Finite Group Databasebeta

← #36378586

Subgroups — #36378586

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×#19150738C211yesmaximal, characteristic, complemented
2#41×D353C211yesmaximal, characteristic, complemented
3#26015024C212yesmaximal, complemented
4#26015023C211yesmaximal, characteristic, complemented
5#26015020C211yesmaximal, characteristic, complemented
6#26015019C212yesmaximal, complemented
7#26015017C211yesmaximal, characteristic, complemented
8#26015015C212yesmaximal, complemented
9#26015013C212yesmaximal, complemented
10M16×D353C211yesmaximal, characteristic, complemented
11#137×C353C211yesmaximal, characteristic, complemented
12#19150738—21no—
13#19150738—21no—
14#19150738C2212yes—
15#19150738C2212yescomplemented
16#19150738C2211yescharacteristic, complemented
17#19150737—21no—
18#19150737C2211yescharacteristic
19#19150736—21no—
20#19150736C2211yescharacteristic, complemented
21Q8×D353C2211yescharacteristic, complemented
22C2×D1412—21no—
23C2×D1412C2211yescharacteristic, complemented
24D4×D353—21no—
25D4×D353C2211yescharacteristic, complemented
26C2×C4×D353—21no—
27C2×C4×D353—21no—
28C2×C4×D353C2211yescharacteristic
29#19150727C2212yescomplemented
30#19150726C2212yescomplemented
31#19150724C2212yescomplemented
32#19150724C2212yescomplemented
33D2824C2212yescomplemented
34#19150722C2212yescomplemented
35#19150721C2211yescharacteristic, complemented
36#19150720C2212yescomplemented
37C8×D353C2212yescomplemented
38#41×C353C2211yescharacteristic
39C2×Q8×C353C2211yescharacteristic, complemented
40Q16×C353C2212yescomplemented
41SD16×C353C2212yescomplemented
42M16×C353C2211yescharacteristic, complemented
43C22×D353—21no—
44C22×D353—21no—
45C22×D353D411yescharacteristic
46Dic706—21no—
47Dic706C2311yescharacteristic
48D1412—21no—
49D1412—22no—
50D1412—22no—
51D1412—21no—
52D1412C2312yes—
53D1412C2311yescharacteristic
54C22⋊D353—21no—
55C22⋊D353—21no—
56C22⋊D353—21no—
57C2×Dic353—21no—
58C2×Dic353D411yescharacteristic
59C4×D353—21no—
60C4×D353—21no—
61C4×D353—21no—
62C4×D353—22no—
63C4×D353—22no—
64C4×D353D411yescharacteristic
65C4×D353D411yescharacteristic
66C4×D353C2311yescharacteristic
67C4×D353C2311yescharacteristic
68#15667725C2312yes—
69Q8×C353—21no—
70Q8×C353C2312yes—
71Q8×C353C2311yescharacteristic
72D4×C353—21no—
73D4×C353C2311yescharacteristic
74C2×C4×C353—21no—
75C2×C4×C353—21no—
76C2×C4×C353C2311yescharacteristic
77C8×C353C2312yes—
78C22×C353—21no—
79C22×C353C2×D411yescharacteristic
80C2×D353—21no—
81C2×D353—22no—
82C2×D353—41no—
83C2×D353—42no—
84C2×D353—42no—
85C2×D353C2×D411yescharacteristic
86C2×D353C2×D411yescharacteristic
87C4×C353—21no—
88C4×C353—22no—
89C4×C353C2×D411yescharacteristic
90C4×C353C2411yescharacteristic
91Dic353—21no—
92Dic353C2×D411yescharacteristic
93Dic353C2×D411yescharacteristic
94C2×C353—21no—
95C2×C353—41no—
96C2×C353C22×D411yescharacteristic
97D353—21no—
98D353—21no—
99D353—41no—
100D353—42no—
101C353#57511yescharacteristic, complemented, Sylow
102#575—3531nomaximal, Sylow
103C2×#41—3531no—
104C2×#41—3531no—
