Finite Group Databasebeta

← #29593404

Subgroups — #29593404

Order 15616.

406 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 59 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#21457618C211yesmaximal, characteristic, complemented
2#21447892C212yesmaximal, complemented
3#21443856C211yesmaximal, characteristic, complemented
4#21443793C212yesmaximal, complemented
5C61×#1712C211yesmaximal, characteristic, complemented
6#16840563—22no—
7#16840198—22no—
8#16840029—22no—
9#16839929—22no—
10#16839880C2212yes—
11#16839583C2212yescomplemented
12#16839575—22no—
13C61×#493—22no—
14C61×#493C2211yescharacteristic
15C61×#493C2211yescharacteristic, complemented
16C82×C61C2211yescharacteristic, complemented
17#14519822—22no—
18#14519803—24no—
19#14519801—44no—
20#14519799D412yes—
21#14519796—42no—
22D8×D61—42no—
23#14519783—22no—
24#14519721—42no—
25#14519718—42no—
26#14519717—22no—
27#14519713D412yes—
28#14519705—22no—
29#14519694—22no—
30C2×D8×C61—22no—
31C2×D8×C61—22no—
32C2×D8×C61—44no—
33#127×C61—21no—
34#127×C61D412yes—
35C4×C8×C61—21no—
36C4×C8×C61C2311yescharacteristic
37C2×D4×C61—22no—
38C2×D4×C61—42no—
39C2×D4×C61—42no—
40C2×D4×C61—41no—
41C2×C22⋊D61—82no—
42#11758224—44no—
43D4×D61—44no—
44#11758222—22no—
45D8×C61—42no—
46D8×C61—42no—
47D8×C61—42no—
48D8×C61—42no—
49D8×C61—88no—
50C2×C8×C61—22no—
51C2×C8×C61C2×D412yes—
52C42×C61C2×D411yescharacteristic
53#11758200—44no—
54#11758199—44no—
55#11758194—22no—
56Dic244D812yescomplemented
57D488D812yescomplemented
58#11758190—22no—
59#11758189—42no—
60C8×D61—42no—
61#11758185—42no—
62C23×C61—42no—
63C23×C61—81no—
64C22×D61—82no—
65D4×C61—44no—
66D4×C61—84no—
67D4×C61—84no—
68D4×C61—82no—
69C2×C4×C61—21no—
70C2×C4×C61#12011yescharacteristic
71C22⋊D61—42no—
72C22⋊D61—84no—
73C2×Dic61—82no—
74D244—22no—
75C4×D61—42no—
76Dic122—22no—
77C8×C61—24no—
78C8×C61C2×D812yes—
79C8×C61AGL(1,ℤ/8)12yes—
80#92418—42no—
81#12310—611nomaximal, Sylow
82C22×C61—82no—
83C22×C61—162no—
84C22×C61—162no—
85C22×C61—161no—
86C22×C61#45311yescharacteristic
87C2×D61—42no—
88C2×D61—322no—
89C4×C61—22no—
90C4×C61#44712yes—
91Dic61—42no—
92#3292—611no—
93#2213—612no—
94#1712D6111yescharacteristic, complemented
95#1357—611no—
96#1335—612no—
97C2×C61—21no—
98C2×C61—162no—
99C2×C61—321no—
100C2×C61#161911yescharacteristic
101D61—82no—
102#576—1222no—
103#509—1222no—
104#493—22no—
105#493C2×D6111yescharacteristic
106#493C2×D6111yescharacteristic, complemented
107#472—1222no—
108#453—1222no—
109#443—612no—
110#364—612no—
111#361—1222no—
112C82C2×D6111yescharacteristic, complemented
113C61#1231011yescharacteristic, complemented, Sylow
114H(5,𝔽2)—1222no—
115AGL(1,ℤ/8)—1224no—
116AGL(1,ℤ/8)—2444no—
117#135—612no—
118#135—2442no—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—44no—
122C2×D8—2442no—
123#131—1222no—
124#127—21no—
125#127C22⋊D6112yes—
126SD32—2442no—
127D16—2442no—
128M32—1222no—
129#108—612no—
130#104—1222no—
131#100—1222no—
132C4×C8—21no—
133C4×C8C22×D6111yescharacteristic
134#41—1222no—
135#41—2444no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—41no—
140C2×D4—4882no—
141C2×D4—2444no—
142Q16—612no—
143SD16—1222no—
144SD16—2444no—
145D8—42no—
146D8—42no—
147D8—42no—
148D8—42no—
149D8—88no—
150D8—612no—
151D8—2444no—
152M16—1222no—
153M16—2442no—
154C2×C8—22no—
155C2×C8—2442no—
156C2×C8D4×D6112yes—
157C42C2×C22⋊D6111yescharacteristic
158C16—2442no—
159C23—42no—
160C23—81no—
161C23—4882no—
162Q8—1222no—
163D4—44no—
164D4—84no—
165D4—84no—
166D4—82no—
167D4—1222no—
168D4—4884no—
169D4—2442no—
170C2×C4—21no—
171C2×C4—4882no—
172C2×C4—2442no—
173C2×C4#1451974511yescharacteristic
174C8—24no—
175C8—2442no—
176C8D8×D6112yes—
177C8#1451980112yes—
178C22—82no—
179C22—162no—
180C22—162no—
181C22—161no—
182C22—19522no—
183C22—2442no—
184C22#1683992911yescharacteristic
185C4—22no—
186C4—2442no—
187C4#1683989412yes—
188C2—21no—
189C2—162no—
190C2—321no—
191C2—4882no—
192C2#2144487111yescharacteristic
subgroupG/H
centerC2#21444871
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1712C2
socleC2×C61#1619
hypercenter#1712D61
hypocenterC61#12310
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1712

Complements

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

NH ≅ G/Ntype
1. #21457618C2semidirect
2. #21447892C2semidirect
3. #21443856C2semidirect
4. #21443793C2semidirect
5. C61×#1712C2semidirect
11. #16839583C22semidirect
15. C61×#493C22semidirect
16. C82×C61C22semidirect
56. Dic244D8semidirect
57. D488D8semidirect
94. #1712D61semidirect
106. #493C2×D61semidirect
112. C82C2×D61semidirect
113. C61#12310semidirect

Maximal subgroups

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

subgroupindex
#214576182
#214478922
#214438562
#214437932
C61×#17122
#1231061