Finite Group Databasebeta

← #29593357

Subgroups — #29593357

Order 15616.

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

normal subgroups only

This group has 342 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#21456684C211yesmaximal, characteristic
2#21448034C212yesmaximal, complemented
3#21447372C211yesmaximal, characteristic, complemented
4#21443805C212yesmaximal, complemented
5C61×#1321C211yesmaximal, characteristic, complemented
6#16840326—21no—
7#16840087—22no—
8#16840015C2212yes—
9#16839998—21no—
10#16839898—22no—
11#16839811C2211yescharacteristic, complemented
12#16839788—21no—
13#16839779—21no—
14#16839585—22no—
15#16839576—22no—
16#16839519C2211yescharacteristic, complemented
17C61×#408C2211yescharacteristic
18C61×#349C2212yescomplemented
19C22×D244—21no—
20C2×D488—21no—
21C2×D488D412yes—
22C2×#11758194—21no—
23#14519773—41no—
24#14519766—41no—
25#14519751—22no—
26#14519751—22no—
27#14519745—41no—
28#14519745—41no—
29#14519736—21no—
30#14519736—21no—
31#14519713—22no—
32#14519711—21no—
33#14519709D412yes—
34#14519706—42no—
35#14519704—42no—
36#14519699—22no—
37#14519699—22no—
38C2×#41×C61D411yescharacteristic
39C2×M16×C61C2311yescharacteristic
40C22×C8×C61D411yescharacteristic
41M32×C61—22no—
42C22⋊C8×C61—22no—
43C23×D61—41no—
44#41×C61—41no—
45#41×C61—41no—
46C2×Q8×C61SD1611yescharacteristic
47C2×D4×C61—21no—
48C2×D4×C61SD1611yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61C2×D411yescharacteristic
51C2×C22⋊D61—42no—
52C2×C22⋊D61—42no—
53C2×D244—22no—
54C2×D244—41no—
55C2×D244—81no—
56C2×D244—81no—
57M16×C61—41no—
58C2×C8×C61—21no—
59C2×C8×C61—41no—
60C2×C8×C61C2×D412yes—
61C22⋊Dic61—41no—
62C22⋊Dic61—41no—
63#11758198—42no—
64#11758198—42no—
65C4⋊Dic61—22no—
66C4⋊Dic61—41no—
67C4×Dic61—22no—
68C4×Dic61—41no—
69#11758194—41no—
70#11758194—41no—
71C2×#92418—41no—
72D488—42no—
73D488—81no—
74C16×C61—42no—
75C23×C61—21no—
76C23×C61AGL(1,ℤ/8)11yescharacteristic
77C22×D61—42no—
78C22×D61—82no—
79C22×D61—162no—
80Q8×C61—41no—
81D4×C61—41no—
82D4×C61—81no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61#12011yescharacteristic
89C2×C4×C61C2×SD1611yescharacteristic
90C22⋊D61—162no—
91C22⋊D61—162no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94D244—42no—
95D244—42no—
96D244—82no—
97C8×C61—22no—
98C8×C61—41no—
99#92418—81no—
100#12301—611nomaximal, Sylow
101C22×C61—21no—
102C22×C61—41no—
103C22×C61—42no—
104C22×C61—41no—
105C22×C61#45011yescharacteristic
106C2×D61—82no—
107C2×D61—82no—
108C2×D61—82no—
109C2×D61—164no—
110C4×C61—21no—
111C4×C61—41no—
112C4×C61#45311yescharacteristic
113C4×C61#45811yescharacteristic
114Dic61—82no—
115Dic61—82no—
116#3199—611no—
117#2238—612no—
118#2098—611no—
119#1339—612no—
120#1321D6111yescharacteristic, complemented
121C2×C61—21no—
122C2×C61—41no—
123C2×C61—81no—
124C2×C61#161311yescharacteristic
125D61—162no—
126#535—1221no—
127#482—1222no—
128#469—612no—
129#466—1221no—
130#447—1222no—
131#430—611no—
132#424—1221no—
133#421—1221no—
134#408C2×D6111yescharacteristic
135#365—1222no—
136#361—1222no—
137#349C2×D6112yescomplemented
138H(3,ℤ/4)—611no—
139C61#1230111yescharacteristic, complemented, Sylow
140C2×#41D24411yescharacteristic
141C22×D4—1221no—
142C2×D8—612no—
143C2×D8—1221no—
144C2×M16—1221no—
145C2×M16C22×D6111yescharacteristic
146C22×C8D24411yescharacteristic
147#127—2441no—
148#124—2441no—
149#121—1222no—
150#121—1222no—
151#120—2441no—
152#120—2441no—
153#117—1221no—
154#117—1221no—
155M32—22no—
156#108—1222no—
157#107—1221no—
158#106—612no—
159C4⋊C8—2442no—
160#104—2442no—
161#102—1222no—
162#102—1222no—
163C22⋊C8—22no—
164C24—2441no—
165#41—41no—
166#41—41no—
167C2×Q8#1175819011yescharacteristic
168C2×D4—21no—
169C2×D4—1222no—
170C2×D4—4881no—
171C2×D4—4881no—
172C2×D4—2441no—
173C2×D4—2442no—
174C2×D4—2442no—
175C2×D4#1175819011yescharacteristic
176C22×C4—21no—
177C22×C4C2×D24411yescharacteristic
178D8—4881no—
179D8—2442no—
180M16—41no—
181M16—2441no—
182M16—2441no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—2441no—
186C2×C8D4×D6112yes—
187C4⋊C4—1222no—
188C4⋊C4—2441no—
189C22⋊C4—2442no—
190C22⋊C4—2441no—
191C22⋊C4—2441no—
192C22⋊C4—2442no—
193C42—1222no—
194C42—2441no—
195C16—42no—
196C23—21no—
197C23—9762no—
198C23—4882no—
199C23—2442no—
200C23#1451980511yescharacteristic
201Q8—41no—
202D4—41no—
203D4—81no—
204D4—9762no—
205D4—9762no—
206D4—4882no—
207D4—2442no—
208D4—2442no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—2442no—
215C2×C4—2442no—
216C2×C4#1451974711yescharacteristic
217C2×C4C2×#1175819011yescharacteristic
218C8—22no—
219C8—41no—
220C8—4881no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—9764no—
226C22—4882no—
227C22—4882no—
228C22—4882no—
229C22#1683991511yescharacteristic
230C4—21no—
231C4—41no—
232C4—4882no—
233C4—4882no—
234C4#1683993211yescharacteristic
235C4#1683995311yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—9762no—
240C2#2144484111yescharacteristic
subgroupG/H
centerC2#21444841
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#1321C2
socleC2×C61#1613
prime radical#21456684C2
hypercenter#1321D61
hypocenterC61#12301
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #21448034C2semidirect
3. #21447372C2semidirect
4. #21443805C2semidirect
5. C61×#1321C2semidirect
11. #16839811C22semidirect
16. #16839519C22semidirect
18. C61×#349C22semidirect
120. #1321D61semidirect
137. #349C2×D61semidirect
139. C61#12301semidirect

Maximal subgroups

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

subgroupindex
#214566842
#214480342
#214473722
#214438052
C61×#13212
#1230161