
9#23778799C212yesmaximal, complemented
10M16×D317C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C317C211yesmaximal, characteristic, complemented
12#19099575C2211yescharacteristic, complemented
13#19099574—21no—
14#19099574C2211yescharacteristic
15#19099573—21no—
16#19099573—21no—
17#19099573—21no—
18#19099573C2212yes—
19#19099573C2211yescharacteristic, complemented
20#19099573C2212yescomplemented
21C2×Dic634C2211yescharacteristic, complemented
22Q8×D317C2211yescharacteristic, complemented
23C2×C22⋊D317—21no—
24D4×D317—21no—
25D4×D317C2211yescharacteristic, complemented
26C22×Dic317—21no—
27C2×C4×D317—21no—
28C2×C4×D317C2211yescharacteristic
29Dic1268C2212yescomplemented
30#19099564C2212yescomplemented
31#19099563C2212yescomplemented
32#19099562C2212yescomplemented
33#19099562C2212yescomplemented
34#19099559C2212yescomplemented
35#19099558C2211yescharacteristic, complemented
36#19099557C2212yescomplemented
37C8×D317C2212yescomplemented
38#41×C317C2211yescharacteristic
39C2×D4×C317C2211yescharacteristic, complemented
40SD16×C317C2212yescomplemented
41D8×C317C2212yescomplemented
42M16×C317C2211yescharacteristic, complemented
43C22×D317—21no—
44C22×D317D411yescharacteristic
45Dic634—21no—
46Dic634—21no—
47Dic634C2311yescharacteristic
48Dic634C2312yes—
49D1268—21no—
50D1268C2311yescharacteristic
51C22⋊D317—21no—
52C22⋊D317—21no—
53C22⋊D317—22no—
54C22⋊D317—22no—
55C22⋊D317—21no—
56C2×Dic317—21no—
57C2×Dic317—21no—
58C2×Dic317—22no—
59C2×Dic317—22no—
60C2×Dic317D411yescharacteristic
61C4×D317—21no—
62C4×D317—21no—
63C4×D317—21no—
64C4×D317D411yescharacteristic
65C4×D317D411yescharacteristic
66C4×D317C2311yescharacteristic
67C4×D317C2311yescharacteristic
68#14579261C2312yes—
69C23×C317—21no—
70Q8×C317C2311yescharacteristic
71D4×C317—21no—
72D4×C317—21no—
73D4×C317C2311yescharacteristic
74D4×C317C2312yes—
75C2×C4×C317—21no—
76C2×C4×C317C2311yescharacteristic
77C8×C317C2312yes—
78C22×C317—21no—
79C22×C317—22no—
80C22×C317—42no—
81C22×C317C2×D411yescharacteristic
82C2×D317—21no—
83C2×D317—41no—
84C2×D317—42no—
85C2×D317C2×D411yescharacteristic
86C2×D317C2×D411yescharacteristic
87C4×C317—21no—
88C4×C317C2×D411yescharacteristic
89C4×C317C2411yescharacteristic
90Dic317—21no—
91Dic317—22no—
92Dic317C2×D411yescharacteristic
93Dic317C2×D411yescharacteristic
94C2×C317—21no—
95C2×C317—41no—
96C2×C317—42no—
97C2×C317C22×D411yescharacteristic
98D317—21no—
99D317—21no—
100D317—41no—
101C317#57511yescharacteristic, complemented, Sylow
102#575—3171nomaximal, Sylow
103C2×#41—3171no—
104C2×#41—3171no—
105#137—3171no—
106#137—3172no—
107#137—3171no—
108AGL(1,ℤ/8)—3171no—
109AGL(1,ℤ/8)—3172no—
110AGL(1,ℤ/8)D31711yescharacteristic, complemented
111#135—3172no—
112#135—3172no—
113C2×M16—3171no—
114#41—3171no—
115#41—3172no—
116#41—3171no—
117#41—3172no—
118#41—3171no—
119#41—6341no—
