Finite Group Databasebeta

← #137×D349 #36374934

Subgroups — #137×D349 #36374934

Order 22336.

258 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#41×D349C211yesmaximal, characteristic
2C2×Q8×D349C211yesmaximal, characteristic, complemented
3#26013672C211yesmaximal, characteristic, complemented
4#26013671C212yesmaximal, complemented
5#26013670C211yesmaximal, characteristic, complemented
6#26013669C211yesmaximal, characteristic, complemented
7#26013668C212yesmaximal, complemented
8Q16×D349C212yesmaximal, complemented
9SD16×D349C212yesmaximal, complemented
10M16×D349C211yesmaximal, characteristic, complemented
11#137×C349C211yesmaximal, characteristic, complemented
12#19150234C2211yescharacteristic
13#19150233—21no—
14#19150233C2211yescharacteristic
15#19150232—21no—
16#19150232C2211yescharacteristic
17C2×Dic698—21no—
18C2×Dic698C2211yescharacteristic, complemented
19Q8×D349—21no—
20Q8×D349—21no—
21Q8×D349C2212yes—
22Q8×D349C2211yescharacteristic
23Q8×D349C2212yescomplemented
24D4×D349—21no—
25D4×D349C2211yescharacteristic
26C2×C4×D349—21no—
27C2×C4×D349—21no—
28C2×C4×D349C2211yescharacteristic
29Dic1396C2212yescomplemented
30#19150223C2212yescomplemented
31#19150223C2212yescomplemented
32#19150222C2212yescomplemented
33#19150221C2212yescomplemented
34#19150220C2212yescomplemented
35#19150217C2211yescharacteristic, complemented
36#19150216C2212yescomplemented
37C8×D349C2212yes—
38#41×C349C2211yescharacteristic
39C2×Q8×C349C2211yescharacteristic, complemented
40Q16×C349C2212yescomplemented
41SD16×C349C2212yescomplemented
42M16×C349C2211yescharacteristic, complemented
43C22×D349—21no—
44C22×D349D411yescharacteristic
45Dic698—21no—
46Dic698—22no—
47Dic698—22no—
48Dic698—21no—
49Dic698C2312yes—
50Dic698C2311yescharacteristic
51D1396—21no—
52D1396C2311yescharacteristic
53C22⋊D349—21no—
54C22⋊D349—21no—
55C22⋊D349—21no—
56C2×Dic349—21no—
57C2×Dic349—21no—
58C2×Dic349D411yescharacteristic
59C4×D349—21no—
60C4×D349—22no—
61C4×D349—22no—
62C4×D349—21no—
63C4×D349—21no—
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—22no—
81C2×D349—21no—
82C2×D349—42no—
83C2×D349C2×D411yescharacteristic
84C2×D349C2×D411yescharacteristic
85C4×C349—21no—
86C4×C349—22no—
87C4×C349C2×D411yescharacteristic
88C4×C349C2411yescharacteristic
89Dic349—21no—
90Dic349—22no—
91Dic349C2×D411yescharacteristic
92Dic349C2×D411yescharacteristic
93C2×C349—21no—
94C2×C349—41no—
95C2×C349C22×D411yescharacteristic
96D349—21no—
97D349—41no—
98D349#13712yesdirect product factor
99C349C2×#13711yescharacteristic, complemented, Sylow
100C2×#137—3491nomaximal, Sylow
101C2×#41—3491no—
102C22×Q8—3491no—
103#137—3491no—
104#137—3491no—
105#137—3492no—
106#137—3492no—
107#137—3491no—
108#137D34911yescharacteristic, direct product factor
109C2×Q16—3492no—
110C2×SD16—3492no—
111C2×M16—3491no—
112#41—3491no—
113#41—3491no—
114#41—3491no—
115#41—6981no—
116#41—6981no—
117#41C2×D34911yescharacteristic
118C2×Q8—3492no—
119C2×Q8—3491no—
120C2×Q8—3492no—
121C2×Q8—3491no—
122C2×Q8—6981no—
123C2×Q8—6981no—
124C2×Q8—6981no—
125C2×Q8C2×D34911yescharacteristic, complemented
126C2×D4—3491no—
127C2×D4—6981no—
128C22×C4—3491no—
129C22×C4—6981no—
130C22×C4—6981no—
131Q16—3492no—
132Q16—3492no—
133Q16—3492no—
134Q16C2×D34912yescomplemented
135SD16—3492no—
136SD16—3492no—
137SD16—3492no—
138SD16C2×D34912yescomplemented
139M16—3491no—
140M16—3492no—
141M16C2×D34911yescharacteristic, complemented
142C2×C8—3492no—
143C23—3491no—
144C23—6981no—
145Q8—21no—
146Q8—3492no—
147Q8—3491no—
148Q8—6982no—
149Q8—6982no—
150Q8—6981no—
151Q8—6981no—
152Q8C22×D34912yes—
153Q8C22×D34911yescharacteristic
154D4—21no—
155D4—3491no—
156D4—6981no—
157D4—6981no—
158D4—6981no—
159D4—6981no—
160D4C22×D34911yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—3491no—
164C2×C4—3491no—
165C2×C4—3491no—
166C2×C4—3491no—
167C2×C4—3491no—
168C2×C4—6981no—
169C2×C4—6981no—
170C2×C4—6982no—
171C2×C4—6982no—
172C2×C4—6981no—
173C2×C4—6981no—
174C2×C4—6981no—
175C2×C4C22×D34911yescharacteristic
176C8—3492no—
177C8C22×D34912yes—
178C22—21no—
179C22—3491no—
180C22—3491no—
181C22—13962no—
182C22—6982no—
183C22—6981no—
184C22D4×D34911yescharacteristic
185C4—21no—
186C4—22no—
187C4—3491no—
188C4—3491no—
189C4—6981no—
190C4—6982no—
191C4D4×D34911yescharacteristic
192C4C23×D34911yescharacteristic
193C2—21no—
194C2—41no—
195C2—3492no—
196C2—13961no—
197C2—6981no—
198C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC2C2×D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
Fitting#137×C349C2
socleC2×C349C22×D4
prime radical#41×D349C2
hypercenter#137D349
hypocenterC349C2×#137
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
2. C2×Q8×D349C2semidirect
3. #26013672C2semidirect
4. #26013671C2semidirect
5. #26013670C2semidirect
6. #26013669C2semidirect
7. #26013668C2semidirect
8. Q16×D349C2semidirect
9. SD16×D349C2semidirect
10. M16×D349C2semidirect
11. #137×C349C2semidirect
18. C2×Dic698C22semidirect
23. Q8×D349C22semidirect
29. Dic1396C22semidirect
30. #19150223C22semidirect
31. #19150223C22semidirect
32. #19150222C22semidirect
33. #19150221C22semidirect
34. #19150220C22semidirect
35. #19150217C22semidirect
36. #19150216C22semidirect
39. C2×Q8×C349C22semidirect
40. Q16×C349C22semidirect
41. SD16×C349C22semidirect
42. M16×C349C22semidirect
98. D349#137direct (N × H)
99. C349C2×#137semidirect
108. #137D349direct (N × H)
125. C2×Q8C2×D349semidirect
134. Q16C2×D349semidirect
138. SD16C2×D349semidirect
141. M16C2×D349semidirect

Maximal subgroups

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

subgroupindex
#41×D3492
C2×Q8×D3492
#260136722
#260136712
#260136702
#260136692
#260136682
Q16×D3492
SD16×D3492
M16×D3492
#137×C3492
C2×#137349