Finite Group Databasebeta

← #36374943

Subgroups — #36374943

Order 22336.

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×#19150234C211yesmaximal, characteristic, complemented
2#41×D349C211yesmaximal, characteristic, complemented
3#26013671C212yesmaximal, complemented
4#26013670C211yesmaximal, characteristic, complemented
5#26013667C211yesmaximal, characteristic, complemented
6#26013666C212yesmaximal, complemented
7#26013664C211yesmaximal, characteristic, complemented
8#26013662C212yesmaximal, complemented
9#26013660C212yesmaximal, complemented
10M16×D349C211yesmaximal, characteristic, complemented
11#137×C349C211yesmaximal, characteristic, complemented
12#19150234—21no—
13#19150234—21no—
14#19150234C2212yes—
15#19150234C2212yescomplemented
16#19150234C2211yescharacteristic, complemented
17#19150233—21no—
18#19150233C2211yescharacteristic
19#19150232—21no—
20#19150232C2211yescharacteristic, complemented
21Q8×D349C2211yescharacteristic, complemented
22C2×D1396—21no—
23C2×D1396C2211yescharacteristic, complemented
24D4×D349—21no—
25D4×D349C2211yescharacteristic, complemented
26C2×C4×D349—21no—
27C2×C4×D349—21no—
28C2×C4×D349C2211yescharacteristic
29#19150223C2212yescomplemented
30#19150222C2212yescomplemented
31#19150220C2212yescomplemented
32#19150220C2212yescomplemented
33D2792C2212yescomplemented
34#19150218C2212yescomplemented
35#19150217C2211yescharacteristic, complemented
36#19150216C2212yescomplemented
37C8×D349C2212yescomplemented
38#41×C349C2211yescharacteristic
39C2×Q8×C349C2211yescharacteristic, complemented
40Q16×C349C2212yescomplemented
41SD16×C349C2212yescomplemented
42M16×C349C2211yescharacteristic, complemented
43C22×D349—21no—
44C22×D349—21no—
45C22×D349D411yescharacteristic
46Dic698—21no—
47Dic698C2311yescharacteristic
48D1396—21no—
49D1396—22no—
50D1396—22no—
51D1396—21no—
52D1396C2312yes—
53D1396C2311yescharacteristic
54C22⋊D349—21no—
55C22⋊D349—21no—
56C22⋊D349—21no—
57C2×Dic349—21no—
58C2×Dic349D411yescharacteristic
59C4×D349—21no—
60C4×D349—21no—
61C4×D349—21no—
62C4×D349—22no—
63C4×D349—22no—
64C4×D349D411yescharacteristic
65C4×D349D411yescharacteristic
66C4×D349C2311yescharacteristic
67C4×D349C2311yescharacteristic
68#14584095C2312yes—
69Q8×C349—21no—
70Q8×C349C2312yes—
71Q8×C349C2311yescharacteristic
72D4×C349—21no—
73D4×C349C2311yescharacteristic
74C2×C4×C349—21no—
75C2×C4×C349—21no—
76C2×C4×C349C2311yescharacteristic
77C8×C349C2312yes—
78C22×C349—21no—
79C22×C349C2×D411yescharacteristic
80C2×D349—21no—
81C2×D349—22no—
82C2×D349—41no—
83C2×D349—42no—
84C2×D349—42no—
85C2×D349C2×D411yescharacteristic
86C2×D349C2×D411yescharacteristic
87C4×C349—21no—
88C4×C349—22no—
89C4×C349C2×D411yescharacteristic
90C4×C349C2411yescharacteristic
91Dic349—21no—
92Dic349C2×D411yescharacteristic
93Dic349C2×D411yescharacteristic
94C2×C349—21no—
95C2×C349—41no—
96C2×C349C22×D411yescharacteristic
97D349—21no—
98D349—21no—
99D349—41no—
100D349—42no—
101C349#57511yescharacteristic, complemented, Sylow
102#575—3491nomaximal, Sylow
103C2×#41—3491no—
104C2×#41—3491no—
105#137—3491no—
106#137—3492no—
107#137D34911yescharacteristic, complemented
