Finite Group Databasebeta

← #36378583

Subgroups — #36378583

Order 22592.

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×#19150736C211yesmaximal, characteristic, complemented
2#41×D353C211yesmaximal, characteristic, complemented
3#26015025C211yesmaximal, characteristic, complemented
4#26015022C211yesmaximal, characteristic, complemented
5#26015021C212yesmaximal, complemented
6#26015018C211yesmaximal, characteristic, complemented
7#26015016C212yesmaximal, complemented
8#26015013C212yesmaximal, complemented
9#26015011C212yesmaximal, complemented
10M16×D353C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C353C211yesmaximal, characteristic, complemented
12#19150738C2211yescharacteristic, complemented
13#19150737—21no—
14#19150737C2211yescharacteristic
15#19150736—21no—
16#19150736—21no—
17#19150736—21no—
18#19150736C2212yes—
19#19150736C2211yescharacteristic, complemented
20#19150736C2212yescomplemented
21C2×Dic706C2211yescharacteristic, complemented
22Q8×D353C2211yescharacteristic, complemented
23C2×C22⋊D353—21no—
24D4×D353—21no—
25D4×D353C2211yescharacteristic, complemented
26C22×Dic353—21no—
27C2×C4×D353—21no—
28C2×C4×D353C2211yescharacteristic
29Dic1412C2212yescomplemented
30#19150727C2212yescomplemented
31#19150726C2212yescomplemented
32#19150725C2212yescomplemented
33#19150725C2212yescomplemented
34#19150722C2212yescomplemented
35#19150721C2211yescharacteristic, complemented
36#19150720C2212yescomplemented
37C8×D353C2212yescomplemented
38#41×C353C2211yescharacteristic
39C2×D4×C353C2211yescharacteristic, complemented
40SD16×C353C2212yescomplemented
41D8×C353C2212yescomplemented
42M16×C353C2211yescharacteristic, complemented
43C22×D353—21no—
44C22×D353D411yescharacteristic
45Dic706—21no—
46Dic706—21no—
47Dic706C2311yescharacteristic
48Dic706C2312yes—
49D1412—21no—
50D1412C2311yescharacteristic
51C22⋊D353—21no—
52C22⋊D353—21no—
53C22⋊D353—22no—
54C22⋊D353—22no—
55C22⋊D353—21no—
56C2×Dic353—21no—
57C2×Dic353—21no—
58C2×Dic353—22no—
59C2×Dic353—22no—
60C2×Dic353D411yescharacteristic
61C4×D353—21no—
62C4×D353—21no—
63C4×D353—21no—
64C4×D353D411yescharacteristic
65C4×D353D411yescharacteristic
66C4×D353C2311yescharacteristic
67C4×D353C2311yescharacteristic
68#15667725C2312yes—
69C23×C353—21no—
70Q8×C353C2311yescharacteristic
71D4×C353—21no—
72D4×C353—21no—
73D4×C353C2311yescharacteristic
74D4×C353C2312yes—
75C2×C4×C353—21no—
76C2×C4×C353C2311yescharacteristic
77C8×C353C2312yes—
78C22×C353—21no—
79C22×C353—22no—
80C22×C353—42no—
81C22×C353C2×D411yescharacteristic
82C2×D353—21no—
83C2×D353—41no—
84C2×D353—42no—
85C2×D353C2×D411yescharacteristic
86C2×D353C2×D411yescharacteristic
87C4×C353—21no—
88C4×C353C2×D411yescharacteristic
89C4×C353C2411yescharacteristic
90Dic353—21no—
91Dic353—22no—
92Dic353C2×D411yescharacteristic
93Dic353C2×D411yescharacteristic
94C2×C353—21no—
95C2×C353—41no—
96C2×C353—42no—
97C2×C353C22×D411yescharacteristic
98D353—21no—
99D353—21no—
100D353—41no—
101C353#57511yescharacteristic, complemented, Sylow
102#575—3531nomaximal, Sylow
103C2×#41—3531no—
104C2×#41—3531no—
105#137—3531no—
106#137—3532no—
