Finite Group Databasebeta

← #28502453

Subgroups — #28502453

Order 15104.

346 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426267C212yesmaximal, complemented
2#21423695C211yesmaximal, characteristic, complemented
3#21422137C211yesmaximal, characteristic, complemented
4#21422069C212yesmaximal, complemented
5C59×#3291C211yesmaximal, characteristic
6#16837800C2211yescharacteristic, complemented
7#16837754—21no—
8#16837748—21no—
9#16837226—22no—
10#16837223C2212yescomplemented
11#16837212—22no—
12C8×C59⋊C8C2211yescharacteristic, complemented
13C59×#577—22no—
14C59×#492C2211yescharacteristic
15C59×#471—22no—
16C59×#456—21no—
17C59×#453—21no—
18C59×#443C2212yes—
19#14273727—21no—
20#14273667—22no—
21#14273659—22no—
22#14273650—42no—
23#14273648—42no—
24C4×C59⋊C8—21no—
252−1+4×C59—21no—
26H(5,𝔽2)×C59—21no—
27#137×C59—22no—
28#137×C59—42no—
29AGL(1,ℤ/8)×C59—22no—
30AGL(1,ℤ/8)×C59—42no—
31#135×C59—42no—
32#135×C59D412yes—
33C2×SD16×C59—22no—
34C2×SD16×C59—42no—
35#131×C59—22no—
36C4⋊Q8×C59D411yescharacteristic
37#127×C59D411yescharacteristic
38#108×C59D412yes—
39#104×C59—22no—
40#101×C59—21no—
41#100×C59—21no—
42C4×C8×C59C2311yescharacteristic
43#41×C59—22no—
44#41×C59—42no—
45#41×C59—42no—
46#41×C59—81no—
47C2×Q8×C59—21no—
48C2×Q8×C59—42no—
49C2×D4×C59—21no—
50C2×D4×C59—41no—
51C2×D4×C59—42no—
52C2×D4×C59—81no—
53C2×Dic118—41no—
54Q16×C59—42no—
55Q16×C59SD1612yes—
56SD16×C59—22no—
57SD16×C59—42no—
58SD16×C59—42no—
59SD16×C59—42no—
60SD16×C59—42no—
61D8×C59—42no—
62D8×C59SD1612yes—
63M16×C59—22no—
64M16×C59—42no—
65C2×C8×C59—42no—
66C2×C8×C59C2×D412yes—
67C4⋊C4×C59—41no—
68C42×C59C2×D411yescharacteristic
69C4⋊Dic59—41no—
70C4⋊Dic59—41no—
71C2×C59⋊C8—22no—
72C59⋊C16—42no—
73C23×C59—41no—
74C23×C59—81no—
75Q8×C59—22no—
76Q8×C59—42no—
77Q8×C59—81no—
78D4×C59—22no—
79D4×C59—42no—
80D4×C59—42no—
81D4×C59—82no—
82D4×C59—82no—
83D4×C59—81no—
84C2×C4×C59—21no—
85C2×C4×C59—42no—
86C2×C4×C59—41no—
87C2×C4×C59—81no—
88C2×C4×C59—82no—
89C2×C4×C59#12011yescharacteristic
90C2×Dic59—81no—
91Dic118—82no—
92C8×C59—42no—
93C8×C59C2×SD1612yes—
94C8×C59AGL(1,ℤ/8)12yes—
95C59⋊C8—42no—
96#12365—591nomaximal, Sylow
97C22×C59—42no—
98C22×C59—81no—
99C22×C59—161no—
100C22×C59—321no—
101C22×C59#45311yescharacteristic
102C4×C59—22no—
103C4×C59—42no—
104C4×C59—81no—
105C4×C59#45012yes—
106Dic59—161no—
107#3291D5911yescharacteristic
108#2230—592no—
109#1708—591no—
110#1359—591no—
111#1335—592no—
112C2×C59—21no—
113C2×C59—82no—
114C2×C59—161no—
115C2×C59#162911yescharacteristic
116#577—22no—
117#498—591no—
118#492C2×D5911yescharacteristic
119#487—1181no—
120#486—1181no—
121#471—22no—
122#456—21no—
123#453—21no—
124#443C2×D5912yes—
125#365—1182no—
126#364—592no—
127#360—1182no—
128C82—591no—
129C59#1236511yescharacteristic, complemented, Sylow
1302−1+4—21no—
131H(5,𝔽2)—21no—
132#137—22no—
133#137—42no—
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)—42no—
136#135—42no—
137#135C22⋊D5912yes—
138C2×SD16—22no—
139C2×SD16—42no—
140#131—22no—
141C4⋊Q8—1181no—
142C4⋊Q8C22⋊D5911yescharacteristic
143#127C22⋊D5911yescharacteristic
144M32—1182no—
145#108C22⋊D5912yes—
146#106—1182no—
147#104—22no—
148#103—2362no—
149#102—2362no—
150#101—21no—
151#100—21no—
152C4×C8—1181no—
153C4×C8C22×D5911yescharacteristic
154#41—22no—
155#41—42no—
156#41—42no—
157#41—81no—
158C2×Q8—21no—
159C2×Q8—42no—
160C2×Q8—2361no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—81no—
165Q16—42no—
166Q16#1174567012yes—
167SD16—22no—
168SD16—42no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172D8—42no—
173D8#1174567012yes—
174M16—22no—
175M16—42no—
176C2×C8—42no—
177C2×C8—1182no—
178C2×C8C2×C22⋊D5912yes—
179C4⋊C4—41no—
180C4⋊C4—2361no—
181C4⋊C4—2361no—
182C42C2×C22⋊D5911yescharacteristic
183C16—2362no—
184C23—41no—
185C23—81no—
186Q8—22no—
187Q8—42no—
188Q8—81no—
189Q8—4722no—
190D4—22no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—4721no—
202C2×C4#1427369611yescharacteristic
203C8—42no—
204C8—2362no—
205C8C2×#1174567012yes—
206C8#1427375312yes—
207C22—42no—
208C22—81no—
209C22—161no—
210C22—321no—
211C22#1683757011yescharacteristic
212C4—22no—
213C4—42no—
214C4—81no—
215C4—9441no—
216C4#1683755312yes—
217C2—21no—
218C2—82no—
219C2—161no—
220C2#2142321211yescharacteristic
subgroupG/H
centerC2#21423212
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
Fitting, prime radicalC59×#3291C2
socleC2×C59#1629
hypercenter#3291D59
hypocenterC59#12365
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #21426267C2semidirect
2. #21423695C2semidirect
3. #21422137C2semidirect
4. #21422069C2semidirect
6. #16837800C22semidirect
10. #16837223C22semidirect
12. C8×C59⋊C8C22semidirect
129. C59#12365semidirect

Maximal subgroups

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

subgroupindex
#214262672
#214236952
#214221372
#214220692
C59×#32912
#1236559