Finite Group Databasebeta

← #29838161

Subgroups — #29838161

Order 15616.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#21458957C212yesmaximal, complemented
2#21448133C214yesmaximal, complemented
3#21448131C212yesmaximal, complemented
4#21448129C212yesmaximal, complemented
5C2×D1952C211yesmaximal, characteristic, complemented
6D32×D61C212yesmaximal, complemented
7#21448112C211yesmaximal, characteristic, complemented
8C61×#2262C211yesmaximal, characteristic, complemented
9#16840564—22no—
10#16840199—24no—
11#16840197—22no—
12#16840195C2214yescomplemented
13C2×D976C2212yescomplemented
14D16×D61C2214yescomplemented
15#16840178C2211yescharacteristic
16#16839599C2214yescomplemented
17D1952C2212yescomplemented
18D1952C2212yescomplemented
19#16839597C2214yescomplemented
20#16839596C2211yescharacteristic, complemented
21#16839595C2212yescomplemented
22C32×D61C2212yescomplemented
23C61×#508C2212yescomplemented
24C61×Q64C2211yescharacteristic, complemented
25C61×SD64C2212yescomplemented
26C61×D32C2211yescharacteristic, complemented
27C2×C32×C61C2211yescharacteristic, complemented
28#14519823—42no—
29#14519804—44no—
30#14519802—42no—
31#14519800—24no—
32C2×D488—22no—
33D8×D61—24no—
34#14519783D411yescharacteristic
35#14519720—24no—
36D976—22no—
37D976C2314yes—
38#14519718—24no—
39#14519717D411yescharacteristic
40#14519716D412yes—
41C16×D61C2312yes—
42#14519683C2312yes—
43#135×C61—22no—
44Q32×C61C2312yes—
45SD32×C61—22no—
46D16×C61C2312yes—
47C2×C16×C61C2311yescharacteristic
48C32×C61C2311yescharacteristic
49C32×C61C2311yescharacteristic
50#41×C61—42no—
51#11758226—44no—
52D4×D61—44no—
53D4×D61—84no—
54#11758222—82no—
55#11758222D811yescharacteristic
56C2×D244—42no—
57C2×D244—82no—
58Q16×C61—22no—
59SD16×C61—42no—
60D8×C61—22no—
61C2×C8×C61C2×D411yescharacteristic
62#11758201—44no—
63#11758199—44no—
64#11758194D811yescharacteristic
65D488—24no—
66D488—42no—
67#11758189D812yes—
68C8×D61C2×D412yes—
69C16×C61C2×D411yescharacteristic
70C16×C61C2411yescharacteristic
71#11758185C2×D412yes—
72C22×D61—82no—
73C22×D61—84no—
74Q8×C61—42no—
75D4×C61—42no—
76D4×C61—82no—
77C2×C4×C61—82no—
78C2×C4×C61C2×D811yescharacteristic
79C22⋊D61—84no—
80C22⋊D61D1612yes—
81D244—44no—
82D244—84no—
83D244—82no—
84D244D1611yescharacteristic
85C4×D61—84no—
86C4×D61C2×D812yes—
87Dic122D1611yescharacteristic
88C8×C61C2×D811yescharacteristic
89C8×C61C22×D411yescharacteristic
90#92418C2×D812yes—
91#33982—611nomaximal, Sylow
92C22×C61—82no—
93C22×C61C2×D1611yescharacteristic
94C2×D61—84no—
95C2×D61—324no—
96C2×D61—322no—
97C2×D61C2×D1612yes—
98C4×C61—82no—
99C4×C61C2×D1611yescharacteristic
100C4×C61C22×D811yescharacteristic
101Dic61C2×D1612yes—
102#3415—612no—
103#2263—612no—
104#2263—614no—
105#2262—612no—
106#2262D6111yescharacteristic, complemented
107C2×D32—612no—
108C2×D32—611no—
109#2258—611no—
110C2×C61—21no—
111C2×C61—162no—
112C2×C61C22×D1611yescharacteristic
113D61—22no—
114D61—164no—
115#576—1222no—
116#509—1224no—
117#509—1222no—
118#508—614no—