107#137—3531no—
108AGL(1,ℤ/8)—3531no—
109AGL(1,ℤ/8)—3532no—
110AGL(1,ℤ/8)D35311yescharacteristic, complemented
111#135—3532no—
112#135—3532no—
113C2×M16—3531no—
114#41—3531no—
115#41—3532no—
116#41—3531no—
117#41—3532no—
118#41—3531no—
119#41—7061no—
120#41—7061no—
121#41—7061no—
122#41—7061no—
123#41C2×D35311yescharacteristic
124C2×Q8—3531no—
125C2×Q8—3531no—
126C2×D4—3531no—
127C2×D4—7061no—
128C2×D4—7061no—
129C2×D4C2×D35311yescharacteristic, complemented
130C22×C4—3531no—
131C22×C4—7061no—
132C22×C4—7061no—
133Q16—3532no—
134Q16—3532no—
135SD16—3532no—
136SD16—3532no—
137SD16—3532no—
138SD16C2×D35312yescomplemented
139D8—3532no—
140D8C2×D35312yescomplemented
141M16—3531no—
142M16—3532no—
143M16C2×D35311yescharacteristic, complemented
144C2×C8—3532no—
145C23—21no—
146C23—3531no—
147C23—7061no—
148Q8—3531no—
149Q8—3532no—
150Q8—7061no—
151Q8—7061no—
152Q8C22×D35311yescharacteristic
153D4—21no—
154D4—21no—
155D4—3531no—
156D4—7061no—
157D4—7061no—
158D4—7061no—
159D4—7062no—
160D4—7062no—
161D4—7061no—
162D4C22×D35312yes—
163D4C22×D35311yescharacteristic
164C2×C4—21no—
165C2×C4—3531no—
166C2×C4—3531no—
167C2×C4—3531no—
168C2×C4—3531no—
169C2×C4—3531no—
170C2×C4—7061no—
171C2×C4—7061no—
172C2×C4—7061no—
173C2×C4—7062no—
174C2×C4—7062no—
175C2×C4—7061no—
176C2×C4—7061no—
177C2×C4C22×D35311yescharacteristic
178C8—3532no—
179C8C22×D35312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—3531no—
184C22—3531no—
185C22—14121no—
186C22—14122no—
187C22—7061no—
188C22D4×D35311yescharacteristic
189C4—21no—
190C4—3531no—
191C4—3531no—
192C4—7061no—
193C4—7062no—
194C4D4×D35311yescharacteristic
195C4C23×D35311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—14121no—
200C2—7061no—
201C2—7061no—
202C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
FittingAGL(1,ℤ/8)×C353C2
socleC2×C353C22×D4
hypercenterAGL(1,ℤ/8)D353
hypocenterC353#575
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19150736C2semidirect
2. #41×D353C2semidirect
3. #26015025C2semidirect
4. #26015022C2semidirect
5. #26015021C2semidirect
6. #26015018C2semidirect
7. #26015016C2semidirect
8. #26015013C2semidirect
9. #26015011C2semidirect
10. M16×D353C2semidirect
11. AGL(1,ℤ/8)×C353C2semidirect
12. #19150738C22semidirect
19. #19150736C22semidirect
20. #19150736C22semidirect
21. C2×Dic706C22semidirect
22. Q8×D353C22semidirect
25. D4×D353C22semidirect
29. Dic1412C22semidirect
30. #19150727C22semidirect
31. #19150726C22semidirect
32. #19150725C22semidirect
33. #19150725C22semidirect
34. #19150722C22semidirect
35. #19150721C22semidirect
36. #19150720C22semidirect
37. C8×D353C22semidirect
39. C2×D4×C353C22semidirect
40. SD16×C353C22semidirect
41. D8×C353C22semidirect
42. M16×C353C22semidirect
101. C353#575semidirect
110. AGL(1,ℤ/8)D353semidirect
129. C2×D4C2×D353semidirect
138. SD16C2×D353semidirect
140. D8C2×D353semidirect
143. M16C2×D353semidirect

Maximal subgroups

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

subgroupindex
C2×#191507362
#41×D3532
#260150252
#260150222
#260150212
#260150182
#260150162
#260150132
#260150112
M16×D3532
AGL(1,ℤ/8)×C3532
#575353