Finite Group Databasebeta

← #29567757

Subgroups — #29567757

Order 15616.

298 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446208C211yesmaximal, characteristic
2C2×#16839502C211yesmaximal, characteristic
3C2×C61×#355C211yesmaximal, characteristic
4#16840237C411yescharacteristic
5#16839771—21no—
6C2×#14519694C411yescharacteristic
7C2×#14519688—21no—
8#16839522—21no—
9#16839502—21no—
10#16839502C412yescomplemented
11C2×#124×C61C2211yescharacteristic
12C2×#101×C61—21no—
13C61×#355—21no—
14C61×#355C412yescomplemented
15C2×#11758194—21no—
16C22×#92418—41no—
17D4×Dic61—42no—
18#14519736—21no—
19C2×C22⋊Dic61—41no—
20#14519701—42no—
21#14519694—41no—
22#14519688—22no—
23#14519688—41no—
24C22×Q8×C61D411yescharacteristic
25C22×D4×C61C2×C411yescharacteristic
26C2×M16×C61—41no—
27#124×C61—41no—
28#124×C61C2×C411yescharacteristic
29#124×C61C2×C411yescharacteristic
30#124×C61D411yescharacteristic
31#124×C61Q811yescharacteristic
32C2×C22⋊C4×C61—41no—
33C2×C42×C61D411yescharacteristic
34#101×C61—22no—
35#101×C61—41no—
36C24×C61—21no—
37C2×Q8×C61—21no—
38C2×Q8×C61—41no—
39C2×Q8×C61—41no—
40C2×Q8×C61C22⋊C411yescharacteristic
41C2×Q8×C61C4⋊C411yescharacteristic
42C2×D4×C61—21no—
43C2×D4×C61—41no—
44C2×D4×C61—41no—
45C2×D4×C61C4211yescharacteristic
46C2×D4×C61C4⋊C411yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61C22⋊C411yescharacteristic
50C22×Dic61—42no—
51M16×C61—42no—
52M16×C61—81no—
53C2×C8×C61—81no—
54C22⋊C4×C61—41no—
55C22⋊C4×C61—41no—
56C22⋊C4×C61—41no—
57C22⋊C4×C61—41no—
58C42×C61—21no—
59C42×C61C22⋊C411yescharacteristic
60C42×C61C4⋊C411yescharacteristic
61C22⋊Dic61—42no—
62C22⋊Dic61—41no—
63C4⋊Dic61—22no—
64C4×Dic61—41no—
65#11758194—41no—
66#11758194—41no—
67C2×#92418—42no—
68C2×#92418—44no—
69C2×#92418—41no—
70C23×C61—21no—
71C23×C61—21no—
72C23×C61—41no—
73C23×C61—41no—
74C23×C61—82no—
75C23×C61#9911yescharacteristic
76Q8×C61—41no—
77Q8×C61—41no—
78Q8×C61—81no—
79D4×C61—41no—
80D4×C61—41no—
81D4×C61—42no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—42no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61#9511yescharacteristic
92C2×C4×C61#9911yescharacteristic
93C2×Dic61—42no—
94C2×Dic61—162no—
95C8×C61—82no—
96#92418—44no—
97#92418—81no—
98#7487—611nomaximal, Sylow
99C22×C61—22no—
100C22×C61—41no—
101C22×C61—41no—
102C22×C61—81no—
103C22×C61—81no—
104C22×C61—82no—
105C22×C61—161no—
106C22×C61#34211yescharacteristic
107C22×C61AΣL(1,𝔽16)11yescharacteristic
108C22×C61#35211yescharacteristic
109C4×C61—21no—
110C4×C61—21no—
111C4×C61—41no—
112C4×C61—41no—
113C4×C61—41no—
114C4×C61—41no—
115Dic61—82no—
116C2×#355D6111yescharacteristic
117#1885—611no—
118C2×#329—611no—
119C2×C61—21no—
120C2×C61—21no—
121C2×C61—81no—
122C2×C61—81no—
123C2×C61#139011yescharacteristic
124C2×C61#140312yes—
125C2×#124C2×D6111yescharacteristic
126#518—611no—
127#419—1221no—
128C2×#101—21no—
129C2×#100—611no—
130C2×#97—1221no—
131#355—21no—
132#355Dic6112yescomplemented
133#339—1221no—
134#329—612no—
135#329—1221no—
136C61#748711yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D6111yescharacteristic
138C22×D4C4×D6111yescharacteristic
139C2×M16—41no—
140C2×M16—1221no—
141C22×C8—2441no—
142#124—41no—
143#124Dic12211yescharacteristic
144#124C4×D6111yescharacteristic
145#124C2×Dic6111yescharacteristic
146#124C22⋊D6111yescharacteristic
147C4×D4—2442no—
148#117—1221no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—2441no—
151C2×C42D24411yescharacteristic
152#103—2442no—
153#101—22no—
154#101—41no—
155#100—2441no—
156#97—1222no—
157#97—2441no—
158C24—21no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8#1175819611yescharacteristic
163C2×Q8C22⋊Dic6111yescharacteristic
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4C4×Dic6111yescharacteristic
168C2×D4#1175819611yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—2442no—
172C22×C4#1175819811yescharacteristic
173M16—42no—
174M16—81no—
175M16—2441no—
176M16—2441no—
177C2×C8—81no—
178C2×C8—2441no—
179C2×C8—2442no—
180C2×C8—2444no—
181C4⋊C4—1222no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—2442no—
187C22⋊C4—2441no—
188C42—21no—
189C42—2441no—
190C42C4⋊Dic6111yescharacteristic
191C42#1175819811yescharacteristic
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—82no—
197C23#1451969111yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—81no—
201D4—41no—
202D4—41no—
203D4—42no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—9762no—
214C2×C4—2442no—
215C2×C4#1451968411yescharacteristic
216C2×C4#1451969111yescharacteristic
217C8—82no—
218C8—4881no—
219C8—2444no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—82no—
226C22—161no—
227C22#1683952811yescharacteristic
228C22#1683954811yescharacteristic
229C22#1683955111yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—4882no—
237C2—21no—
238C2—21no—
239C2—81no—
240C2—81no—
241C2#2144393611yescharacteristic
242C2#2144396612yes—
subgroupG/H
centerC22#16839548
commutatorC2×D4×C61C42
FrattiniC2×#124C2×D61
FittingC2×C61×#355C2
socleC22×C61AΣL(1,𝔽16)
prime radicalC22×D4×C61C2×C4
cosocleC2×#124×C61C22
hypercenterC2×#355D61
hypocenterC61#7487
LCSG ▷ C2×D4×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
10. #16839502C4semidirect
14. C61×#355C4semidirect
132. #355Dic61semidirect
136. C61#7487semidirect

Maximal subgroups

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

subgroupindex
#214462082
C2×#168395022
C2×C61×#3552
#748761