Finite Group Databasebeta

← #29567367

Subgroups — #29567367

Order 15616.

452 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 39 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#21445739C211yesmaximal, characteristic
2#21443755C211yesmaximal, characteristic, complemented
3C61×#1320C211yesmaximal, characteristic, complemented
4C23×D244C411yescharacteristic
5C2×#14519693—21no—
6C2×#14519693—21no—
7#16839726C411yescharacteristic
8#16839544—21no—
9C61×#407C2211yescharacteristic
10C61×#349—21no—
11C24×D61—21no—
12C22×D244—21no—
13C22×D244—42no—
14C22×D244—42no—
15C22×D244—41no—
16C22×D244—41no—
17C22×D244C812yes—
18C2×#11758194—21no—
19#14519717—41no—
20#14519693—42no—
21#14519693—42no—
22C22⋊#92418—41no—
23C23×C4×C61C2×C411yescharacteristic
24C2×M16×C61D411yescharacteristic
25C2×M16×C61D411yescharacteristic
26M32×C61—41no—
27C22⋊C8×C61—41no—
28C24×C61M1611yescharacteristic
29C23×D61—22no—
30C23×D61—44no—
31C23×D61—42no—
32C23×D61—84no—
33C22×C4×C61—21no—
34C22×C4×C61—21no—
35C22×C4×C61—41no—
36C22×C4×C61—41no—
37C22×C4×C61C22⋊C411yescharacteristic
38C22×C4×C61C2×C811yescharacteristic
39C2×D244—22no—
40C2×D244—22no—
41C2×D244—41no—
42C2×D244—41no—
43C2×D244—42no—
44C2×D244—48no—
45C2×D244—42no—
46C2×D244—42no—
47C2×D244—41no—
48C2×D244—41no—
49C2×D244—44no—
50C2×D244—42no—
51C2×D244—42no—
52M16×C61—41no—
53M16×C61—41no—
54C2×C8×C61—21no—
55C2×C8×C61—21no—
56#11758194—41no—
57#11758194—41no—
58C2×#92418—41no—
59C16×C61—81no—
60#11758185—81no—
61C23×C61—21no—
62C23×C61—41no—
63C23×C61—42no—
64C23×C61C22⋊C811yescharacteristic
65C22×D61—24no—
66C22×D61—42no—
67C22×D61—44no—
68C22×D61—44no—
69C22×D61—42no—
70C22×D61—84no—
71C22×D61—84no—
72C22×D61—88no—
73C22×D61—88no—
74C22×D61—84no—
75C2×C4×C61—21no—
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61—42no—
79C2×C4×C61—41no—
80C2×C4×C61—41no—
81C2×C4×C61—41no—
82C2×C4×C61#9911yescharacteristic
83C2×C4×C61#10011yescharacteristic
84D244—42no—
85D244—42no—
86D244—44no—
87D244—44no—
88D244—42no—
89D244—42no—
90C8×C61—41no—
91C8×C61—41no—
92#92418—81no—
93#7342—611nomaximal, Sylow
94C22×C61—22no—
95C22×C61—21no—
96C22×C61—41no—
97C22×C61—41no—
98C22×C61—42no—
99C22×C61—42no—
100C22×C61—41no—
101C22×C61#32311yescharacteristic
102C2×D61—44no—
103C2×D61—84no—
104C2×D61—84no—
105C2×D61—88no—
106C2×D61—88no—
107C2×D61—84no—
108C4×C61—21no—
109C4×C61—41no—
110C4×C61AΣL(1,𝔽16)11yescharacteristic
111C4×C61AΣL(1,𝔽16)11yescharacteristic
112#1783—611no—
113#1320—611no—
114#1320D6111yescharacteristic, complemented
115C2×C61—21no—
116C2×C61—42no—
117C2×C61—41no—
118C2×C61#131611yescharacteristic
119D61—84no—
120C23×D4—611no—
121C2×#100—1221no—
122C2×#100—1221no—
123#407—611no—
124#407C2×D6111yescharacteristic
125#349—21no—
126#349—1221no—
127C61#734211yescharacteristic, complemented, Sylow
128C25—1221no—
129C22×D4—612no—
130C22×D4—1221no—
131C22×D4—2441no—
132C22×D4—2441no—
133C22×D4—2442no—
134C22×D4—2442no—
135C23×C4C4×D6111yescharacteristic
136C2×M16—1221no—
137C2×M16D24411yescharacteristic
138C2×M16C22⋊D6111yescharacteristic
139M32—41no—
140M32—2441no—
141#100—2442no—
142#100—2442no—
143C22⋊C8—41no—
144C22⋊C8—2441no—
145C24—1222no—
146C24—4884no—
147C24—2442no—
148C24—2444no—
149C24#1175818911yescharacteristic
150C2×D4—1222no—
151C2×D4—1222no—
152C2×D4—2442no—
153C2×D4—2441no—
154C2×D4—2441no—
155C2×D4—2448no—
156C2×D4—2442no—
157C2×D4—2442no—
158C2×D4—2444no—
159C2×D4—2442no—
160C2×D4—2442no—
161C2×D4—2441no—
162C2×D4—2441no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C8×D6111yescharacteristic
168C22×C4#1175819811yescharacteristic
169M16—41no—
170M16—41no—
171M16—2441no—
172M16—2441no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—2441no—
176C16—81no—
177C16—4881no—
178C23—21no—
179C23—41no—
180C23—42no—
181C23—1224no—
182C23—4884no—
183C23—4884no—
184C23—4884no—
185C23—4888no—
186C23—4888no—
187C23—2442no—
188C23—2444no—
189C23—2444no—
190C23—2442no—
191C23#1451968911yescharacteristic
192D4—2442no—
193D4—2442no—
194D4—2444no—
195D4—2444no—
196D4—2442no—
197D4—2442no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4#1451969111yescharacteristic
206C2×C4#1451969311yescharacteristic
207C8—41no—
208C8—41no—
209C8—4881no—
210C22—22no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—4884no—
218C22—4884no—
219C22—4888no—
220C22—4888no—
221C22—4884no—
222C22—2444no—
223C22#1683948511yescharacteristic
224C4—21no—
225C4—41no—
226C4#1683954811yescharacteristic
227C4#1683955011yescharacteristic
228C2—21no—
229C2—42no—
230C2—41no—
231C2—4884no—
232C2#2144374611yescharacteristic
subgroupG/H
centerC2#21443746
commutatorC22×C4×C61C2×C8
Frattini#407C2×D61
FittingC61×#1320C2
socleC2×C61#1316
prime radicalC23×D244C4
cosocleC61×#407C22
hypercenter#1320D61
hypocenterC61#7342
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #21443755C2semidirect
3. C61×#1320C2semidirect
114. #1320D61semidirect
127. C61#7342semidirect

Maximal subgroups

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

subgroupindex
#214457392
#214437552
C61×#13202
#734261