119#508C2×D6112yescomplemented
120C2×D16—614no—
121C2×D16—612no—
122#504—611no—
123Q64C2×D6111yescharacteristic, complemented
124SD64—614no—
125SD64C2×D6112yescomplemented
126D32—614no—
127D32—612no—
128D32—612no—
129D32C2×D6111yescharacteristic, complemented
130M64—611no—
131M64—612no—
132C2×C32—612no—
133C2×C32C2×D6111yescharacteristic, complemented
134C61#3398211yescharacteristic, complemented, Sylow
135H(5,𝔽2)—2442no—
136AGL(1,ℤ/8)—2444no—
137AGL(1,ℤ/8)—2442no—
138#135—22no—
139#135—1224no—
140C2×D8—1224no—
141C2×D8—1222no—
142#131—611no—
143Q32C22×D6112yes—
144SD32—22no—
145SD32—1224no—
146D16—614no—
147D16—1224no—
148D16—1222no—
149D16C22×D6112yes—
150M32—611no—
151M32—612no—
152C2×C16—612no—
153C2×C16C22×D6111yescharacteristic
154C32—612no—
155C32C22×D6111yescharacteristic
156C32C22×D6111yescharacteristic
157#41—42no—
158#41—611no—
159#41—4882no—
160#41—2444no—
161C2×D4—4882no—
162C2×D4—4884no—
163C2×D4—2444no—
164C2×D4—2442no—
165Q16—22no—
166SD16—42no—
167SD16—2444no—
168D8—22no—
169D8—1224no—
170D8—2444no—
171D8—2442no—
172M16—611no—
173M16—612no—
174C2×C8—612no—
175C2×C8D4×D6111yescharacteristic
176C16—612no—
177C16D4×D6111yescharacteristic
178C16C23×D6111yescharacteristic
179C23—4884no—
180C23—4882no—
181Q8—42no—
182Q8—611no—
183D4—42no—
184D4—82no—
185D4—612no—
186D4—611no—
187D4—4884no—
188D4—4884no—
189D4—4882no—
190D4—2444no—
191C2×C4—82no—
192C2×C4—612no—
193C2×C4—4884no—
194C2×C4D8×D6111yescharacteristic
195C8—612no—
196C8D8×D6111yescharacteristic
197C8C2×D4×D6111yescharacteristic
198C22—82no—
199C22—612no—
200C22—19522no—
201C22—19524no—
202C22—4884no—
203C22D16×D6111yescharacteristic
204C4—82no—
205C4—612no—
206C4D16×D6111yescharacteristic
207C4C2×D8×D6111yescharacteristic
208C2—21no—
209C2—162no—
210C2—1222no—
211C2—9764no—
212C2C2×D16×D6111yescharacteristic
subgroupG/H
centerC2C2×D16×D61
commutator, cosocleC16×C61C24
FrattiniC16C23×D61
FittingC61×#2262C2
socleC2×C61C22×D16
hypercenter#2262D61
hypocenterC61#33982
LCSG ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #21458957C2semidirect
2. #21448133C2semidirect
3. #21448131C2semidirect
4. #21448129C2semidirect
5. C2×D1952C2semidirect
6. D32×D61C2semidirect
7. #21448112C2semidirect
8. C61×#2262C2semidirect
12. #16840195C22semidirect
13. C2×D976C22semidirect
14. D16×D61C22semidirect
16. #16839599C22semidirect
17. D1952C22semidirect
18. D1952C22semidirect
19. #16839597C22semidirect
20. #16839596C22semidirect
21. #16839595C22semidirect
22. C32×D61C22semidirect
23. C61×#508C22semidirect
24. C61×Q64C22semidirect
25. C61×SD64C22semidirect
26. C61×D32C22semidirect
27. C2×C32×C61C22semidirect
106. #2262D61semidirect
119. #508C2×D61semidirect
123. Q64C2×D61semidirect
125. SD64C2×D61semidirect
129. D32C2×D61semidirect
133. C2×C32C2×D61semidirect
134. C61#33982semidirect

Maximal subgroups

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

subgroupindex
#214589572
#214481332
#214481312
#214481292
C2×D19522
D32×D612
#214481122
C61×#22622
#3398261