Finite Group Databasebeta

← #26012290

Subgroups — #26012290

Order 11072.

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×#14583827C211yesmaximal, characteristic, complemented
2#41×D173C211yesmaximal, characteristic, complemented
3#19149711C212yesmaximal, complemented
4#19149710C211yesmaximal, characteristic, complemented
5#19149707C211yesmaximal, characteristic, complemented
6#19149706C212yesmaximal, complemented
7#19149704C211yesmaximal, characteristic, complemented
8#19149702C212yesmaximal, complemented
9#19149700C212yesmaximal, complemented
10M16×D173C211yesmaximal, characteristic, complemented
11#137×C173C211yesmaximal, characteristic, complemented
12#14583827—21no—
13#14583827—21no—
14#14583827C2212yes—
15#14583827C2212yescomplemented
16#14583827C2211yescharacteristic, complemented
17#14583826—21no—
18#14583826C2211yescharacteristic
19#14583825—21no—
20#14583825C2211yescharacteristic, complemented
21Q8×D173C2211yescharacteristic, complemented
22C2×D692—21no—
23C2×D692C2211yescharacteristic, complemented
24D4×D173—21no—
25D4×D173C2211yescharacteristic, complemented
26C2×C4×D173—21no—
27C2×C4×D173—21no—
28C2×C4×D173C2211yescharacteristic
29#14583816C2212yescomplemented
30#14583815C2212yescomplemented
31#14583813C2212yescomplemented
32#14583813C2212yescomplemented
33D1384C2212yescomplemented
34#14583811C2212yescomplemented
35#14583810C2211yescharacteristic, complemented
36#14583809C2212yescomplemented
37C8×D173C2212yescomplemented
38#41×C173C2211yescharacteristic
39C2×Q8×C173C2211yescharacteristic, complemented
40Q16×C173C2212yescomplemented
41SD16×C173C2212yescomplemented
42M16×C173C2211yescharacteristic, complemented
43C22×D173—21no—
44C22×D173—21no—
45C22×D173D411yescharacteristic
46Dic346—21no—
47Dic346C2311yescharacteristic
48D692—21no—
49D692—22no—
50D692—22no—
51D692—21no—
52D692C2312yes—
53D692C2311yescharacteristic
54C22⋊D173—21no—
55C22⋊D173—21no—
56C22⋊D173—21no—
57C2×Dic173—21no—
58C2×Dic173D411yescharacteristic
59C4×D173—21no—
60C4×D173—21no—
61C4×D173—21no—
62C4×D173—22no—
63C4×D173—22no—
64C4×D173D411yescharacteristic
65C4×D173D411yescharacteristic
66C4×D173C2311yescharacteristic
67C4×D173C2311yescharacteristic
68#13063436C2312yes—
69Q8×C173—21no—
70Q8×C173C2311yescharacteristic
71Q8×C173C2312yes—
72D4×C173—21no—
73D4×C173C2311yescharacteristic
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173C2311yescharacteristic
77C8×C173C2312yes—
78C22×C173—21no—
79C22×C173C2×D411yescharacteristic
80C2×D173—21no—
81C2×D173—22no—
82C2×D173—41no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173C2×D411yescharacteristic
86C2×D173C2×D411yescharacteristic
87C4×C173—21no—
88C4×C173—22no—
89C4×C173C2×D411yescharacteristic
90C4×C173C2411yescharacteristic
91Dic173—21no—
92Dic173C2×D411yescharacteristic
93Dic173C2×D411yescharacteristic
94C2×C173—21no—
95C2×C173—41no—
96C2×C173C22×D411yescharacteristic
97D173—21no—
98D173—21no—
99D173—41no—
100D173—42no—
101C173#57511yescharacteristic, complemented, Sylow
102#575—1731nomaximal, Sylow
103C2×#41—1731no—
104C2×#41—1731no—
105#137—1731no—