108AGL(1,ℤ/8)—3491no—
109AGL(1,ℤ/8)—3492no—
110AGL(1,ℤ/8)—3491no—
111#135—3492no—
112#135—3492no—
113C2×M16—3491no—
114#41—3491no—
115#41—3492no—
116#41—3492no—
117#41—3491no—
118#41—3491no—
119#41—6981no—
120#41—6981no—
121#41—6981no—
122#41—6981no—
123#41C2×D34911yescharacteristic
124C2×Q8—3491no—
125C2×Q8C2×D34911yescharacteristic, complemented
126C2×D4—3491no—
127C2×D4—3491no—
128C2×D4—6981no—
129C2×D4—6981no—
130C22×C4—3491no—
131C22×C4—6981no—
132C22×C4—6981no—
133Q16—3492no—
134Q16C2×D34912yescomplemented
135SD16—3492no—
136SD16—3492no—
137SD16—3492no—
138SD16C2×D34912yescomplemented
139D8—3492no—
140D8—3492no—
141M16—3491no—
142M16—3492no—
143M16C2×D34911yescharacteristic, complemented
144C2×C8—3492no—
145C23—3491no—
146C23—6981no—
147C23—6981no—
148Q8—21no—
149Q8—3491no—
150Q8—6981no—
151Q8C22×D34911yescharacteristic
152Q8C22×D34912yes—
153D4—21no—
154D4—3492no—
155D4—3491no—
156D4—6981no—
157D4—6982no—
158D4—6982no—
159D4—6981no—
160D4—6981no—
161D4—6981no—
162D4—6981no—
163D4C22×D34911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3491no—
167C2×C4—3491no—
168C2×C4—3491no—
169C2×C4—3491no—
170C2×C4—3491no—
171C2×C4—6981no—
172C2×C4—6982no—
173C2×C4—6982no—
174C2×C4—6981no—
175C2×C4—6981no—
176C2×C4—6981no—
177C2×C4C22×D34911yescharacteristic
178C8—3492no—
179C8C22×D34912yes—
180C22—21no—
181C22—3491no—
182C22—3491no—
183C22—13962no—
184C22—13961no—
185C22—13962no—
186C22—6982no—
187C22—6981no—
188C22D4×D34911yescharacteristic
189C4—21no—
190C4—22no—
191C4—3491no—
192C4—3491no—
193C4—6981no—
194C4D4×D34911yescharacteristic
195C4C23×D34911yescharacteristic
196C2—21no—
197C2—41no—
198C2—13962no—
199C2—13961no—
200C2—6981no—
201C2—6981no—
202C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC2C2×D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
Fitting#137×C349C2
socleC2×C349C22×D4
hypercenter#137D349
hypocenterC349#575
LCSG ▷ C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#19150234C2semidirect
2. #41×D349C2semidirect
3. #26013671C2semidirect
4. #26013670C2semidirect
5. #26013667C2semidirect
6. #26013666C2semidirect
7. #26013664C2semidirect
8. #26013662C2semidirect
9. #26013660C2semidirect
10. M16×D349C2semidirect
11. #137×C349C2semidirect
15. #19150234C22semidirect
16. #19150234C22semidirect
20. #19150232C22semidirect
21. Q8×D349C22semidirect
23. C2×D1396C22semidirect
25. D4×D349C22semidirect
29. #19150223C22semidirect
30. #19150222C22semidirect
31. #19150220C22semidirect
32. #19150220C22semidirect
33. D2792C22semidirect
34. #19150218C22semidirect
35. #19150217C22semidirect
36. #19150216C22semidirect
37. C8×D349C22semidirect
39. C2×Q8×C349C22semidirect
40. Q16×C349C22semidirect
41. SD16×C349C22semidirect
42. M16×C349C22semidirect
101. C349#575semidirect
107. #137D349semidirect
125. C2×Q8C2×D349semidirect
134. Q16C2×D349semidirect
138. SD16C2×D349semidirect
143. M16C2×D349semidirect

Maximal subgroups

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

subgroupindex
C2×#191502342
#41×D3492
#260136712
#260136702
#260136672
#260136662
#260136642
#260136622
#260136602
M16×D3492
#137×C3492
#575349