Finite Group Databasebeta

← #29567739

Subgroups — #29567739

Order 15616.

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

normal subgroups only

This group has 284 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#21446223C211yesmaximal, characteristic
2#21444443C211yesmaximal, characteristic
3C2×C61×#354C211yesmaximal, characteristic
4#16840240C411yescharacteristic, complemented
5#16839819—21no—
6#16839767—21no—
7C2×#14519696C411yescharacteristic, complemented
8#16839522—21no—
9#16839502—21no—
10#16839502C412yescomplemented
11C2×#124×C61C2211yescharacteristic
12C2×#99×C61—21no—
13C61×#354—21no—
14C61×#354C412yescomplemented
15C2×#11758194—21no—
16Q8×Dic61—42no—
17#14519736—21no—
18#14519736—41no—
19C4⋊#92418—41no—
20#14519698—42no—
21#14519696—41no—
22C22⋊#92418—41no—
23#14519688—22no—
24C22×Q8×C61C2×C411yescharacteristic
25C22×D4×C61D411yescharacteristic
26#124×C61—41no—
27#124×C61C2×C411yescharacteristic
28#124×C61C2×C411yescharacteristic
29#124×C61D411yescharacteristic
30#124×C61Q811yescharacteristic
31C2×C22⋊C4×C61—41no—
32C2×C22⋊C4×C61—41no—
33C2×C42×C61D411yescharacteristic
34#99×C61—22no—
35#99×C61—41no—
36C24×C61—21no—
37C2×Q8×C61—21no—
38C2×Q8×C61—41no—
39C2×Q8×C61—41no—
40C2×Q8×C61C4211yescharacteristic
41C2×Q8×C61C4⋊C411yescharacteristic
42C2×D4×C61—21no—
43C2×D4×C61—41no—
44C2×D4×C61—41no—
45C2×D4×C61C22⋊C411yescharacteristic
46C2×D4×C61C4⋊C411yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—81no—
50C22×C4×C61C22⋊C411yescharacteristic
51C22⋊C4×C61—41no—
52C22⋊C4×C61—41no—
53C22⋊C4×C61—41no—
54C22⋊C4×C61—41no—
55C22⋊C4×C61—42no—
56C22⋊C4×C61—81no—
57C42×C61—21no—
58C42×C61C22⋊C411yescharacteristic
59C42×C61C4⋊C411yescharacteristic
60C22⋊Dic61—41no—
61C4⋊Dic61—22no—
62C4⋊Dic61—42no—
63C4×Dic61—41no—
64C4×Dic61—42no—
65#11758194—41no—
66#11758194—41no—
67C2×#92418—42no—
68C2×#92418—41no—
69C23×C61—21no—
70C23×C61—21no—
71C23×C61—41no—
72C23×C61—41no—
73C23×C61—82no—
74C23×C61#9911yescharacteristic
75Q8×C61—42no—
76Q8×C61—41no—
77Q8×C61—41no—
78D4×C61—41no—
79D4×C61—41no—
80D4×C61—81no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—81no—
90C2×C4×C61—82no—
91C2×C4×C61—162no—
92C2×C4×C61#9511yescharacteristic
93C2×C4×C61#9911yescharacteristic
94C2×Dic61—42no—
95#92418—81no—
96#92418—82no—
97#7481—611nomaximal, Sylow
98C22×C61—41no—
99C22×C61—41no—
100C22×C61—41no—
101C22×C61—81no—
102C22×C61—81no—
103C22×C61—82no—
104C22×C61—161no—
105C22×C61#34211yescharacteristic
106C22×C61AΣL(1,𝔽16)11yescharacteristic
107C22×C61#35211yescharacteristic
108C4×C61—21no—
109C4×C61—21no—
110C4×C61—41no—
111C4×C61—41no—
112C4×C61—41no—
113C4×C61—41no—
114C4×C61—162no—
115Dic61—82no—
116C2×#354D6111yescharacteristic
117#1888—611no—
118#1533—611no—
119C2×C61—21no—
120C2×C61—21no—
121C2×C61—21no—
122C2×C61—81no—
123C2×C61—81no—
124C2×C61#139011yescharacteristic
125C2×#124C2×D6111yescharacteristic
126#519—611no—
127#432—1221no—
128#418—1221no—
129C2×#101—611no—
130C2×#99—21no—
131#354—21no—
132#354Dic6112yescomplemented
133#339—1221no—
134#329—612no—
135#329—1221no—
136C61#748111yescharacteristic, complemented, Sylow
137C22×Q8C4×D6111yescharacteristic
138C22×D4C22⋊D6111yescharacteristic
139C2×M16—1221no—
140#124—41no—
141#124Dic12211yescharacteristic
142#124C4×D6111yescharacteristic
143#124C2×Dic6111yescharacteristic
144#124C22⋊D6111yescharacteristic
145C4×Q8—2442no—
146#117—1221no—
147#117—2441no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42D24411yescharacteristic
151C4⋊C8—2441no—
152#102—2442no—
153#101—2441no—
154#99—22no—
155#99—41no—
156C22⋊C8—2441no—
157#97—1222no—
158C24—21no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8C4×Dic6111yescharacteristic
163C2×Q8#1175819611yescharacteristic
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4#1175819611yescharacteristic
168C2×D4C22⋊Dic6111yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—81no—
172C22×C4#1175819811yescharacteristic
173M16—2441no—
174M16—2441no—
175C2×C8—2442no—
176C2×C8—2441no—
177C4⋊C4—1222no—
178C4⋊C4—2442no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—81no—
185C22⋊C4—2441no—
186C42—21no—
187C42—2441no—
188C42—2442no—
189C42C4⋊Dic6111yescharacteristic
190C42#1175819811yescharacteristic
191C23—21no—
192C23—21no—
193C23—41no—
194C23—41no—
195C23—82no—
196C23#1451969111yescharacteristic
197Q8—41no—
198Q8—41no—
199Q8—42no—
200D4—41no—
201D4—41no—
202D4—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—162no—
214C2×C4—2442no—
215C2×C4#1451968411yescharacteristic
216C2×C4#1451969111yescharacteristic
217C8—4881no—
218C8—4882no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—161no—
226C22#1683952811yescharacteristic
227C22#1683954811yescharacteristic
228C22#1683955111yescharacteristic
229C4—21no—
230C4—21no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—162no—
236C4—4882no—
237C2—21no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—81no—
242C2#2144393611yescharacteristic
subgroupG/H
centerC2#21443936
commutatorC2×Q8×C61C42
FrattiniC2×#124C2×D61
FittingC2×C61×#354C2
socleC2×C61#1390
prime radicalC22×D4×C61D4
cosocleC2×#124×C61C22
hypercenterC2×#354D61
hypocenterC61#7481
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
4. #16840240C4semidirect
7. C2×#14519696C4semidirect
10. #16839502C4semidirect
14. C61×#354C4semidirect
132. #354Dic61semidirect
136. C61#7481semidirect

Maximal subgroups

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

subgroupindex
#214462232
#214444432
C2×C61×#3542
#748161