Finite Group Databasebeta

← #29600712

Subgroups — #29600712

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
2#21448136C211yesmaximal, characteristic, complemented
3C2×#16839601C211yesmaximal, characteristic, complemented
4#21447945C211yesmaximal, characteristic, complemented
5#21444008C211yesmaximal, characteristic, complemented
6#21443960C211yesmaximal, characteristic, complemented
7C61×#1417C211yesmaximal, characteristic, complemented
8#16840564—21no—
9#16840199—22no—
10#16840197—21no—
11#16840195—21no—
12C2×Dic488C2211yescharacteristic, complemented
13C2×D976C2211yescharacteristic
14D16×D61—21no—
15#16840178C2211yescharacteristic
16#16840038—21no—
17#16839602—21no—
18#16839601—22no—
19#16839600—21no—
20#16839596C2211yescharacteristic, complemented
21#16839577—21no—
22C61×#508C2211yescharacteristic
23C2×C32×C61C2211yescharacteristic, complemented
24C61×#368C2211yescharacteristic, complemented
25#14519823—41no—
26#14519810—41no—
27#14519804—42no—
28#14519802—41no—
29#14519800—22no—
30C2×Dic244—21no—
31C2×D488—21no—
32D8×D61—22no—
33#14519783D411yescharacteristic
34Dic488—21no—
35Dic488—21no—
36#14519720—22no—
37D976—21no—
38D976D412yes—
39#14519718—22no—
40#14519717D411yescharacteristic
41#14519716—21no—
42C16×D61—21no—
43#14519695—41no—
44#14519683—21no—
45#135×C61—21no—
46Q32×C61D411yescharacteristic
47SD32×C61—21no—
48D16×C61D411yescharacteristic
49C2×C16×C61C2311yescharacteristic
50#108×C61—21no—
51C32×C61—21no—
52#41×C61—41no—
53#11758226—42no—
54D4×D61—42no—
55D4×D61—82no—
56#11758222—81no—
57#11758222D811yescharacteristic
58C2×D244—41no—
59C2×D244—81no—
60C2×Dic122—41no—
61Q16×C61—21no—
62SD16×C61—41no—
63D8×C61—21no—
64M16×C61—41no—
65C2×C8×C61C2×D411yescharacteristic
66#11758201—42no—
67#11758199—42no—
68#11758194D811yescharacteristic
69Dic244—41no—
70Dic244—41no—
71Dic244—81no—
72D488—22no—
73D488—41no—
74#11758190—81no—
75#11758189—21no—
76C8×D61—21no—
77C16×C61C2×D411yescharacteristic
78C16×C61C2×D411yescharacteristic
79#11758185—21no—
80C22×D61—82no—
81C22×D61—81no—
82Q8×C61—41no—
83D4×C61—41no—
84D4×C61—81no—
85C2×C4×C61—81no—
86C2×C4×C61C2×D811yescharacteristic
87C22⋊D61—21no—
88C22⋊D61—82no—
89C2×Dic61—81no—
90D244—42no—
91D244—82no—
92D244—81no—
93D244D1611yescharacteristic
94C4×D61—21no—
95C4×D61—82no—
96Dic122—81no—
97Dic122—81no—
98Dic122D1611yescharacteristic
99C8×C61—81no—
100C8×C61#12011yescharacteristic
101C8×C61AGL(1,ℤ/8)11yescharacteristic
102#92418—21no—
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#44711yescharacteristic
112C4×C61C2×D1611yescharacteristic
113Dic61—21no—
114Dic61—161no—
115#3415—611no—
116#2264—611no—
117C2×SD64—611no—
118#2222—611no—
119#1419—611no—
120#1417D6111yescharacteristic, complemented
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×Q32—611no—
133C2×D16—611no—
134C2×D16—1221no—
135#504—611no—
136#473—1221no—
137Q64—1221no—
138SD64—1222no—
139SD64—1221no—
140M64—611no—
141C2×C32C2×D6111yescharacteristic, complemented
142#368C2×D6111yescharacteristic, complemented
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—1221no—
152C2×D8—1222no—
153C2×D8—1221no—
154#131—611no—
155Q32—1221no—
156Q32—1221no—
157Q32D24411yescharacteristic
158SD32—21no—
159SD32—1222no—
160D16—612no—
161D16—1222no—
162D16—1221no—
163D16D24411yescharacteristic
164M32—611no—
165M32—1221no—
166C2×C16—1221no—
167C2×C16C22×D6111yescharacteristic
168#108—21no—
169#101—2441no—
170C32—21no—
171C32—1221no—
172#41—41no—
173#41—611no—
174#41—4881no—
175#41—2442no—
176C2×Q8—2441no—
177C2×D4—4881no—
178C2×D4—4882no—
179C2×D4—2442no—
180C2×D4—2441no—
181Q16—21no—
182Q16—4881no—
183Q16—2441no—
184Q16—2441no—
185SD16—41no—
186SD16—4881no—
187SD16—2442no—
188D8—21no—
189D8—1222no—
190D8—2442no—
191D8—2441no—
192M16—41no—
193M16—611no—
194M16—1221no—
195C2×C8—1221no—
196C2×C8D4×D6111yescharacteristic
197C16—1221no—
198C16C2×D24411yescharacteristic
199C16D4×D6111yescharacteristic
200C23—4882no—
201C23—4881no—
202Q8—41no—
203Q8—611no—
204Q8—4881no—
205Q8—4881no—
206D4—41no—
207D4—81no—
208D4—611no—
209D4—1221no—
210D4—4882no—
211D4—4882no—
212D4—4881no—
213D4—2442no—
214C2×C4—81no—
215C2×C4—1221no—
216C2×C4—4882no—
217C2×C4—4881no—
218C2×C4D8×D6111yescharacteristic
219C8—81no—
220C8—1221no—
221C8#1451974711yescharacteristic
222C8#1451980411yescharacteristic
223C22—81no—
224C22—1221no—
225C22—19521no—
226C22—19522no—
227C22—4882no—
228C22#1684019911yescharacteristic
229C4—81no—
230C4—1221no—
231C4—9761no—
232C4#1683989711yescharacteristic
233C4D16×D6111yescharacteristic
234C2—21no—
235C2—161no—
236C2—9762no—
237C2—2441no—
238C2#2144773211yescharacteristic
subgroupG/H
centerC2#21447732
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC61×#1417C2
socleC2×C61#2184
prime radical#21458957C2
hypercenter#1417D61
hypocenterC61#13671
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1417

Complements

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

NH ≅ G/Ntype
2. #21448136C2semidirect
3. C2×#16839601C2semidirect
4. #21447945C2semidirect
5. #21444008C2semidirect
6. #21443960C2semidirect
7. C61×#1417C2semidirect
12. C2×Dic488C22semidirect
20. #16839596C22semidirect
23. C2×C32×C61C22semidirect
24. C61×#368C22semidirect
120. #1417D61semidirect
141. C2×C32C2×D61semidirect
142. #368C2×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
#214481362
C2×#168396012
#214479452
#214440082
#214439602
C61×#14172
#1367161