Finite Group Databasebeta

← #28498185

Subgroups — #28498185

Order 15104.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422440C212yesmaximal
2#21422040C212yesmaximal
3#21422030C212yesmaximal
4C59×#2327C211yesmaximal, characteristic
5#16837459—22no—
6#16837128—22no—
7#16837128C2214yes—
8C2×C4⋊Q8×C59C411yescharacteristic
9C2×C4×Q8×C59C411yescharacteristic
10C2×C4×Q8×C59C2211yescharacteristic
11C59×#391C411yescharacteristic
12C59×#391C2211yescharacteristic
13C59×#387—22no—
14C59×#387—22no—
15C59×#382C2211yescharacteristic
16C59×#376C411yescharacteristic
17C4⋊(C59⋊C8)—42no—
18C22⋊(C59⋊C8)—24no—
19#14273638—42no—
20C22×Q8×C59C2×C411yescharacteristic
21C22×Q8×C59C2×C411yescharacteristic
22C4⋊Q8×C59—22no—
23C4⋊Q8×C59—22no—
24C4×Q8×C59—42no—
25C4×Q8×C59—42no—
26C2×C4⋊C4×C59—22no—
27C2×C4⋊C4×C59—21no—
28C2×C4⋊C4×C59C2×C411yescharacteristic
29C2×C4⋊C4×C59D412yes—
30C2×C4⋊C4×C59C2311yescharacteristic
31C2×C42×C59—21no—
32C2×C42×C59C2×C411yescharacteristic
33C2×C42×C59C2×C411yescharacteristic
34C2×C42×C59C2×C411yescharacteristic
35C2×C42×C59D412yes—
36#95×C59—21no—
37#95×C59—21no—
38#95×C59—22no—
39#95×C59—22no—
40#95×C59—22no—
41C2×Q8×C59—21no—
42C2×Q8×C59—21no—
43C2×Q8×C59—21no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59M1612yes—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59C22⋊C412yes—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D412yes—
58C4⋊C4×C59—28no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—42no—
63C42×C59—22no—
64C42×C59—22no—
65C42×C59—22no—
66C42×C59—21no—
67C42×C59—42no—
68C42×C59—41no—
69C42×C59M1612yes—
70C2×C59⋊C8—44no—
71C23×C59C2×C22⋊C411yescharacteristic
72Q8×C59—42no—
73Q8×C59—42no—
74Q8×C59—42no—
75Q8×C59—42no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59#9912yes—
98C2×C4×C59C2×M1611yescharacteristic
99C2×C4×C59C2×M1611yescharacteristic
100C59⋊C8—84no—
101#11536—591nomaximal, Sylow
102C22×C59—21no—
103C22×C59—21no—
104C22×C59#40711yescharacteristic
105C22×C59C2×#9911yescharacteristic
106C22×C59#41311yescharacteristic
107C4×C59—22no—
108C4×C59—21no—
109C4×C59—21no—
110C4×C59—42no—
111C4×C59—42no—
112C4×C59—41no—
113C4×C59—42no—
114C4×C59—42no—
115C4×C59—81no—
116C4×C59—81no—
117#2327D5911yescharacteristic
118#1467—592no—
119#1324—592no—
120#1319—592no—
121C2×C59—21no—
122C2×C59—21no—
123C2×C59#146511yescharacteristic
124C2×C59#212911yescharacteristic
125C2×C59#213311yescharacteristic
126C2×C4⋊Q8Dic5911yescharacteristic
127C2×C4×Q8Dic5911yescharacteristic
128C2×C4×Q8C2×D5911yescharacteristic
129#432—1182no—
130#391Dic5911yescharacteristic
131#391C2×D5911yescharacteristic
132#387—22no—
133#387—22no—
134#382C2×D5911yescharacteristic
135#376Dic5911yescharacteristic
136#324—594no—
137#324—1182no—
138C59#1153611yescharacteristic, complemented, Sylow
139C22×Q8C2×Dic5911yescharacteristic
140C22×Q8C2×Dic5911yescharacteristic
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4×Q8—42no—
144C4×Q8—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic5911yescharacteristic
148C2×C4⋊C4C22⋊D5912yes—
149C2×C4⋊C4C22×D5911yescharacteristic
150C2×C42—21no—
151C2×C42C2×Dic5911yescharacteristic
152C2×C42C2×Dic5911yescharacteristic
153C2×C42C2×Dic5911yescharacteristic
154C2×C42C22⋊D5912yes—
155C4⋊C8—2362no—
156C22⋊C8—1184no—
157#97—2362no—
158#95—21no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8#1174566412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C22⋊Dic5912yes—
176C22×C4C22⋊Dic5911yescharacteristic
177C22×C4C22⋊Dic5911yescharacteristic
178C22×C4C22×Dic5911yescharacteristic
179C22×C4C2×C22⋊D5912yes—
180C2×C8—2364no—
181C4⋊C4—28no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—21no—
190C42—41no—
191C42—42no—
192C42#1174566412yes—
193C23C2×C22⋊Dic5911yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4#1427364212yes—
220C2×C4C2×#1174566411yescharacteristic
221C2×C4C2×#1174566411yescharacteristic
222C8—4724no—
223C22—21no—
224C22—21no—
225C22#1683736611yescharacteristic
226C22C2×#1427364211yescharacteristic
227C22#1683738611yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2#2142243411yescharacteristic
241C2#2142577111yescharacteristic
242C2#2142578511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2327C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2327D59
hypocenterC59#11536
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
138. C59#11536semidirect

Maximal subgroups

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

subgroupindex
#214224402
#214220402
#214220302
C59×#23272
#1153659