Finite Group Databasebeta

← #29593426

Subgroups — #29593426

Order 15616.

384 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 67 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#21447895C212yesmaximal, complemented
2#21445483C211yesmaximal, characteristic, complemented
3#21443860C211yesmaximal, characteristic, complemented
4#21443795C212yesmaximal, complemented
5C61×#3291C211yesmaximal, characteristic, complemented
6#16840197—22no—
7#16840139C2211yescharacteristic, complemented
8#16840138—21no—
9#16840110—21no—
10#16839583C2212yescomplemented
11#16839574—22no—
12#16839483C2211yescharacteristic, complemented
13C61×#577—22no—
14C61×#492C2211yescharacteristic
15C61×#471—22no—
16C61×#456—21no—
17C61×#453—21no—
18C61×#443C2212yes—
19C2×D488—22no—
20C2×#11758199—42no—
21#14519774—21no—
22#14519720—42no—
23#14519718—42no—
24#14519717—22no—
25#14519697—42no—
26C4×#92418—21no—
272−1+4×C61—21no—
28H(5,𝔽2)×C61—21no—
29#137×C61—22no—
30#137×C61—42no—
31AGL(1,ℤ/8)×C61—22no—
32AGL(1,ℤ/8)×C61—42no—
33#135×C61—42no—
34#135×C61D412yes—
35C2×SD16×C61—22no—
36C2×SD16×C61—42no—
37#131×C61—22no—
38C4⋊Q8×C61D411yescharacteristic
39#127×C61D411yescharacteristic
40#108×C61D412yes—
41#104×C61—22no—
42#101×C61—21no—
43#100×C61—21no—
44C4×C8×C61C2311yescharacteristic
45#41×C61—22no—
46#41×C61—42no—
47#41×C61—42no—
48#41×C61—81no—
49C2×Q8×C61—21no—
50C2×Q8×C61—42no—
51C2×D4×C61—21no—
52C2×D4×C61—41no—
53C2×D4×C61—42no—
54C2×D4×C61—81no—
55C2×D244—41no—
56C2×D244—41no—
57C2×D244—41no—
58Q16×C61—42no—
59Q16×C61D812yescomplemented
60SD16×C61—22no—
61SD16×C61—42no—
62SD16×C61—42no—
63SD16×C61—42no—
64SD16×C61—42no—
65D8×C61—42no—
66D8×C61D812yescomplemented
67M16×C61—22no—
68M16×C61—42no—
69C2×C8×C61—42no—
70C2×C8×C61C2×D412yes—
71C4⋊C4×C61—41no—
72C42×C61C2×D411yescharacteristic
73#11758199—84no—
74C2×#92418—22no—
75D488—42no—
76D488—42no—
77#11758185—42no—
78C23×C61—41no—
79C23×C61—81no—
80C22×D61—81no—
81Q8×C61—22no—
82Q8×C61—42no—
83Q8×C61—81no—
84D4×C61—22no—
85D4×C61—42no—
86D4×C61—42no—
87D4×C61—82no—
88D4×C61—82no—
89D4×C61—81no—
90C2×C4×C61—21no—
91C2×C4×C61—42no—
92C2×C4×C61—41no—
93C2×C4×C61—81no—
94C2×C4×C61—82no—
95C2×C4×C61#12011yescharacteristic
96D244—82no—
97D244—82no—
98D244—82no—
99C8×C61—42no—
100C8×C61C2×D812yes—
101C8×C61AGL(1,ℤ/8)12yes—
102#92418—42no—
103#12314—611nomaximal, Sylow
104C22×C61—42no—
105C22×C61—81no—
106C22×C61—161no—
107C22×C61—321no—
108C22×C61#45311yescharacteristic
109C2×D61—161no—
110C2×D61—161no—
111C2×D61—161no—
112C4×C61—22no—
113C4×C61—42no—
114C4×C61—81no—
115C4×C61#44712yes—
116#3291D6111yescharacteristic, complemented
117#2213—612no—
118#1721—611no—
119#1359—611no—
120#1336—612no—
121C2×C61—21no—
122C2×C61—82no—
123C2×C61—161no—
124C2×C61#161911yescharacteristic
125D61—321no—
126#577—22no—
127#509—1222no—
128#493—611no—
129#493—1221no—
130#492C2×D6111yescharacteristic
131#486—1221no—
132#471—22no—
133#456—21no—
134#453—21no—
135#443C2×D6112yes—
136#364—612no—
137#361—1222no—
138#322—611no—
139C61#1231411yescharacteristic, complemented, Sylow
1402−1+4—21no—
141H(5,𝔽2)—21no—
142#137—22no—
143#137—42no—
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—42no—
146#135—42no—
147#135C22⋊D6112yes—
148C2×SD16—22no—
149C2×SD16—42no—
150C2×D8—1222no—
151C2×D8—2442no—
152#131—22no—
153C4⋊Q8C22⋊D6111yescharacteristic
154#127—1221no—
155#127C22⋊D6111yescharacteristic
156SD32—2442no—
157D16—2442no—
158M32—1222no—
159#108C22⋊D6112yes—
160#104—22no—
161#102—2442no—
162#101—21no—
163#100—21no—
164C4×C8—1221no—
165C4×C8C22×D6111yescharacteristic
166#41—22no—
167#41—42no—
168#41—42no—
169#41—81no—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—81no—
176C2×D4—2441no—
177C2×D4—2441no—
178C2×D4—2441no—
179Q16—42no—
180Q16#1175819912yescomplemented
181SD16—22no—
182SD16—42no—
183SD16—42no—
184SD16—42no—
185SD16—42no—
186D8—42no—
187D8—4884no—
188D8—2442no—
189D8—2442no—
190D8#1175819912yescomplemented
191M16—22no—
192M16—42no—
193C2×C8—42no—
194C2×C8—1222no—
195C2×C8C2×C22⋊D6112yes—
196C4⋊C4—41no—
197C42C2×C22⋊D6111yescharacteristic
198C16—2442no—
199C23—41no—
200C23—81no—
201C23—4881no—
202Q8—22no—
203Q8—42no—
204Q8—81no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—81no—
211D4—4882no—
212D4—4882no—
213D4—4882no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4#1451974611yescharacteristic
220C8—42no—
221C8—2442no—
222C8C2×#1175819912yes—
223C8#1451980312yes—
224C22—42no—
225C22—81no—
226C22—161no—
227C22—321no—
228C22—9761no—
229C22—9761no—
230C22—9761no—
231C22#1683993011yescharacteristic
232C4—22no—
233C4—42no—
234C4—81no—
235C4#1683989612yes—
236C2—21no—
237C2—82no—
238C2—161no—
239C2—19521no—
240C2#2144487311yescharacteristic
subgroupG/H
centerC2#21444873
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#3291C2
socleC2×C61#1619
hypercenter#3291D61
hypocenterC61#12314
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #21447895C2semidirect
2. #21445483C2semidirect
3. #21443860C2semidirect
4. #21443795C2semidirect
5. C61×#3291C2semidirect
7. #16840139C22semidirect
10. #16839583C22semidirect
12. #16839483C22semidirect
59. Q16×C61D8semidirect
66. D8×C61D8semidirect
116. #3291D61semidirect
139. C61#12314semidirect
180. Q16#11758199semidirect
190. D8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214478952
#214454832
#214438602
#214437952
C61×#32912
#1231461