Finite Group Databasebeta

← #28497805

Subgroups — #28497805

Order 15104.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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#21422410C212yesmaximal
2#21422030C214yesmaximal, complemented
3C59×#2310C211yesmaximal, characteristic
4#16837455—22no—
5#16837128C2214yes—
6#16837126—22no—
7C23×Q8×C59C411yescharacteristic
8C2×#117×C59C411yescharacteristic
9C2×#117×C59C2211yescharacteristic
10C59×#391—22no—
11C59×#391C412yes—
12C59×#391C2212yes—
13C59×#381—22no—
14C4⋊(C59⋊C8)—42no—
15C22⋊(C59⋊C8)—24no—
16C4×C59⋊C8—42no—
17C22×Q8×C59—21no—
18C22×Q8×C59—22no—
19C22×Q8×C59—42no—
20C22×Q8×C59—42no—
21C22×Q8×C59C814yescomplemented
22C22×Q8×C59C2×C412yes—
23C23×C4×C59—21no—
24C23×C4×C59C2×C411yescharacteristic
25#117×C59—42no—
26#117×C59—42no—
27C2×C4⋊C4×C59—21no—
28C2×C4⋊C4×C59C2×C411yescharacteristic
29C2×C4⋊C4×C59C2311yescharacteristic
30C2×C22⋊C4×C59—21no—
31C2×C22⋊C4×C59D412yes—
32C2×C42×C59C2×C412yes—
33C2×C42×C59D412yes—
34#95×C59—24no—
35#95×C59—24no—
36C24×C59C22⋊C411yescharacteristic
37C2×Q8×C59—24no—
38C2×Q8×C59—22no—
39C2×Q8×C59—48no—
40C2×Q8×C59—48no—
41C2×Q8×C59—42no—
42C2×Q8×C59—42no—
43C2×Q8×C59—42no—
44C2×Q8×C59—42no—
45C2×Q8×C59C2×C814yes—
46C22×C4×C59—22no—
47C22×C4×C59—22no—
48C22×C4×C59—24no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—42no—
52C22×C4×C59—42no—
53C22×C4×C59C22⋊C412yes—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C2×C812yes—
56C22×C4×C59M1612yes—
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D412yes—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—42no—
62C22⋊C4×C59—42no—
63C22⋊C4×C59—42no—
64C42×C59—22no—
65C42×C59—22no—
66C42×C59—42no—
67C2×C59⋊C8—44no—
68C23×C59—21no—
69C23×C59—21no—
70C23×C59—41no—
71C23×C59—41no—
72C23×C59C22⋊C812yes—
73C23×C59C2×C22⋊C411yescharacteristic
74Q8×C59—44no—
75Q8×C59—44no—
76Q8×C59—48no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—24no—
80C2×C4×C59—24no—
81C2×C4×C59—24no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—48no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59C22⋊C812yes—
93C2×C4×C59#9912yes—
94C2×C4×C59#10012yes—
95C2×C4×C59C22×C811yescharacteristic
96C2×C4×C59C2×M1611yescharacteristic
97C59⋊C8—84no—
98#11434—591nomaximal, Sylow
99C22×C59—21no—
100C22×C59—21no—
101C22×C59—22no—
102C22×C59—42no—
103C22×C59—42no—
104C22×C59—41no—
105C22×C59—41no—
106C22×C59C2×C22⋊C811yescharacteristic
107C22×C59C2×#9911yescharacteristic
108C22×C59C2×#10011yescharacteristic
109C4×C59—21no—
110C4×C59—21no—
111C4×C59—42no—
112C4×C59—42no—
113C4×C59—44no—
114C4×C59—82no—
115C4×C59#32314yes—
116#2310D5911yescharacteristic
117#1458—592no—
118#1319—594no—
119C2×C59—21no—
120C2×C59—21no—
121C2×C59—42no—
122C2×C59C2×#32311yescharacteristic
123C2×C59#212911yescharacteristic
124C2×C59#213511yescharacteristic
125C23×Q8Dic5911yescharacteristic
126C2×#117Dic5911yescharacteristic
127C2×#117C2×D5911yescharacteristic
128#431—1182no—
129#391—22no—
130#391Dic5912yes—
131#391C2×D5912yes—
132#381—22no—
133#324—594no—
134#323—1182no—
135C59#1143411yescharacteristic, complemented, Sylow
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C59⋊C814yescomplemented
141C22×Q8C2×Dic5912yes—
142C23×C4—21no—
143C23×C4C2×Dic5911yescharacteristic
144#117—42no—
145#117—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic5911yescharacteristic
148C2×C4⋊C4C22×D5911yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4C22⋊D5912yes—
151C2×C42C2×Dic5912yes—
152C2×C42C22⋊D5912yes—
153C4⋊C8—2362no—
154C22⋊C8—1184no—
155C4×C8—2362no—
156#95—24no—
157#95—24no—
158C24C22⋊Dic5911yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C2×C59⋊C814yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×C59⋊C812yes—
176C22×C4#1174566412yes—
177C22×C4C22⋊Dic5912yes—
178C22×C4C22⋊Dic5911yescharacteristic
179C22×C4C22×Dic5911yescharacteristic
180C22×C4C2×C22⋊D5912yes—
181C2×C8—2364no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C42—22no—
188C42—22no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊(C59⋊C8)12yes—
195C23C2×C22⋊Dic5911yescharacteristic
196Q8—44no—
197Q8—44no—
198Q8—48no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—48no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4C22⋊(C59⋊C8)12yes—
215C2×C4#1427364212yes—
216C2×C4#1427364412yes—
217C2×C4C22×C59⋊C811yescharacteristic
218C2×C4C2×#1174566411yescharacteristic
219C8—4724no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×C22⋊(C59⋊C8)11yescharacteristic
228C22C2×#1427364211yescharacteristic
229C22C2×#1427364411yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—42no—
235C4—82no—
236C4#1683712614yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1683712611yescharacteristic
241C2#2142577111yescharacteristic
242C2#2142579111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2310C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2310D59
hypocenterC59#11434
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #21422030C2semidirect
21. C22×Q8×C59C8semidirect
135. C59#11434semidirect
140. C22×Q8C59⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214224102
#214220302
C59×#23102
#1143459