Finite Group Databasebeta

← #29600708

Subgroups — #29600708

Order 15616.

284 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 55 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#21458957C211yesmaximal, characteristic
2C2×#16839599C211yesmaximal, characteristic, complemented
3#21447941C211yesmaximal, characteristic, complemented
4#21444010C211yesmaximal, characteristic, complemented
5#21444003C211yesmaximal, characteristic, complemented
6#21443962C211yesmaximal, characteristic, complemented
7C61×#2264C211yesmaximal, characteristic, complemented
8#16840564—21no—
9#16840199—22no—
10#16840197—21no—
11#16840195—21no—
12C2×D976C2211yescharacteristic
13D16×D61—21no—
14#16840178C2211yescharacteristic
15#16840034—21no—
16#16839599—22no—
17C2×#14519683C2211yescharacteristic, complemented
18#16839592C2211yescharacteristic, complemented
19#16839578—21no—
20C61×#508C2211yescharacteristic
21C2×Q32×C61C2211yescharacteristic, complemented
22C61×Q64—21no—
23C61×SD64—21no—
24C61×M64C2211yescharacteristic, complemented
25#14519823—41no—
26#14519808—41no—
27#14519804—42no—
28#14519802—41no—
29#14519800—22no—
30C2×D488—21no—
31D8×D61—22no—
32#14519783D411yescharacteristic
33#14519720—22no—
34D976—21no—
35D976D412yes—
36#14519718—22no—
37#14519717D411yescharacteristic
38#14519716—21no—
39C16×D61—21no—
40#14519713—21no—
41#14519696—41no—
42#14519683—21no—
43#135×C61—21no—
44C2×Q16×C61—21no—
45Q32×C61—21no—
46Q32×C61—21no—
47Q32×C61D411yescharacteristic
48SD32×C61—21no—
49D16×C61D411yescharacteristic
50C2×C16×C61C2311yescharacteristic
51C32×C61—21no—
52#41×C61—41no—
53C2×Q8×C61—41no—
54#11758226—42no—
55D4×D61—42no—
56D4×D61—82no—
57#11758222—81no—
58#11758222D811yescharacteristic
59C2×D244—41no—
60C2×D244—81no—
61Q16×C61—21no—
62Q16×C61—41no—
63Q16×C61—41no—
64SD16×C61—41no—
65D8×C61—21no—
66C2×C8×C61C2×D411yescharacteristic
67#11758202—81no—
68#11758201—42no—
69#11758201—81no—
70#11758199—42no—
71#11758194—41no—
72#11758194D811yescharacteristic
73D488—22no—
74D488—41no—
75#11758189—21no—
76C8×D61—21no—
77C16×C61C2×D411yescharacteristic
78C16×C61C2×D411yescharacteristic
79#11758185—21no—
80C22×D61—82no—
81C22×D61—81no—
82Q8×C61—41no—
83Q8×C61—81no—
84Q8×C61—81no—
85D4×C61—41no—
86D4×C61—81no—
87C2×C4×C61—81no—
88C2×C4×C61—81no—
89C2×C4×C61C2×D811yescharacteristic
90C22⋊D61—21no—
91C22⋊D61—82no—
92D244—42no—
93D244—82no—
94D244—81no—
95D244D1611yescharacteristic
96C4×D61—21no—
97C4×D61—82no—
98Dic122D1611yescharacteristic
99C8×C61#12011yescharacteristic
100C8×C61AGL(1,ℤ/8)11yescharacteristic
101#92418—21no—
102#92418—81no—
103#13671—611nomaximal, Sylow
104C22×C61—81no—
105C22×C61#50911yescharacteristic
106C2×D61—21no—
107C2×D61—82no—
108C2×D61—322no—
109C2×D61—321no—
110C4×C61—81no—
111C4×C61—161no—
112C4×C61#44711yescharacteristic
113C4×C61C2×D1611yescharacteristic
114Dic61—21no—
115#3415—611no—
116#2264D6111yescharacteristic, complemented
117C2×SD64—611no—
118#2222—611no—
119#1419—611no—
120#1417—611no—
