Finite Group Databasebeta

← #29566693

Subgroups — #29566693

Order 15616.

286 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 49 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#21446829C211yesmaximal, characteristic
2#21443687C211yesmaximal, characteristic
3C61×#1314C211yesmaximal, characteristic
4C22×Dic244C411yescharacteristic, complemented
5C2×#14519756—21no—
6C2×#14519702—21no—
7#16839622—21no—
8#16839525C411yescharacteristic, complemented
9C2×C22⋊C8×C61C2211yescharacteristic
10C61×C22⋊C16—21no—
11C22×Dic122—21no—
12C2×Dic244—41no—
13C2×Dic244—42no—
14#14519756—42no—
15#14519756—42no—
16C2×C4⋊Dic61—21no—
17C2×C4⋊Dic61—41no—
18C2×#11758196—41no—
19C2×C22⋊Dic61—41no—
20C2×C22⋊Dic61—41no—
21#14519702—42no—
22#14519684—22no—
23C23×C4×C61D411yescharacteristic
24C22×C8×C61C2×C411yescharacteristic
25C22×C8×C61D411yescharacteristic
26C2×C16×C61—41no—
27C22⋊C8×C61—41no—
28C24×C61D811yescharacteristic
29C22×C4×C61—21no—
30C22×C4×C61—21no—
31C22×C4×C61—41no—
32C22×C4×C61—41no—
33C22×C4×C61C22⋊C411yescharacteristic
34C22×C4×C61SD1611yescharacteristic
35C22×Dic61—41no—
36C22×Dic61—41no—
37C22×Dic61—42no—
38C2×Dic122—22no—
39C2×Dic122—41no—
40C2×Dic122—41no—
41C2×Dic122—41no—
42C2×Dic122—41no—
43C2×Dic122—41no—
44C2×C8×C61—21no—
45C2×C8×C61—21no—
46C2×C8×C61—41no—
47C2×C8×C61—41no—
48C22⋊Dic61—44no—
49C22⋊Dic61—81no—
50C22⋊Dic61—81no—
51C4⋊Dic61—22no—
52C4⋊Dic61—41no—
53C4⋊Dic61—81no—
54C4⋊Dic61—81no—
55#11758196—44no—
56Dic244—42no—
57Dic244—42no—
58C16×C61—81no—
59C23×C61—21no—
60C23×C61—21no—
61C23×C61—41no—
62C23×C61—41no—
63C23×C61#10211yescharacteristic
64C23×C61SD3211yescharacteristic
65C23×C61Q3211yescharacteristic
66C2×C4×C61—21no—
67C2×C4×C61—21no—
68C2×C4×C61—21no—
69C2×C4×C61—41no—
70C2×C4×C61—41no—
71C2×C4×C61—41no—
72C2×C4×C61—81no—
73C2×C4×C61#9911yescharacteristic
74C2×C4×C61#10411yescharacteristic
75C2×Dic61—42no—
76C2×Dic61—42no—
77C2×Dic61—81no—
78C2×Dic61—81no—
79C2×Dic61—81no—
80C2×Dic61—81no—
81C2×Dic61—82no—
82C2×Dic61—82no—
83C2×Dic61—82no—
84Dic122—42no—
85Dic122—42no—
86Dic122—81no—
87Dic122—81no—
88C8×C61—41no—
89C8×C61—41no—
90#7078—611nomaximal, Sylow
91C22×C61—21no—
92C22×C61—21no—
93C22×C61—21no—
94C22×C61—21no—
95C22×C61—41no—
96C22×C61—41no—
97C22×C61—41no—
98C22×C61—41no—
99C22×C61—81no—
100C22×C61#32711yescharacteristic
101C22×C61#35811yescharacteristic
102C22×C61#36011yescharacteristic
103C4×C61—21no—
104C4×C61—41no—
105C4×C61AΣL(1,𝔽16)11yescharacteristic
106C4×C61#35311yescharacteristic
107Dic61—82no—
108Dic61—82no—
109Dic61—162no—
110#2001—611no—
111#1314D6111yescharacteristic
112#1289—611no—
113C2×C61—21no—
114C2×C61—21no—
115C2×C61—41no—
116C2×C61—41no—
117C2×C61#134011yescharacteristic
118C2×C61#134311yescharacteristic
119C2×C61#134811yescharacteristic
120C22×Q16—611no—
121C2×C22⋊Q8—1221no—
122C2×#103—1221no—
123C2×C22⋊C8C2×D6111yescharacteristic
124#380—1221no—
125C22⋊C16—21no—
126#340—611no—
127C61#707811yescharacteristic, complemented, Sylow
128C22×Q8—1221no—
129C23×C4D24411yescharacteristic
130C2×Q16—2441no—
131C2×Q16—2442no—
132C22×C8C4×D6111yescharacteristic
133C22×C8C22⋊D6111yescharacteristic
134C22⋊Q8—2442no—
135C22⋊Q8—2442no—
136C2×C4⋊C4—1221no—
137C2×C4⋊C4—2441no—
138C2×C4⋊C4—2441no—
139C2×C22⋊C4—2441no—
140C2×C22⋊C4—2441no—
141C2×C16—41no—
142#103—2442no—
143C22⋊C8—41no—
144#95—1222no—
145C24D48811yescharacteristic
146C2×Q8—1222no—
147C2×Q8—2441no—
148C2×Q8—2441no—
149C2×Q8—2441no—
150C2×Q8—2441no—
151C2×Q8—2441no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—2442no—
157C22×C4—2441no—
158C22×C4—2441no—
159C22×C4#1175819011yescharacteristic
160C22×C4#1175819811yescharacteristic
161Q16—2442no—
162Q16—2442no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—1222no—
168C4⋊C4—4881no—
169C4⋊C4—4881no—
170C4⋊C4—2441no—
171C4⋊C4—2444no—
172C22⋊C4—4881no—
173C22⋊C4—4881no—
174C22⋊C4—2444no—
175C16—81no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#1451969911yescharacteristic
181C23#1451972211yescharacteristic
182C23Dic48811yescharacteristic
183Q8—4881no—
184Q8—4881no—
185Q8—2442no—
186Q8—2442no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—4882no—
195C2×C4—4882no—
196C2×C4—4882no—
197C2×C4—4881no—
198C2×C4—4881no—
199C2×C4—4881no—
200C2×C4—4881no—
201C2×C4—2442no—
202C2×C4—2442no—
203C2×C4#1451969111yescharacteristic
204C2×C4#1451970311yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#1683949711yescharacteristic
217C22#1683956711yescharacteristic
218C22#1683957311yescharacteristic
219C4—21no—
220C4—41no—
221C4—9762no—
222C4—4882no—
223C4—4882no—
224C4#1683955011yescharacteristic
225C4#1683955411yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#2144380811yescharacteristic
231C2#2144381711yescharacteristic
232C2#2144383211yescharacteristic
subgroupG/H
centerC22#16839497
commutatorC22×C8×C61C2×C4
FrattiniC2×C22⋊C8C2×D61
FittingC61×#1314C2
socleC22×C61#327
prime radicalC24×C61D8
cosocleC2×C22⋊C8×C61C22
hypercenter#1314D61
hypocenterC61#7078
LCSG ▷ C22×C8×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
4. C22×Dic244C4semidirect
8. #16839525C4semidirect
127. C61#7078semidirect

Maximal subgroups

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

subgroupindex
#214468292
#214436872
C61×#13142
#707861