Finite Group Databasebeta

← #28501686

Subgroups — #28501686

Order 15104.

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

normal subgroups only

This group has 394 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#21429469C211yesmaximal, characteristic
2#21423104C212yesmaximal
3#21422101C212yesmaximal
4#21422042C211yesmaximal, characteristic
5C59×#2060C211yesmaximal, characteristic
6C2×C4×Dic118—21no—
7#16837730—22no—
8#16837332—21no—
9#16837332C2212yes—
10#16837307—22no—
11#16837301—22no—
12#16837139—22no—
13#16837128C2212yes—
14C59×#544C2211yescharacteristic
15C59×#428—21no—
16C59×#384C2211yescharacteristic
17C59×#343C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118—21no—
20C4×Dic118—42no—
21C4×Dic118—42no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59D412yes—
25C2×Dic59⋊C4—22no—
26C2×C4×Dic59—22no—
27#14273658—42no—
28#14273651—42no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C59⋊#95D412yes—
34C2×M16×C59—21no—
35C4⋊Q8×C59—41no—
36#125×C59—41no—
37C22⋊Q8×C59—22no—
38#117×C59—21no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C42×C59D411yescharacteristic
43#107×C59—41no—
44#106×C59—41no—
45C2×Q8×C59—22no—
46C22×C4×C59—21no—
47C22×C4×C59—21no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D412yes—
50C22×Dic59—22no—
51C22×Dic59—22no—
52C2×Dic118—24no—
53C2×Dic118—42no—
54C2×Dic118—42no—
55M16×C59—22no—
56M16×C59—41no—
57C2×C8×C59—41no—
58C4⋊C4×C59—21no—
59C4⋊C4×C59—21no—
60C4⋊C4×C59—22no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—44no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—41no—
65C22⋊C4×C59—41no—
66C42×C59—21no—
67C42×C59—41no—
68C42×C59SD1612yes—
69C4⋊Dic59—42no—
70C4⋊Dic59—42no—
71C4⋊Dic59—81no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74C4×Dic59—44no—
75C2×C59⋊C8—42no—
76C23×C59#12011yescharacteristic
77Q8×C59—82no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59C2×SD1611yescharacteristic
90C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
91C2×Dic59—24no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—42no—
97Dic118—48no—
98C8×C59—42no—
99C59⋊C8—82no—
100#12247—591nomaximal, Sylow
101C22×C59—21no—
102C22×C59—21no—
103C22×C59#45011yescharacteristic
104C22×C59#45611yescharacteristic
105C22×C59B(4,𝔽2)11yescharacteristic
106C4×C59—24no—
107C4×C59—21no—
108C4×C59—21no—
109C4×C59—42no—
110C4×C59—81no—
111C4×C59—81no—
112C4×C59—81no—
113Dic59—44no—
114Dic59—82no—
115#2582—591no—
116#2060D5911yescharacteristic
117#1611—592no—
118#1346—592no—
119#1325—591no—
120C2×C59—21no—
121C2×C59—21no—
122C2×C59#160411yescharacteristic
123C2×C59#219811yescharacteristic
124C2×C59#220511yescharacteristic
125#544C2×D5911yescharacteristic
126C2×C4×Q8—1181no—
127#483—1182no—
128#428—21no—
129#398—592no—
130#398—1181no—
131#391—1182no—
132#389—1182no—
133#384C2×D5911yescharacteristic
134#343C2×D5911yescharacteristic
135#339—21no—
136#328—1182no—
137#324—592no—
138C59#1224711yescharacteristic, complemented, Sylow
139C22×Q8—1181no—
140C2×M16—21no—
141C4⋊Q8—41no—
142#125—41no—
143C22⋊Q8—22no—
144C4×Q8—2362no—
145C4×Q8—2362no—
146#117—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—592no—
149C2×C4⋊C4—1181no—
150C2×C4⋊C4—1181no—
151C2×C4⋊C4—1182no—
152C2×C4⋊C4C22⋊D5911yescharacteristic
153C2×C4⋊C4C22×D5911yescharacteristic
154C2×C42—1182no—
155C2×C42C22⋊D5911yescharacteristic
156#107—41no—
157#106—41no—
158#106—2362no—
159#103—2362no—
160C22⋊C8—1182no—
161#95—592no—
162#95—1182no—
163#95—1181no—
164#95—1182no—
165C2×Q8—22no—
166C2×Q8—1184no—
167C2×Q8—2362no—
168C2×Q8—2362no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1182no—
172C22×C4—1182no—
173C22×C4D4×D5912yes—
174C22×C4C2×C22⋊D5911yescharacteristic
175M16—22no—
176M16—41no—
177C2×C8—41no—
178C2×C8—2362no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—4721no—
187C4⋊C4—2362no—
188C4⋊C4—2362no—
189C4⋊C4—2362no—
190C4⋊C4—2362no—
191C22⋊C4—41no—
192C42—21no—
193C42—41no—
194C42—2364no—
195C42#1174567112yes—
196C23#1427369511yescharacteristic
197Q8—82no—
198Q8—2368no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1184no—
211C2×C4—2362no—
212C2×C4—2362no—
213C2×C4—2362no—
214C2×C4—2362no—
215C2×C4—2362no—
216C2×C4C2×#1174567111yescharacteristic
217C2×C4#1427375211yescharacteristic
218C8—42no—
219C8—4722no—
220C22—21no—
221C22—21no—
222C22#1683755211yescharacteristic
223C22#1683758411yescharacteristic
224C22#1683758811yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—4722no—
233C4—2364no—
234C2—21no—
235C2—21no—
236C2#2142306411yescharacteristic
237C2#2142609111yescharacteristic
238C2#2142612511yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2060C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2060D59
hypocenterC59#12247
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
138. C59#12247semidirect

Maximal subgroups

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

subgroupindex
#214294692
#214231042
#214221012
#214220422
C59×#20602
#1224759