121#1401—611no—
122C2×C61—21no—
123C2×C61—161no—
124C2×C61#218411yescharacteristic
125D61—41no—
126D61—162no—
127#576—1221no—
128#509—1221no—
129#509—1222no—
130#508—1221no—
131#508C2×D6111yescharacteristic
132C2×Q32C2×D6111yescharacteristic, complemented
133C2×D16—611no—
134C2×D16—1221no—
135#504—611no—
136#473—1221no—
137Q64—21no—
138SD64—21no—
139SD64—1222no—
140M64C2×D6111yescharacteristic, complemented
141C2×C32—611no—
142#368—611no—
143#362—1221no—
144C61#1367111yescharacteristic, complemented, Sylow
145H(5,𝔽2)—2441no—
146#137—2441no—
147AGL(1,ℤ/8)—2441no—
148AGL(1,ℤ/8)—2442no—
149#135—21no—
150#135—1222no—
151C2×Q16—21no—
152C2×D8—1222no—
153C2×D8—1221no—
154#131—611no—
155Q32—21no—
156Q32—21no—
157Q32C22⋊D6111yescharacteristic
158SD32—21no—
159SD32—1222no—
160D16—612no—
161D16—1222no—
162D16—1221no—
163D16C22⋊D6111yescharacteristic
164M32—611no—
165M32—1221no—
166C2×C16—1221no—
167C2×C16C22×D6111yescharacteristic
168#108—1221no—
169#101—2441no—
170C32—21no—
171C32—1221no—
172#41—41no—
173#41—611no—
174#41—4881no—
175#41—2442no—
176C2×Q8—41no—
177C2×D4—4881no—
178C2×D4—4882no—
179C2×D4—2442no—
180C2×D4—2441no—
181Q16—21no—
182Q16—41no—
183Q16—41no—
184Q16—4881no—
185SD16—41no—
186SD16—4881no—
187SD16—2442no—
188D8—21no—
189D8—1222no—
190D8—2442no—
191D8—2441no—
192M16—611no—
193M16—1221no—
194M16—2441no—
195C2×C8—1221no—
196C2×C8D4×D6111yescharacteristic
197C16—1221no—
198C16D4×D6111yescharacteristic
199C16C2×C22⋊D6111yescharacteristic
200C23—4882no—
201C23—4881no—
202Q8—41no—
203Q8—81no—
204Q8—81no—
205Q8—611no—
206D4—41no—
207D4—81no—
208D4—611no—
209D4—1221no—
210D4—4882no—
211D4—4882no—
212D4—4881no—
213D4—2442no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—1221no—
217C2×C4—4882no—
218C2×C4D8×D6111yescharacteristic
219C8—1221no—
220C8—4881no—
221C8#1451974511yescharacteristic
222C8#1451980111yescharacteristic
223C22—81no—
224C22—1221no—
225C22—19521no—
226C22—19522no—
227C22—4882no—
228C22#1684019611yescharacteristic
229C4—81no—
230C4—161no—
231C4—1221no—
232C4#1683989411yescharacteristic
233C4D16×D6111yescharacteristic
234C2—21no—
235C2—161no—
236C2—9762no—
237C2—2441no—
238C2#2144772911yescharacteristic
subgroupG/H
centerC2#21447729
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC61×#2264C2
socleC2×C61#2184
prime radical#21458957C2
hypercenter#2264D61
hypocenterC61#13671
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
2. C2×#16839599C2semidirect
3. #21447941C2semidirect
4. #21444010C2semidirect
5. #21444003C2semidirect
6. #21443962C2semidirect
7. C61×#2264C2semidirect
17. C2×#14519683C22semidirect
18. #16839592C22semidirect
21. C2×Q32×C61C22semidirect
24. C61×M64C22semidirect
116. #2264D61semidirect
132. C2×Q32C2×D61semidirect
140. M64C2×D61semidirect
144. C61#13671semidirect

Maximal subgroups

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

subgroupindex
#214589572
C2×#168395992
#214479412
#214440102
#214440032
#214439622
C61×#22642
#1367161