106#137—1732no—
107#137D17311yescharacteristic, complemented
108AGL(1,ℤ/8)—1731no—
109AGL(1,ℤ/8)—1732no—
110AGL(1,ℤ/8)—1731no—
111#135—1732no—
112#135—1732no—
113C2×M16—1731no—
114#41—3461no—
115#41—3461no—
116#41—3461no—
117#41—3461no—
118#41—1732no—
119#41—1731no—
120#41—1732no—
121#41—1731no—
122#41—1731no—
123#41C2×D17311yescharacteristic
124C2×Q8—1731no—
125C2×Q8C2×D17311yescharacteristic, complemented
126C2×D4—3461no—
127C2×D4—3461no—
128C2×D4—1731no—
129C2×D4—1731no—
130C22×C4—3461no—
131C22×C4—3461no—
132C22×C4—1731no—
133Q16—1732no—
134Q16C2×D17312yescomplemented
135SD16—1732no—
136SD16—1732no—
137SD16—1732no—
138SD16C2×D17312yescomplemented
139D8—1732no—
140D8—1732no—
141M16—1731no—
142M16—1732no—
143M16C2×D17311yescharacteristic, complemented
144C2×C8—1732no—
145C23—3461no—
146C23—3461no—
147C23—1731no—
148Q8—21no—
149Q8—3461no—
150Q8—1731no—
151Q8C22×D17311yescharacteristic
152Q8C22×D17312yes—
153D4—21no—
154D4—3461no—
155D4—3462no—
156D4—3462no—
157D4—3461no—
158D4—3461no—
159D4—3461no—
160D4—3461no—
161D4—1732no—
162D4—1731no—
163D4C22×D17311yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3461no—
167C2×C4—3461no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3461no—
171C2×C4—3461no—
172C2×C4—1731no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4C22×D17311yescharacteristic
178C8—1732no—
179C8C22×D17312yes—
180C22—21no—
181C22—3462no—
182C22—3461no—
183C22—1731no—
184C22—1731no—
185C22—6921no—
186C22—6922no—
187C22—6922no—
188C22D4×D17311yescharacteristic
189C4—21no—
190C4—22no—
191C4—3461no—
192C4—1731no—
193C4—1731no—
194C4D4×D17311yescharacteristic
195C4C23×D17311yescharacteristic
196C2—21no—
197C2—41no—
198C2—3461no—
199C2—3461no—
200C2—6922no—
201C2—6921no—
202C2C2×D4×D17311yescharacteristic
subgroupG/H
centerC2C2×D4×D173
commutator, cosocleC4×C173C24
FrattiniC4C23×D173
Fitting#137×C173C2
socleC2×C173C22×D4
hypercenter#137D173
hypocenterC173#575
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#14583827C2semidirect
2. #41×D173C2semidirect
3. #19149711C2semidirect
4. #19149710C2semidirect
5. #19149707C2semidirect
6. #19149706C2semidirect
7. #19149704C2semidirect
8. #19149702C2semidirect
9. #19149700C2semidirect
10. M16×D173C2semidirect
11. #137×C173C2semidirect
15. #14583827C22semidirect
16. #14583827C22semidirect
20. #14583825C22semidirect
21. Q8×D173C22semidirect
23. C2×D692C22semidirect
25. D4×D173C22semidirect
29. #14583816C22semidirect
30. #14583815C22semidirect
31. #14583813C22semidirect
32. #14583813C22semidirect
33. D1384C22semidirect
34. #14583811C22semidirect
35. #14583810C22semidirect
36. #14583809C22semidirect
37. C8×D173C22semidirect
39. C2×Q8×C173C22semidirect
40. Q16×C173C22semidirect
41. SD16×C173C22semidirect
42. M16×C173C22semidirect
101. C173#575semidirect
107. #137D173semidirect
125. C2×Q8C2×D173semidirect
134. Q16C2×D173semidirect
138. SD16C2×D173semidirect
143. M16C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#145838272
#41×D1732
#191497112
#191497102
#191497072
#191497062
#191497042
#191497022
#191497002
M16×D1732
#137×C1732
#575173