120#41—6341no—
121#41—6341no—
122#41—6341no—
123#41C2×D31711yescharacteristic
124C2×Q8—3171no—
125C2×Q8—3171no—
126C2×D4—3171no—
127C2×D4—6341no—
128C2×D4—6341no—
129C2×D4C2×D31711yescharacteristic, complemented
130C22×C4—3171no—
131C22×C4—6341no—
132C22×C4—6341no—
133Q16—3172no—
134Q16—3172no—
135SD16—3172no—
136SD16—3172no—
137SD16—3172no—
138SD16C2×D31712yescomplemented
139D8—3172no—
140D8C2×D31712yescomplemented
141M16—3171no—
142M16—3172no—
143M16C2×D31711yescharacteristic, complemented
144C2×C8—3172no—
145C23—21no—
146C23—3171no—
147C23—6341no—
148Q8—3171no—
149Q8—3172no—
150Q8—6341no—
151Q8—6341no—
152Q8C22×D31711yescharacteristic
153D4—21no—
154D4—21no—
155D4—3171no—
156D4—6341no—
157D4—6341no—
158D4—6341no—
159D4—6342no—
160D4—6342no—
161D4—6341no—
162D4C22×D31712yes—
163D4C22×D31711yescharacteristic
164C2×C4—21no—
165C2×C4—3171no—
166C2×C4—3171no—
167C2×C4—3171no—
168C2×C4—3171no—
169C2×C4—3171no—
170C2×C4—6341no—
171C2×C4—6341no—
172C2×C4—6341no—
173C2×C4—6342no—
174C2×C4—6342no—
175C2×C4—6341no—
176C2×C4—6341no—
177C2×C4C22×D31711yescharacteristic
178C8—3172no—
179C8C22×D31712yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—3171no—
184C22—3171no—
185C22—6341no—
186C22—12681no—
187C22—12682no—
188C22D4×D31711yescharacteristic
189C4—21no—
190C4—3171no—
191C4—3171no—
192C4—6341no—
193C4—6342no—
194C4D4×D31711yescharacteristic
195C4C23×D31711yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—6341no—
200C2—6341no—
201C2—12681no—
202C2C2×D4×D31711yescharacteristic
subgroupG/H
centerC2C2×D4×D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
FittingAGL(1,ℤ/8)×C317C2
socleC2×C317C22×D4
hypercenterAGL(1,ℤ/8)D317
hypocenterC317#575
LCSG ▷ C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19099573C2semidirect
2. #41×D317C2semidirect
3. #23778813C2semidirect
4. #23778810C2semidirect
5. #23778809C2semidirect
6. #23778806C2semidirect
7. #23778804C2semidirect
8. #23778801C2semidirect
9. #23778799C2semidirect
10. M16×D317C2semidirect
11. AGL(1,ℤ/8)×C317C2semidirect
12. #19099575C22semidirect
19. #19099573C22semidirect
20. #19099573C22semidirect
21. C2×Dic634C22semidirect
22. Q8×D317C22semidirect
25. D4×D317C22semidirect
29. Dic1268C22semidirect
30. #19099564C22semidirect
31. #19099563C22semidirect
32. #19099562C22semidirect
33. #19099562C22semidirect
34. #19099559C22semidirect
35. #19099558C22semidirect
36. #19099557C22semidirect
37. C8×D317C22semidirect
39. C2×D4×C317C22semidirect
40. SD16×C317C22semidirect
41. D8×C317C22semidirect
42. M16×C317C22semidirect
101. C317#575semidirect
110. AGL(1,ℤ/8)D317semidirect
129. C2×D4C2×D317semidirect
138. SD16C2×D317semidirect
140. D8C2×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×#190995732
#41×D3172
#237788132
#237788102
#237788092
#237788062
#237788042
#237788012
#237787992
M16×D3172
AGL(1,ℤ/8)×C3172
#575317