105#137—3531no—
106#137—3532no—
107#137D35311yescharacteristic, complemented
108AGL(1,ℤ/8)—3531no—
109AGL(1,ℤ/8)—3532no—
110AGL(1,ℤ/8)—3531no—
111#135—3532no—
112#135—3532no—
113C2×M16—3531no—
114#41—3531no—
115#41—3532no—
116#41—3532no—
117#41—3531no—
118#41—3531no—
119#41—7061no—
120#41—7061no—
121#41—7061no—
122#41—7061no—
123#41C2×D35311yescharacteristic
124C2×Q8—3531no—
125C2×Q8C2×D35311yescharacteristic, complemented
126C2×D4—3531no—
127C2×D4—3531no—
128C2×D4—7061no—
129C2×D4—7061no—
130C22×C4—3531no—
131C22×C4—7061no—
132C22×C4—7061no—
133Q16—3532no—
134Q16C2×D35312yescomplemented
135SD16—3532no—
136SD16—3532no—
137SD16—3532no—
138SD16C2×D35312yescomplemented
139D8—3532no—
140D8—3532no—
141M16—3531no—
142M16—3532no—
143M16C2×D35311yescharacteristic, complemented
144C2×C8—3532no—
145C23—3531no—
146C23—7061no—
147C23—7061no—
148Q8—21no—
149Q8—3531no—
150Q8—7061no—
151Q8C22×D35311yescharacteristic
152Q8C22×D35312yes—
153D4—21no—
154D4—3532no—
155D4—3531no—
156D4—7061no—
157D4—7062no—
158D4—7062no—
159D4—7061no—
160D4—7061no—
161D4—7061no—
162D4—7061no—
163D4C22×D35311yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3531no—
167C2×C4—3531no—
168C2×C4—3531no—
169C2×C4—3531no—
170C2×C4—3531no—
171C2×C4—7061no—
172C2×C4—7062no—
173C2×C4—7062no—
174C2×C4—7061no—
175C2×C4—7061no—
176C2×C4—7061no—
177C2×C4C22×D35311yescharacteristic
178C8—3532no—
179C8C22×D35312yes—
180C22—21no—
181C22—3531no—
182C22—3531no—
183C22—14122no—
184C22—14121no—
185C22—14122no—
186C22—7062no—
187C22—7061no—
188C22D4×D35311yescharacteristic
189C4—21no—
190C4—22no—
191C4—3531no—
192C4—3531no—
193C4—7061no—
194C4D4×D35311yescharacteristic
195C4C23×D35311yescharacteristic
196C2—21no—
197C2—41no—
198C2—14122no—
199C2—14121no—
200C2—7061no—
201C2—7061no—
202C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
Fitting#137×C353C2
socleC2×C353C22×D4
hypercenter#137D353
hypocenterC353#575
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#19150738C2semidirect
2. #41×D353C2semidirect
3. #26015024C2semidirect
4. #26015023C2semidirect
5. #26015020C2semidirect
6. #26015019C2semidirect
7. #26015017C2semidirect
8. #26015015C2semidirect
9. #26015013C2semidirect
10. M16×D353C2semidirect
11. #137×C353C2semidirect
15. #19150738C22semidirect
16. #19150738C22semidirect
20. #19150736C22semidirect
21. Q8×D353C22semidirect
23. C2×D1412C22semidirect
25. D4×D353C22semidirect
29. #19150727C22semidirect
30. #19150726C22semidirect
31. #19150724C22semidirect
32. #19150724C22semidirect
33. D2824C22semidirect
34. #19150722C22semidirect
35. #19150721C22semidirect
36. #19150720C22semidirect
37. C8×D353C22semidirect
39. C2×Q8×C353C22semidirect
40. Q16×C353C22semidirect
41. SD16×C353C22semidirect
42. M16×C353C22semidirect
101. C353#575semidirect
107. #137D353semidirect
125. C2×Q8C2×D353semidirect
134. Q16C2×D353semidirect
138. SD16C2×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×#191507382
#41×D3532
#260150242
#260150232
#260150202
#260150192
#260150172
#260150152
#260150132
M16×D3532
#137×C3532
#575353