Finite Group Databasebeta

← #28501185

Subgroups — #28501185

Order 15104.

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

normal subgroups only

This group has 330 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#21435550C211yesmaximal, characteristic
2#21426030C212yesmaximal
3#21422113C212yesmaximal
4#21422038C211yesmaximal, characteristic
5C59×#2047C211yesmaximal, characteristic
6#16838014—21no—
7C2×#14273673—22no—
8#16837710—22no—
9#16837660—21no—
10#16837643C2212yes—
11#16837613—22no—
12#16837206—22no—
13C22⋊(C59⋊C16)C2212yes—
14C22×Q16×C59C2211yescharacteristic
15C59×#424C2211yescharacteristic
16C59×#419—21no—
17C2×#101×C59—21no—
18C59×#341C2211yescharacteristic
19C2×Dic236—21no—
20C2×Dic236D412yes—
21#14273727—21no—
22#14273720—42no—
23#14273719—21no—
24#14273717—42no—
25#14273706—22no—
26C4×Dic118—22no—
27#14273673—44no—
28C2×C59⋊C16—22no—
29#14273661D412yes—
30#14273659—21no—
31#14273652—22no—
32#14273652—22no—
33C22×Q8×C59—21no—
34C2×Q16×C59—22no—
35C2×Q16×C59—42no—
36C2×M16×C59—21no—
37C2×M16×C59D411yescharacteristic
38C22×C8×C59C2311yescharacteristic
39#117×C59D411yescharacteristic
40C4⋊C8×C59—22no—
41#103×C59—42no—
42#101×C59—42no—
43C2×Q8×C59—22no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C22×C4×C59—41no—
50C22×C4×C59C2×D411yescharacteristic
51C2×Dic118—22no—
52Q16×C59—24no—
53Q16×C59—42no—
54M16×C59—41no—
55M16×C59—41no—
56M16×C59—41no—
57C2×C8×C59—21no—
58C2×C8×C59—41no—
59C2×C8×C59—41no—
60C2×C8×C59C2×D412yes—
61C4⋊C4×C59D812yes—
62C22⋊C4×C59—21no—
63C42×C59—21no—
64C22⋊Dic59—42no—
65C4⋊Dic59—22no—
66C4⋊Dic59—42no—
67Dic59⋊C4—42no—
68Dic59⋊C4—42no—
69Dic59⋊C4—42no—
70C4×Dic59—42no—
71Dic236—42no—
72Dic236—81no—
73C59⋊C16—42no—
74C23×C59AGL(1,ℤ/8)11yescharacteristic
75Q8×C59—42no—
76Q8×C59—42no—
77Q8×C59—81no—
78Q8×C59—81no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—22no—
82C2×C4×C59—42no—
83C2×C4×C59—81no—
84C2×C4×C59—81no—
85C2×C4×C59#12011yescharacteristic
86C2×C4×C59C2×D811yescharacteristic
87C2×Dic59—42no—
88C2×Dic59—42no—
89Dic118—42no—
90Dic118—42no—
91C8×C59—22no—
92C8×C59—41no—
93C8×C59—81no—
94#12168—591nomaximal, Sylow
95C22×C59—21no—
96C22×C59—22no—
97C22×C59#44711yescharacteristic
98C4×C59—21no—
99C4×C59—42no—
100C4×C59—82no—
101C4×C59#45411yescharacteristic
102C4×C59B(4,𝔽2)11yescharacteristic
103Dic59—82no—
104Dic59—82no—
105#3261—591no—
106#2188—592no—
107#2047D5911yescharacteristic
108#1350—592no—
109#1323—591no—
110C2×C59—41no—
111C2×C59#159811yescharacteristic
112C2×C59#219512yes—
113C22×Q16C2×D5911yescharacteristic
114#541—1181no—
115C2×Q32—1182no—
116#479—1182no—
117#470—1181no—
118#467—592no—
119#462—1182no—
120#424C2×D5911yescharacteristic
121#419—21no—
122C2×#101—21no—
123#358—1182no—
124C22⋊C16—592no—
125#341C2×D5911yescharacteristic
126C59#1216811yescharacteristic, complemented, Sylow
127C22×Q8—21no—
128C2×Q16—22no—
129C2×Q16—42no—
130C2×Q16—592no—
131C2×Q16—1181no—
132C2×M16—21no—
133C2×M16C22⋊D5911yescharacteristic
134C22×C8C22×D5911yescharacteristic
135C4⋊Q8—1181no—
136#126—2362no—
137#125—1181no—
138#125—2362no—
139C22⋊Q8—1182no—
140C4×Q8—1182no—
141#117C22⋊D5911yescharacteristic
142Q32—2364no—
143C2×C16—1182no—
144#107—592no—
145#106—1181no—
146C4⋊C8—22no—
147#103—42no—
148#103—1182no—
149#103—1182no—
150#101—42no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—1182no—
158C22×C4—41no—
159C22×C4C2×C22⋊D5911yescharacteristic
160Q16—24no—
161Q16—42no—
162Q16—4721no—
163Q16—2362no—
164M16—41no—
165M16—41no—
166M16—41no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D5912yes—
171C4⋊C4—1182no—
172C4⋊C4—2362no—
173C4⋊C4—2362no—
174C4⋊C4—2362no—
175C4⋊C4—2362no—
176C4⋊C4#1174566912yes—
177C22⋊C4—21no—
178C22⋊C4—2362no—
179C42—21no—
180C42—2362no—
181C16—2362no—
182C23#1427375211yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—81no—
186Q8—81no—
187Q8—2362no—
188Q8—2362no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—2362no—
196C2×C4—2362no—
197C2×C4#1427369511yescharacteristic
198C2×C4C2×#1174566911yescharacteristic
199C8—22no—
200C8—41no—
201C8—81no—
202C22—22no—
203C22—21no—
204C22#1683753511yescharacteristic
205C4—21no—
206C4—42no—
207C4—82no—
208C4—4722no—
209C4—4722no—
210C4#1683757411yescharacteristic
211C4#1683758811yescharacteristic
212C2—41no—
213C2#2142303011yescharacteristic
214C2#2142607212yes—
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2047C2
socleC22×C59#447
prime radicalC23×C59AGL(1,ℤ/8)
hypercenter#2047D59
hypocenterC59#12168
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #2047

Complements

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

NH ≅ G/Ntype
126. C59#12168semidirect

Maximal subgroups

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

subgroupindex
#214355502
#214260302
#214221132
#214220382
C59×#20472
#1216859