Finite Group Databasebeta

← #29600710

Subgroups — #29600710

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#21458956C211yesmaximal, characteristic
2#21448137C211yesmaximal, characteristic, complemented
3#21447941C211yesmaximal, characteristic, complemented
4#21444009C211yesmaximal, characteristic, complemented
5#21444004C211yesmaximal, characteristic, complemented
6#21443960C211yesmaximal, characteristic, complemented
7C2×C61×SD64C211yesmaximal, characteristic, complemented
8#16840563—21no—
9#16840198—21no—
10#16840196—22no—
11#16840194C2211yescharacteristic
12#16840191—21no—
13D16×D61—21no—
14#16840178C2211yescharacteristic
15#16840034—21no—
16#16839602—21no—
17#16839599—21no—
18#16839596C2211yescharacteristic, complemented
19#16839592C2211yescharacteristic, complemented
20#16839578—21no—
21C2×Q32×C61C2211yescharacteristic, complemented
22C2×D16×C61C2211yescharacteristic
23C61×SD64—22no—
24C2×C32×C61C2211yescharacteristic, complemented
25#14519822—41no—
26#14519808—41no—
27#14519803—41no—
28#14519801—42no—
29#14519799—21no—
30#14519796—22no—
31D8×D61—22no—
32#14519783D411yescharacteristic
33Dic488D411yescharacteristic
34#14519722—21no—
35#14519721—22no—
36D976D411yescharacteristic
37#14519718—22no—
38#14519717D411yescharacteristic
39#14519716—21no—
40C16×D61—21no—
41#14519713—21no—
42#14519696—41no—
43#14519683—21no—
44C2×Q16×C61—21no—
45C2×D8×C61—21no—
46Q32×C61—21no—
47Q32×C61—21no—
48D16×C61—21no—
49D16×C61D412yes—
50C2×C16×C61C2311yescharacteristic
51C32×C61—21no—
52C2×Q8×C61—41no—
53C2×D4×C61—41no—
54C2×C22⋊D61—82no—
55#11758224—42no—
56#11758224—81no—
57D4×D61—42no—
58D4×D61—81no—
59#11758222—41no—
60#11758222D811yescharacteristic
61Q16×C61—41no—
62Q16×C61—41no—
63D8×C61—22no—
64D8×C61—41no—
65C2×C8×C61C2×D411yescharacteristic
66#11758202—81no—
67#11758201—81no—
68#11758200—42no—
69#11758199—42no—
70#11758194—41no—
71#11758194D811yescharacteristic
72Dic244—21no—
73D488—21no—
74#11758190—41no—
75#11758189—21no—
76C8×D61—21no—
77C16×C61C2×D411yescharacteristic
78C16×C61C2×D411yescharacteristic
79#11758185—21no—
80C23×C61—81no—
81C22×D61—82no—
82Q8×C61—81no—
83Q8×C61—81no—
84D4×C61—42no—
85D4×C61—81no—
86C2×C4×C61—81no—
87C2×C4×C61C2×D811yescharacteristic
88C22⋊D61—21no—
89C22⋊D61—82no—
90C22⋊D61—82no—
91C22⋊D61—81no—
92C2×Dic61—82no—
93D244—41no—
94D244D1611yescharacteristic
95C4×D61—21no—
96C4×D61—81no—
97Dic122—41no—
98Dic122D1611yescharacteristic
99C8×C61#12011yescharacteristic
100C8×C61AGL(1,ℤ/8)11yescharacteristic
101#92418—21no—
102#92418—81no—
103#13671—611nomaximal, Sylow
104C22×C61—82no—
105C22×C61—321no—
106C22×C61#50911yescharacteristic
107C2×D61—21no—
108C2×D61—81no—
109C2×D61—322no—
110C4×C61—161no—
111C4×C61#44711yescharacteristic
112C4×C61C2×D1611yescharacteristic
113Dic61—21no—
114Dic61—81no—
115#3415—611no—
116#2264—611no—
117C2×SD64D6111yescharacteristic, complemented
