Finite Group Databasebeta

← #29593770

Subgroups — #29593770

Order 15616.

358 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 59 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#21457612C211yesmaximal, characteristic
2#21447987C212yesmaximal, complemented
3#21443860C211yesmaximal, characteristic, complemented
4#21443793C212yesmaximal, complemented
5C61×#1705C211yesmaximal, characteristic, complemented
6#16840568—22no—
7#16840022—22no—
8#16839944—21no—
9#16839929—21no—
10#16839880C2212yes—
11#16839584—22no—
12#16839583C2212yescomplemented
13#16839571—22no—
14C61×#498—21no—
15C61×#498C2211yescharacteristic, complemented
16C61×#492—21no—
17C61×#492C2211yescharacteristic
18C82×C61C2211yescharacteristic, complemented
19#14519825—21no—
20#14519822—21no—
21#14519808—22no—
22#14519806—42no—
23#14519804—42no—
24#14519803—22no—
25#14519799D412yes—
26#14519798—42no—
27SD16×D61—42no—
28#14519783—22no—
29#14519717—22no—
30#14519713D412yes—
31#14519705—22no—
32#14519696—21no—
33#14519694—21no—
34C2×SD16×C61—22no—
35C2×SD16×C61—42no—
36C4⋊Q8×C61—21no—
37C4⋊Q8×C61D411yescharacteristic
38#127×C61D411yescharacteristic
39#106×C61—22no—
40#106×C61—42no—
41C4×C8×C61—21no—
42C4×C8×C61C2311yescharacteristic
43C2×Q8×C61—21no—
44C2×Q8×C61—41no—
45C2×D4×C61—21no—
46C2×D4×C61—41no—
47C2×C22⋊D61—81no—
48#11758226—42no—
49Q8×D61—42no—
50#11758224—42no—
51D4×D61—42no—
52#11758222—22no—
53#11758222—81no—
54SD16×C61—42no—
55SD16×C61—42no—
56SD16×C61—84no—
57C2×C8×C61—22no—
58C2×C8×C61C2×D412yes—
59C4⋊C4×C61—41no—
60C4⋊C4×C61—41no—
61C4⋊C4×C61—41no—
62C42×C61C2×D411yescharacteristic
63#11758202—42no—
64#11758201—42no—
65#11758200—42no—
66#11758199—42no—
67#11758194—22no—
68Dic244SD1612yescomplemented
69D488SD1612yescomplemented
70#11758190—22no—
71#11758189—42no—
72C8×D61—42no—
73#11758185—42no—
74C23×C61—41no—
75C22×D61—81no—
76Q8×C61—42no—
77Q8×C61—82no—
78D4×C61—42no—
79D4×C61—82no—
80C2×C4×C61—21no—
81C2×C4×C61—41no—
82C2×C4×C61—81no—
83C2×C4×C61#12011yescharacteristic
84C22⋊D61—42no—
85C22⋊D61—82no—
86C2×Dic61—81no—
87D244—22no—
88D244—81no—
89C4×D61—42no—
90C4×D61—82no—
91Dic122—22no—
92Dic122—81no—
93C8×C61—24no—
94C8×C61C2×SD1612yes—
95C8×C61AGL(1,ℤ/8)12yes—
96#92418—42no—
97#12364—611nomaximal, Sylow
98C22×C61—81no—
99C22×C61—161no—
100C22×C61#45311yescharacteristic
101C2×D61—42no—
102C2×D61—321no—
103C4×C61—22no—
104C4×C61—81no—
105C4×C61—161no—
106C4×C61#45012yes—
107Dic61—42no—
108#3291—611no—
109#2230—612no—
110#1705D6111yescharacteristic, complemented
111#1359—611no—
112#1335—612no—
113C2×C61—21no—
114C2×C61—161no—
115C2×C61#162911yescharacteristic
116D61—82no—
117#577—1222no—
118#498—21no—
119#498C2×D6111yescharacteristic, complemented
120#492—21no—
121#492C2×D6111yescharacteristic
122#471—1222no—
123#456—1221no—
124#453—1221no—
125#443—612no—
126#365—1222no—
127#364—612no—
128#360—1222no—
129C82C2×D6111yescharacteristic, complemented
130C61#1236411yescharacteristic, complemented, Sylow
1312−1+4—1221no—
132H(5,𝔽2)—1221no—
133#137—1222no—
134#137—2442no—
135AGL(1,ℤ/8)—1222no—
136AGL(1,ℤ/8)—2442no—
137#135—612no—
138#135—2442no—
139C2×SD16—22no—
140C2×SD16—42no—
141C2×SD16—2442no—
142#131—1222no—
143C4⋊Q8—21no—
144C4⋊Q8C22⋊D6111yescharacteristic
145#127C22⋊D6111yescharacteristic
146M32—1222no—
147#108—612no—
148#106—22no—
149#106—42no—
150#104—1222no—
151#101—1221no—
152#100—1221no—
153C4×C8—21no—
154C4×C8C22×D6111yescharacteristic
155#41—1222no—
156#41—4881no—
157#41—2442no—
158#41—2442no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—2442no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—4881no—
165C2×D4—2442no—
166Q16—612no—
167Q16—2442no—
168SD16—42no—
169SD16—42no—
170SD16—84no—
171SD16—1222no—
172SD16—2442no—
173SD16—2442no—
174D8—612no—
175D8—2442no—
176M16—1222no—
177M16—2442no—
178C2×C8—22no—
179C2×C8—2442no—
180C2×C8D4×D6112yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C42C2×C22⋊D6111yescharacteristic
185C16—2442no—
186C23—41no—
187C23—4881no—
188Q8—42no—
189Q8—82no—
190Q8—1222no—
191Q8—4881no—
192D4—42no—
193D4—82no—
194D4—1222no—
195D4—4881no—
196D4—4882no—
197D4—2442no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4—4882no—
202C2×C4—4881no—
203C2×C4—2442no—
204C2×C4#1451974511yescharacteristic
205C8—24no—
206C8—2442no—
207C8SD16×D6112yes—
208C8#1451980412yes—
209C22—81no—
210C22—161no—
211C22—19521no—
212C22—2442no—
213C22#1683992911yescharacteristic
214C4—22no—
215C4—81no—
216C4—161no—
217C4—2442no—
218C4#1683991112yes—
219C2—21no—
220C2—161no—
221C2—4882no—
222C2#2144493311yescharacteristic
subgroupG/H
centerC2#21444933
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1705C2
socleC2×C61#1629
prime radical#21457612C2
hypercenter#1705D61
hypocenterC61#12364
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1705

Complements

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

NH ≅ G/Ntype
2. #21447987C2semidirect
3. #21443860C2semidirect
4. #21443793C2semidirect
5. C61×#1705C2semidirect
12. #16839583C22semidirect
15. C61×#498C22semidirect
18. C82×C61C22semidirect
68. Dic244SD16semidirect
69. D488SD16semidirect
110. #1705D61semidirect
119. #498C2×D61semidirect
129. C82C2×D61semidirect
130. C61#12364semidirect

Maximal subgroups

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

subgroupindex
#214576122
#214479872
#214438602
#214437932
C61×#17052
#1236461