Finite Group Databasebeta

← #29590554

Subgroups — #29590554

Order 15616.

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

normal subgroups only

This group has 316 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#21456469C211yesmaximal, characteristic
2#21447635C211yesmaximal, characteristic, complemented
3#21443809C212yesmaximal, complemented
4#21443803C212yesmaximal, complemented
5C61×#1747C211yesmaximal, characteristic, complemented
6#16840367—21no—
7#16840164C411yescharacteristic, complemented
8#16840139C411yescharacteristic, complemented
9#16840127—21no—
10#16840015C2212yes—
11#16839998—21no—
12#16839875C412yescomplemented
13C4×D488—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×D488—21no—
23C2×D488C2×C412yes—
24#14519774—21no—
25#14519769—21no—
26#14519762—42no—
27#14519751—22no—
28#14519750—42no—
29C4×D244—22no—
30#14519717—22no—
31#14519711—21no—
32#14519709C2×C412yes—
33#14519699—22no—
34#14519699—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×C22⋊D61—42no—
48C2×D244—22no—
49C2×D244—42no—
50C2×C4×D61—42no—
51M16×C61—41no—
52M16×C61—41no—
53M16×C61—41no—
54C2×C8×C61—21no—
55C2×C8×C61—21no—
56C2×C8×C61—22no—
57C2×C8×C61C22⋊C411yescharacteristic
58C2×C8×C61C22⋊C411yescharacteristic
59C2×C8×C61C22×C411yescharacteristic
60C2×C8×C61C2×D411yescharacteristic
61C4⋊C4×C61—21no—
62C4⋊C4×C61—41no—
63C22⋊C4×C61—21no—
64C22⋊C4×C61—41no—
65C42×C61—22no—
66C42×C61C22⋊C411yescharacteristic
67C42×C61C22⋊C411yescharacteristic
68#11758198—42no—
69#11758198—42no—
70C4⋊Dic61—22no—
71#11758196—42no—
72D488—42no—
73D488—41no—
74D488—41no—
75#11758185—42no—
76C23×C61AGL(1,ℤ/8)11yescharacteristic
77C22×D61—42no—
78C2×C4×C61—21no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—42no—
83C2×C4×C61C2×C22⋊C411yescharacteristic
84C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
85C22⋊D61—162no—
86C2×Dic61—42no—
87D244—42no—
88D244—42no—
89D244—162no—
90C4×D61—162no—
91C8×C61—21no—
92C8×C61—21no—
93C8×C61—21no—
94C8×C61—42no—
95C8×C61#10411yescharacteristic
96C8×C61#10411yescharacteristic
97#11770—611nomaximal, Sylow
98C22×C61—21no—
99C22×C61—41no—
100C22×C61#41811yescharacteristic
101C2×D61—82no—
102C2×D61—82no—
103C2×D61—82no—
104C4×C61—21no—
105C4×C61—41no—
106C4×C61—41no—
107C4×C61—82no—
108C4×C61#41011yescharacteristic
109C4×C61C2×#10411yescharacteristic
110Dic61—82no—
111#3176—611no—
112#2166—611no—
113#1747D6111yescharacteristic, complemented
114#1341—612no—
115#1339—612no—
116C2×C61—21no—
117C2×C61—41no—
118C2×C61#150511yescharacteristic
119D61—162no—
120#540—1221no—
121#499—611no—
122#493—611no—
123#490—1221no—
124#469—612no—
125#466—1221no—
126#442—612no—
127C4×D8—1221no—
128#405—21no—
129#405C2×D6111yescharacteristic
130#364—1221no—
131#349—612no—
132#347—1221no—
133#338C2×D6111yescharacteristic
134H(3,ℤ/4)—21no—
135H(3,ℤ/4)C2×D6111yescharacteristic
136C61#1177011yescharacteristic, complemented, Sylow
137C2×D8—612no—
138C2×D8—1221no—
139C2×M16—21no—
140C2×M16C22×D6111yescharacteristic
141C22×C8C22⋊D6111yescharacteristic
142#127—1221no—
143#125—1221no—
144#123—2442no—
145#121—1222no—
146#121—2442no—
147C4×D4—1222no—
148#117—21no—
149#117C22⋊D6111yescharacteristic
150M32—1222no—
151#107—1221no—
152#106—612no—
153#102—1222no—
154#102—1222no—
155#97—22no—
156#97C4×D6111yescharacteristic
157#97D24411yescharacteristic
158C4×C8—22no—
159C4×C8C4×D6111yescharacteristic
160C4×C8D24411yescharacteristic
161C2×D4—1222no—
162C2×D4—2442no—
163C2×D4—2442no—
164C22×C4—2442no—
165C22×C4C2×C22⋊D6111yescharacteristic
166D8—2442no—
167D8—2441no—
168D8—2441no—
169M16—41no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8#1175819811yescharacteristic
176C2×C8#1175819811yescharacteristic
177C2×C8C2×C4×D6111yescharacteristic
178C2×C8C2×D24411yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—41no—
181C4⋊C4—1222no—
182C4⋊C4—2442no—
183C22⋊C4—21no—
184C22⋊C4—41no—
185C22⋊C4—2442no—
186C22⋊C4—2442no—
187C42—22no—
188C42#1175819811yescharacteristic
189C42#1175819811yescharacteristic
190C16—2442no—
191C23—2442no—
192C23#1451980211yescharacteristic
193D4—9762no—
194D4—9762no—
195D4—2442no—
196D4—2442no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—9762no—
203C2×C4—2442no—
204C2×C4C2×#1175819811yescharacteristic
205C2×C4#1451980211yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—42no—
210C8#1451970511yescharacteristic
211C8#1451970511yescharacteristic
212C22—21no—
213C22—41no—
214C22—4882no—
215C22—4882no—
216C22—4882no—
217C22#1683976611yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—4882no—
223C4#1683973511yescharacteristic
224C4C2×#1451970511yescharacteristic
225C2—21no—
226C2—41no—
227C2—9762no—
228C2#2144432611yescharacteristic
subgroupG/H
centerC2#21444326
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1747C2
socleC2×C61#1505
prime radical#21456469C2
cosocleC2×M16×C61C23
hypercenter#1747D61
hypocenterC61#11770
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
2. #21447635C2semidirect
3. #21443809C2semidirect
4. #21443803C2semidirect
5. C61×#1747C2semidirect
7. #16840164C4semidirect
8. #16840139C4semidirect
12. #16839875C4semidirect
113. #1747D61semidirect
136. C61#11770semidirect

Maximal subgroups

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

subgroupindex
#214564692
#214476352
#214438092
#214438032
C61×#17472
#1177061