Finite Group Databasebeta

← #29590559

Subgroups — #29590559

Order 15616.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#21456478C211yesmaximal, characteristic
2#21447635C211yesmaximal, characteristic
3#21443812C212yesmaximal
4#21443806C212yesmaximal
5C61×#1747C211yesmaximal, characteristic
6#16840374—21no—
7#16840164C411yescharacteristic, complemented
8#16840142C411yescharacteristic, complemented
9#16840131—21no—
10#16840020C2212yes—
11#16840003—21no—
12#16839870C412yescomplemented
13C4×Dic244—21no—
14#16839583—21no—
15#16839544C2212yes—
16#16839540—21no—
17C61×#405—21no—
18C61×#405C2211yescharacteristic
19C61×#338C2211yescharacteristic
20C61×H(3,ℤ/4)—21no—
21C61×H(3,ℤ/4)C2211yescharacteristic
22C2×Dic244—21no—
23C2×Dic244C2×C412yes—
24#14519777—21no—
25#14519770—42no—
26#14519769—21no—
27#14519767—42no—
28#14519756—22no—
29C4×Dic122—22no—
30#14519717—22no—
31#14519711—21no—
32#14519709C2×C412yes—
33#14519702—22no—
34#14519702—22no—
35C2×M16×C61—21no—
36C2×M16×C61C2311yescharacteristic
37C22×C8×C61D411yescharacteristic
38#117×C61—21no—
39#117×C61D411yescharacteristic
40#97×C61—22no—
41#97×C61C2×C411yescharacteristic
42#97×C61D411yescharacteristic
43C4×C8×C61—22no—
44C4×C8×C61C2×C411yescharacteristic
45C4×C8×C61D411yescharacteristic
46C22×C4×C61C2×D411yescharacteristic
47C2×Dic122—22no—
48M16×C61—41no—
49M16×C61—41no—
50M16×C61—41no—
51C2×C8×C61—21no—
52C2×C8×C61—21no—
53C2×C8×C61—22no—
54C2×C8×C61C22⋊C411yescharacteristic
55C2×C8×C61C22⋊C411yescharacteristic
56C2×C8×C61C22×C411yescharacteristic
57C2×C8×C61C2×D411yescharacteristic
58C4⋊C4×C61—21no—
59C4⋊C4×C61—41no—
60C22⋊C4×C61—21no—
61C22⋊C4×C61—41no—
62C42×C61—22no—
63C42×C61C22⋊C411yescharacteristic
64C42×C61C22⋊C411yescharacteristic
65C22⋊Dic61—42no—
66C4⋊Dic61—22no—
67C4⋊Dic61—42no—
68#11758196—42no—
69#11758196—42no—
70#11758196—42no—
71C4×Dic61—42no—
72Dic244—42no—
73Dic244—41no—
74Dic244—41no—
75#11758185—42no—
76C23×C61AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—42no—
82C2×C4×C61C2×C22⋊C411yescharacteristic
83C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
84C2×Dic61—42no—
85C2×Dic61—42no—
86Dic122—42no—
87Dic122—42no—
88C8×C61—21no—
89C8×C61—21no—
90C8×C61—21no—
91C8×C61—42no—
92C8×C61#10411yescharacteristic
93C8×C61#10411yescharacteristic
94#11771—611nomaximal, Sylow
95C22×C61—21no—
96C22×C61—41no—
97C22×C61#41811yescharacteristic
98C4×C61—21no—
99C4×C61—41no—
100C4×C61—41no—
101C4×C61—82no—
102C4×C61#41011yescharacteristic
103C4×C61C2×#10411yescharacteristic
104Dic61—82no—
105Dic61—82no—
106#3177—611no—
107#2166—611no—
108#1747D6111yescharacteristic
109#1342—612no—
110#1340—612no—
111C2×C61—21no—
112C2×C61—41no—
113C2×C61#150511yescharacteristic
114#541—1221no—
115#499—611no—
116#494—611no—
117#491—1221no—
118#470—612no—
119#467—1221no—
120#441—612no—
121C4×Q16—1221no—
122#405—21no—
123#405C2×D6111yescharacteristic
124#364—1221no—
125#349—612no—
126#347—1221no—
127#338C2×D6111yescharacteristic
128H(3,ℤ/4)—21no—
129H(3,ℤ/4)C2×D6111yescharacteristic
130C61#1177111yescharacteristic, complemented, Sylow
131C2×Q16—612no—
132C2×Q16—1221no—
133C2×M16—21no—
134C2×M16C22×D6111yescharacteristic
135C22×C8C22⋊D6111yescharacteristic
136C4⋊Q8—1221no—
137#126—2442no—
138#125—1221no—
139#125—2442no—
140C22⋊Q8—1222no—
141C4×Q8—1222no—
142#117—21no—
143#117C22⋊D6111yescharacteristic
144M32—1222no—
145#107—1221no—
146#106—612no—
147#103—1222no—
148#103—1222no—
149#97—22no—
150#97C4×D6111yescharacteristic
151#97D24411yescharacteristic
152C4×C8—22no—
153C4×C8C4×D6111yescharacteristic
154C4×C8D24411yescharacteristic
155C2×Q8—1222no—
156C22×C4C2×C22⋊D6111yescharacteristic
157Q16—2442no—
158Q16—2441no—
159Q16—2441no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8#1175819811yescharacteristic
167C2×C8#1175819811yescharacteristic
168C2×C8C2×C4×D6111yescharacteristic
169C2×C8C2×D24411yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—1222no—
173C4⋊C4—2442no—
174C4⋊C4—2442no—
175C4⋊C4—2442no—
176C4⋊C4—2442no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—2442no—
180C42—22no—
181C42—2442no—
182C42#1175819811yescharacteristic
183C42#1175819811yescharacteristic
184C16—2442no—
185C23#1451980211yescharacteristic
186Q8—2442no—
187Q8—2442no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—2442no—
194C2×C4—2442no—
195C2×C4C2×#1175819811yescharacteristic
196C2×C4#1451980211yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8#1451970511yescharacteristic
202C8#1451970511yescharacteristic
203C22—21no—
204C22—41no—
205C22#1683976611yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—4882no—
211C4—4882no—
212C4#1683973511yescharacteristic
213C4C2×#1451970511yescharacteristic
214C2—21no—
215C2—41no—
216C2#2144432611yescharacteristic
subgroupG/H
centerC2#21444326
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1747C2
socleC2×C61#1505
prime radicalC23×C61AGL(1,ℤ/8)
cosocleC2×M16×C61C23
hypercenter#1747D61
hypocenterC61#11771
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
7. #16840164C4semidirect
8. #16840142C4semidirect
12. #16839870C4semidirect
130. C61#11771semidirect

Maximal subgroups

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

subgroupindex
#214564782
#214476352
#214438122
#214438062
C61×#17472
#1177161