118#2222—611no—
119#1419—611no—
120#1417—611no—
121#1401—611no—
122C2×C61—21no—
123C2×C61—162no—
124C2×C61#218411yescharacteristic
125D61—41no—
126D61—161no—
127#576—1221no—
128#509—1221no—
129#509—1222no—
130#508—611no—
131#508—1221no—
132C2×Q32C2×D6111yescharacteristic, complemented
133C2×D16—1221no—
134C2×D16C2×D6111yescharacteristic
135#504—611no—
136#473—1221no—
137Q64—1221no—
138SD64—22no—
139SD64—1221no—
140M64—611no—
141C2×C32C2×D6111yescharacteristic, complemented
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—1222no—
150#135—1221no—
151C2×Q16—21no—
152C2×D8—21no—
153C2×D8—1222no—
154#131—611no—
155Q32—21no—
156Q32—21no—
157Q32—611no—
158SD32—1222no—
159SD32—1221no—
160D16—21no—
161D16—611no—
162D16—1222no—
163D16C22⋊D6112yes—
164M32—611no—
165M32—1221no—
166C2×C16—1221no—
167C2×C16C22×D6111yescharacteristic
168#108—1221no—
169#101—2441no—
170C32—21no—
171C32—1221no—
172#41—611no—
173#41—4881no—
174#41—2442no—
175#41—2441no—
176C2×Q8—41no—
177C2×D4—41no—
178C2×D4—4881no—
179C2×D4—4882no—
180C2×D4—2442no—
181Q16—41no—
182Q16—41no—
183Q16—1221no—
184Q16—4881no—
185SD16—4881no—
186SD16—2442no—
187SD16—2441no—
188D8—22no—
189D8—41no—
190D8—1221no—
191D8—2442no—
192M16—611no—
193M16—1221no—
194M16—2441no—
195C2×C8—1221no—
196C2×C8D4×D6111yescharacteristic
197C16—1221no—
198C16D4×D6111yescharacteristic
199C16C2×C22⋊D6111yescharacteristic
200C23—81no—
201C23—4882no—
202Q8—81no—
203Q8—81no—
204Q8—611no—
205Q8—2441no—
206D4—42no—
207D4—81no—
208D4—611no—
209D4—1221no—
210D4—4882no—
211D4—4882no—
212D4—4881no—
213D4—2441no—
214C2×C4—81no—
215C2×C4—1221no—
216C2×C4—4882no—
217C2×C4—4881no—
218C2×C4D8×D6111yescharacteristic
219C8—1221no—
220C8—4881no—
221C8#1451974511yescharacteristic
222C8#1451980111yescharacteristic
223C22—82no—
224C22—321no—
225C22—1221no—
226C22—19522no—
227C22—4881no—
228C22#1684019611yescharacteristic
229C4—161no—
230C4—1221no—
231C4—4881no—
232C4#1683989411yescharacteristic
233C4D16×D6111yescharacteristic
234C2—21no—
235C2—162no—
236C2—9761no—
237C2—2441no—
238C2#2144772911yescharacteristic
subgroupG/H
centerC2#21447729
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC2×C61×SD64C2
socleC2×C61#2184
prime radical#21458956C2
hypercenterC2×SD64D61
hypocenterC61#13671
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
2. #21448137C2semidirect
3. #21447941C2semidirect
4. #21444009C2semidirect
5. #21444004C2semidirect
6. #21443960C2semidirect
7. C2×C61×SD64C2semidirect
18. #16839596C22semidirect
19. #16839592C22semidirect
21. C2×Q32×C61C22semidirect
24. C2×C32×C61C22semidirect
117. C2×SD64D61semidirect
132. C2×Q32C2×D61semidirect
141. C2×C32C2×D61semidirect
144. C61#13671semidirect

Maximal subgroups

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

subgroupindex
#214589562
#214481372
#214479412
#214440092
#214440042
#214439602
C2×C61×SD642
#1367161