Finite Group Databasebeta

← #28498169

Subgroups — #28498169

Order 15104.

412 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 99 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#21422042C212yesmaximal
3#21422030C212yesmaximal
4C4×C4⋊Q8×C59C211yesmaximal, 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×#389—22no—
14C59×#384C2211yescharacteristic
15C4×C4⋊C4×C59—22no—
16C43×C59C411yescharacteristic
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—41no—
25C4×Q8×C59—41no—
26C4×Q8×C59—41no—
27C4×Q8×C59—41no—
28C2×C4⋊C4×C59—22no—
29C2×C4⋊C4×C59—21no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59D412yes—
33C2×C4⋊C4×C59C2311yescharacteristic
34C2×C42×C59—21no—
35C2×C42×C59—21no—
36C2×C42×C59—21no—
37C2×C42×C59C2×C411yescharacteristic
38C2×C42×C59C2×C411yescharacteristic
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59D412yes—
41#95×C59—22no—
42#95×C59—22no—
43C2×Q8×C59—21no—
44C2×Q8×C59—21no—
45C2×Q8×C59—21no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59M1612yes—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D412yes—
60C4⋊C4×C59—28no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—44no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—41no—
66C42×C59—24no—
67C42×C59—21no—
68C42×C59—21no—
69C42×C59—22no—
70C42×C59—22no—
71C42×C59—21no—
72C42×C59—42no—
73C42×C59—41no—
74C42×C59—41no—
75C42×C59—41no—
76C42×C59M1612yes—
77C2×C59⋊C8—44no—
78C23×C59C2×C22⋊C411yescharacteristic
79Q8×C59—42no—
80Q8×C59—42no—
81Q8×C59—42no—
82Q8×C59—42no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59#9911yescharacteristic
105C2×C4×C59#9911yescharacteristic
106C2×C4×C59C2×M1611yescharacteristic
107C2×C4×C59C2×M1611yescharacteristic
108C59⋊C8—84no—
109#11532—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—21no—
112C22×C59#40711yescharacteristic
113C22×C59C2×#9911yescharacteristic
114C22×C59#41311yescharacteristic
115C4×C59—24no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—21no—
119C4×C59—42no—
120C4×C59—42no—
121C4×C59—41no—
122C4×C59—42no—
123C4×C59—42no—
124C4×C59—81no—
125C4×C59#35612yes—
126C4×C4⋊Q8D5911yescharacteristic
127#1467—592no—
128#1325—592no—
129#1319—592no—
130C2×C59—21no—
131C2×C59—21no—
132C2×C59#146511yescharacteristic
133C2×C59#212611yescharacteristic
134C2×C59C2×#35611yescharacteristic
135C2×C4⋊Q8Dic5911yescharacteristic
136C2×C4×Q8Dic5911yescharacteristic
137C2×C4×Q8C2×D5911yescharacteristic
138#432—1182no—
139#391Dic5911yescharacteristic
140#391C2×D5911yescharacteristic
141#389—22no—
142#384C2×D5911yescharacteristic
143C4×C4⋊C4—22no—
144C43Dic5911yescharacteristic
145#324—594no—
146#324—1182no—
147C59#1153211yescharacteristic, complemented, Sylow
148C22×Q8C2×Dic5911yescharacteristic
149C22×Q8C2×Dic5911yescharacteristic
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C2×Dic5911yescharacteristic
160C2×C4⋊C4C22⋊D5912yes—
161C2×C4⋊C4C22×D5911yescharacteristic
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42C2×Dic5911yescharacteristic
166C2×C42C2×Dic5911yescharacteristic
167C2×C42C2×Dic5911yescharacteristic
168C2×C42C22⋊D5912yes—
169C4⋊C8—2362no—
170C22⋊C8—1184no—
171#97—2362no—
172#95—22no—
173#95—22no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8#1174566412yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4C22⋊Dic5912yes—
187C22×C4C22⋊Dic5911yescharacteristic
188C22×C4C22⋊Dic5911yescharacteristic
189C22×C4C22×Dic5911yescharacteristic
190C22×C4C2×C22⋊D5912yes—
191C2×C8—2364no—
192C4⋊C4—28no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C42—24no—
199C42—22no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—42no—
206C42—41no—
207C42—41no—
208C42#1174566412yes—
209C23C2×C22⋊Dic5911yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4#1427364211yescharacteristic
236C2×C4#1427364211yescharacteristic
237C2×C4C2×#1174566411yescharacteristic
238C2×C4C2×#1174566411yescharacteristic
239C8—4724no—
240C22—21no—
241C22—21no—
242C22#1683736611yescharacteristic
243C22C2×#1427364211yescharacteristic
244C22#1683738611yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—41no—
252C4—42no—
253C4—42no—
254C4—81no—
255C4#1683720112yes—
256C2—21no—
257C2—21no—
258C2#2142243411yescharacteristic
259C2#2142576111yescharacteristic
260C2C2×#1683720111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC4×C4⋊Q8×C59C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenterC4×C4⋊Q8D59
hypocenterC59#11532
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
147. C59#11532semidirect

Maximal subgroups

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

subgroupindex
#214224402
#214220422
#214220302
C4×C4⋊Q8×C592
#1153259