Finite Group Databasebeta

← #28502036

Subgroups — #28502036

Order 15104.

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

normal subgroups only

This group has 310 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#21426313C212yesmaximal, complemented
2#21425649C211yesmaximal, characteristic
3#21422084C212yesmaximal, complemented
4#21422034C211yesmaximal, characteristic
5C59×#3202C211yesmaximal, characteristic
6#16837451C2211yescharacteristic
7#16837428—21no—
8#16837419—21no—
9#16837224—22no—
10#16837218—22no—
11#16837184C2212yes—
12#16837159C2211yescharacteristic
13C59×#536—21no—
14C59×#485—22no—
15C59×#470C2212yes—
16C59×#467—21no—
17C59×#451—22no—
18C59×#408C2211yescharacteristic
19C2×#11745664—21no—
20#14273686—21no—
21#14273667—22no—
22#14273663—22no—
23Dic59⋊C8—42no—
24#14273654—42no—
25C2×#41×C59D411yescharacteristic
26C22×Q8×C59—21no—
27C2×Q16×C59—21no—
28C2×Q16×C59D412yes—
29C2×M16×C59C2311yescharacteristic
30C22×C8×C59D411yescharacteristic
31C4⋊Q8×C59—41no—
32#124×C59—41no—
33#123×C59—41no—
34#123×C59—41no—
35C22⋊Q8×C59—22no—
36C22⋊Q8×C59—22no—
37#117×C59—21no—
38#107×C59—21no—
39#106×C59D412yes—
40#103×C59—22no—
41#103×C59—22no—
42C22⋊C8×C59—22no—
43#41×C59—41no—
44#41×C59—41no—
45C2×Q8×C59—22no—
46C2×Q8×C59—41no—
47C2×Q8×C59—81no—
48C2×Q8×C59—81no—
49C2×Q8×C59SD1611yescharacteristic
50C2×D4×C59—21no—
51C2×D4×C59SD1611yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—41no—
54C22×C4×C59C2×D411yescharacteristic
55Q16×C59—42no—
56Q16×C59—81no—
57M16×C59—41no—
58C2×C8×C59—21no—
59C2×C8×C59—41no—
60C2×C8×C59C2×D412yes—
61C4⋊C4×C59—22no—
62C4⋊C4×C59—42no—
63C4⋊C4×C59—42no—
64C22⋊C4×C59—41no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—41no—
68C22⋊Dic59—41no—
69C4⋊Dic59—41no—
70C4×Dic59—22no—
71#11745664—41no—
72#11745664—41no—
73C2×C59⋊C8—41no—
74C59⋊C16—42no—
75C23×C59—21no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77Q8×C59—41no—
78Q8×C59—42no—
79Q8×C59—42no—
80Q8×C59—82no—
81D4×C59—41no—
82D4×C59—81no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—82no—
91C2×C4×C59#12011yescharacteristic
92C2×C4×C59C2×SD1611yescharacteristic
93C2×Dic59—42no—
94C8×C59—22no—
95C8×C59—41no—
96C59⋊C8—81no—
97#12302—591nomaximal, Sylow
98C22×C59—21no—
99C22×C59—41no—
100C22×C59—42no—
101C22×C59—41no—
102C22×C59#45011yescharacteristic
103C4×C59—21no—
104C4×C59—41no—
105C4×C59—82no—
106C4×C59—82no—
107C4×C59#45311yescharacteristic
108C4×C59#45811yescharacteristic
109Dic59—82no—
110#3202D5911yescharacteristic
111#2239—592no—
112#2098—591no—
113#1340—592no—
114#1321—591no—
115C2×C59—21no—
116C2×C59—41no—
117C2×C59—81no—
118C2×C59#161311yescharacteristic
119#536—21no—
120#485—22no—
121#470C2×D5912yes—
122#467—21no—
123#451—22no—
124#430—591no—
125#424—1181no—
126#421—1181no—
127#408C2×D5911yescharacteristic
128#365—1182no—
129#362—1182no—
130#349—592no—
131H(3,ℤ/4)—591no—
132C59#1230211yescharacteristic, complemented, Sylow
133C2×#41C22⋊D5911yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16C22⋊D5912yes—
137C2×M16—1181no—
138C2×M16C22×D5911yescharacteristic
139C22×C8C22⋊D5911yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—1181no—
148M32—1182no—
149#108—1182no—
150#107—21no—
151#106C22⋊D5912yes—
152C4⋊C8—2362no—
153#104—2362no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#1174567011yescharacteristic
164C2×D4—21no—
165C2×D4#1174567011yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D5911yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—2361no—
173M16—2361no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—2361no—
177C2×C8C2×C22⋊D5912yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—2361no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—2361no—
186C42—41no—
187C42—1182no—
188C16—2362no—
189C23—21no—
190C23#1427375311yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—2362no—
206C2×C4#1427369611yescharacteristic
207C2×C4C2×#1174567011yescharacteristic
208C8—22no—
209C8—41no—
210C8—4721no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1683755311yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—4722no—
221C4#1683757011yescharacteristic
222C4#1683759211yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#2142311611yescharacteristic
subgroupG/H
centerC2#21423116
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#3202C2
socleC2×C59#1613
prime radicalC2×#41×C59D4
hypercenter#3202D59
hypocenterC59#12302
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #21426313C2semidirect
3. #21422084C2semidirect
132. C59#12302semidirect

Maximal subgroups

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

subgroupindex
#214263132
#214256492
#214220842
#214220342
C59×#32022
#1230259