Finite Group Databasebeta

← #29590865

Subgroups — #29590865

Order 15616.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#21447631C211yesmaximal, characteristic
2#21445968C211yesmaximal, characteristic
3#21443813C212yesmaximal
4#21443807C212yesmaximal
5C61×#3185C211yesmaximal, characteristic
6#16839583—21no—
7#16839544C2212yes—
8#16839540—21no—
9#16839533—21no—
10#16839522—21no—
11#16839520C2211yescharacteristic
12#16839519C2211yescharacteristic
13C61×#533—21no—
14C61×#498C411yescharacteristic, complemented
15C61×#494C411yescharacteristic, complemented
16C61×#487—21no—
17C61×#470—21no—
18C61×#470C2212yes—
19C61×#441—21no—
20C61×#441C412yescomplemented
21C4×M16×C61C2211yescharacteristic
22C2×#11758194—21no—
23#14519736—21no—
24#14519717—22no—
25C2×Q16×C61—21no—
26C2×Q16×C61C2×C412yes—
27C2×M16×C61D411yescharacteristic
28C2×M16×C61C2311yescharacteristic
29C4⋊Q8×C61—21no—
30C4⋊Q8×C61—21no—
31#125×C61—41no—
32C22⋊Q8×C61—22no—
33C22⋊Q8×C61—41no—
34C4×Q8×C61—22no—
35C4×Q8×C61—41no—
36#117×C61—41no—
37C2×C42×C61D411yescharacteristic
38#106×C61—21no—
39#106×C61C2×C412yes—
40#103×C61—22no—
41#103×C61—22no—
42#97×C61C2×C411yescharacteristic
43#97×C61D411yescharacteristic
44C4×C8×C61C2×C411yescharacteristic
45C4×C8×C61D411yescharacteristic
46C2×Q8×C61—22no—
47C2×Q8×C61—41no—
48C22×C4×C61—21no—
49C22×C4×C61C2×D411yescharacteristic
50Q16×C61—42no—
51Q16×C61—42no—
52M16×C61—41no—
53M16×C61—41no—
54C2×C8×C61C22⋊C411yescharacteristic
55C2×C8×C61C22⋊C411yescharacteristic
56C2×C8×C61C22×C411yescharacteristic
57C2×C8×C61C2×D411yescharacteristic
58C4⋊C4×C61—22no—
59C4⋊C4×C61—41no—
60C4⋊C4×C61—41no—
61C4⋊C4×C61—42no—
62C4⋊C4×C61—41no—
63C4⋊C4×C61—42no—
64C22⋊C4×C61—42no—
65C42×C61—21no—
66C42×C61—42no—
67C42×C61C22⋊C411yescharacteristic
68C42×C61C22⋊C411yescharacteristic
69C22⋊Dic61—41no—
70C4⋊Dic61—41no—
71C4×Dic61—41no—
72#11758194—41no—
73#11758194—41no—
74C2×#92418—41no—
75#11758185—42no—
76C23×C61AGL(1,ℤ/8)11yescharacteristic
77Q8×C61—42no—
78Q8×C61—42no—
79Q8×C61—82no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61C2×C22⋊C411yescharacteristic
88C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
89C2×Dic61—81no—
90C8×C61—21no—
91C8×C61—21no—
92C8×C61—21no—
93C8×C61—21no—
94#92418—81no—
95#11838—611nomaximal, Sylow
96C22×C61—21no—
97C22×C61—41no—
98C22×C61#41811yescharacteristic
99C4×C61—22no—
100C4×C61—21no—
101C4×C61—41no—
102C4×C61—82no—
103C4×C61—82no—
104C4×C61#41011yescharacteristic
105C4×C61#42111yescharacteristic
106Dic61—161no—
107#3185D6111yescharacteristic
108#2165—611no—
109#1841—611no—
110#1342—612no—
111#1340—612no—
112C2×C61—21no—
113C2×C61—41no—
114C2×C61#151411yescharacteristic
115#533—21no—
116#498Dic6111yescharacteristic, complemented
117#494Dic6111yescharacteristic, complemented
118#487—21no—
119#470—21no—
120#470C2×D6112yes—
121#441—21no—
122#441Dic6112yescomplemented
123C4×M16C2×D6111yescharacteristic
124#364—1221no—
125#349—612no—
126#347—1221no—
127#344—1221no—
128#339—1221no—
129#338—611no—
130H(3,ℤ/4)—611no—
131C61#1183811yescharacteristic, complemented, Sylow
132C2×Q16—21no—
133C2×Q16C2×Dic6112yes—
134C2×M16—1221no—
135C2×M16C22⋊D6111yescharacteristic
136C2×M16C22×D6111yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—41no—
145#117—1221no—
146C2×C42C22⋊D6111yescharacteristic
147M32—1222no—
148#106—21no—
149#106C2×Dic6112yes—
150#103—22no—
151#103—22no—
152#97C2×Dic6111yescharacteristic
153#97C22⋊D6111yescharacteristic
154C4×C8C2×Dic6111yescharacteristic
155C4×C8C22⋊D6111yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C22×C4—21no—
159C22×C4C2×C22⋊D6111yescharacteristic
160Q16—42no—
161Q16—42no—
162M16—41no—
163M16—41no—
164M16—2441no—
165M16—2441no—
166C2×C8—2441no—
167C2×C8C22⋊Dic6111yescharacteristic
168C2×C8C22⋊Dic6111yescharacteristic
169C2×C8C22×Dic6111yescharacteristic
170C2×C8C2×C22⋊D6111yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—2441no—
178C22⋊C4—42no—
179C22⋊C4—2441no—
180C42—21no—
181C42—42no—
182C42—2441no—
183C42C22⋊Dic6111yescharacteristic
184C42C22⋊Dic6111yescharacteristic
185C16—2442no—
186C23#1451980311yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—4881no—
198C2×C4C2×C22⋊Dic6111yescharacteristic
199C2×C4#1451980311yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—4881no—
205C22—21no—
206C22—41no—
207C22#1683976711yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—9761no—
214C4#1683973611yescharacteristic
215C4#1683977911yescharacteristic
216C2—21no—
217C2—41no—
218C2#2144436611yescharacteristic
subgroupG/H
centerC2#21444366
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#3185C2
socleC2×C61#1514
prime radicalC23×C61AGL(1,ℤ/8)
cosocleC2×M16×C61C23
hypercenter#3185D61
hypocenterC61#11838
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
14. C61×#498C4semidirect
15. C61×#494C4semidirect
20. C61×#441C4semidirect
116. #498Dic61semidirect
117. #494Dic61semidirect
122. #441Dic61semidirect
131. C61#11838semidirect

Maximal subgroups

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

subgroupindex
#214476312
#214459682
#214438132
#214438072
C61×#31852
#1183861