Finite Group Databasebeta

← #28503875

Subgroups — #28503875

Order 15104.

464 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16837201C212yesmaximal
2C4×#14273646C212yesmaximal
3#21422223C212yesmaximal
4C59×#2327C211yesmaximal, characteristic
5C2×#14273646C2214yes—
6C4×#11745664—22no—
7#16837201C418yescomplemented
8#16837171—22no—
9C2×C4⋊Q8×C59C2211yescharacteristic
10C2×C4×Q8×C59C411yescharacteristic
11C2×C4×Q8×C59C2211yescharacteristic
12C59×#391C411yescharacteristic
13C59×#391C2211yescharacteristic
14C59×#387—22no—
15C59×#387—22no—
16C59×#382C411yescharacteristic
17C59×#376C411yescharacteristic
18C2×#11745664—24no—
19#14273646—24no—
20#14273646C2×C418yes—
21#14273638—42no—
22C4×C59⋊C8—42no—
23C22×Q8×C59C2×C411yescharacteristic
24C22×Q8×C59C2311yescharacteristic
25C4⋊Q8×C59—22no—
26C4⋊Q8×C59C2×C414yes—
27C4×Q8×C59—42no—
28C4×Q8×C59—42no—
29C2×C4⋊C4×C59—22no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59C2×C411yescharacteristic
33C2×C4⋊C4×C59D412yes—
34C2×C42×C59—21no—
35C2×C42×C59C2×C411yescharacteristic
36C2×C42×C59C2×C411yescharacteristic
37C2×C42×C59C2×C411yescharacteristic
38C2×C42×C59D412yes—
39#95×C59—21no—
40#95×C59—21no—
41#95×C59—22no—
42#95×C59—22no—
43#95×C59—22no—
44C2×Q8×C59—21no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59C4212yes—
50C2×Q8×C59C22×C412yes—
51C2×Q8×C59#4112yes—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59C22⋊C412yes—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D412yes—
61C4⋊C4×C59—28no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—42no—
66C42×C59—22no—
67C42×C59—22no—
68C42×C59—22no—
69C42×C59—21no—
70C42×C59—42no—
71C42×C59—41no—
72C42×C59C4212yes—
73#11745664—28no—
74#11745664—44no—
75C2×C59⋊C8—44no—
76C23×C59C2×C22⋊C411yescharacteristic
77Q8×C59—42no—
78Q8×C59—42no—
79Q8×C59—42no—
80Q8×C59—42no—
81C2×C4×C59—22no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59#9912yes—
102C2×C4×C59C2×C4211yescharacteristic
103C2×C4×C59#11711yescharacteristic
104C2×C4×C59C4×D414yes—
105C59⋊C8—48no—
106#12631—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59C4×C22⋊C411yescharacteristic
110C22×C59C2×#9911yescharacteristic
111C22×C59#41011yescharacteristic
112C4×C59—22no—
113C4×C59—24no—
114C4×C59—21no—
115C4×C59—21no—
116C4×C59—42no—
117C4×C59—41no—
118C4×C59—42no—
119C4×C59—42no—
120C4×C59—81no—
121C4×C59—81no—
122#2327D5911yescharacteristic
123C2×#356—592no—
124C4×#101—592no—
125#1394—592no—
126C2×C59—21no—
127C2×C59—21no—
128C2×C59C4×#9911yescharacteristic
129C2×C59#213311yescharacteristic
130C2×C59#213511yescharacteristic
131C2×C4⋊Q8C2×D5911yescharacteristic
132C2×C4×Q8Dic5911yescharacteristic
133C2×C4×Q8C2×D5911yescharacteristic
134C2×#101—594no—
135C4×M16—1182no—
136#391Dic5911yescharacteristic
137#391C2×D5911yescharacteristic
138#387—22no—
139#387—22no—
140#382Dic5911yescharacteristic
141#376Dic5911yescharacteristic
142#356—598no—
143#343—1182no—
144C59#1263111yescharacteristic, complemented, Sylow
145C22×Q8C2×Dic5911yescharacteristic
146C22×Q8C22×D5911yescharacteristic
147C2×M16—1184no—
148C4⋊Q8—22no—
149C4⋊Q8C4×D5914yes—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×Dic5911yescharacteristic
155C2×C4⋊C4C2×Dic5911yescharacteristic
156C2×C4⋊C4C22⋊D5912yes—
157C2×C42—21no—
158C2×C42C2×Dic5911yescharacteristic
159C2×C42C2×Dic5911yescharacteristic
160C2×C42C2×Dic5911yescharacteristic
161C2×C42C22⋊D5912yes—
162#101—598no—
163#101—1184no—
164#97—2362no—
165C4×C8—2362no—
166#95—21no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95—22no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C4×Dic5912yes—
177C2×Q8C2×C4×D5912yes—
178C2×Q8#1174568512yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4C22⋊Dic5912yes—
184C22×C4C22⋊Dic5911yescharacteristic
185C22×C4C22⋊Dic5911yescharacteristic
186C22×C4C22×Dic5911yescharacteristic
187C22×C4C2×C22⋊D5912yes—
188M16—1188no—
189M16—2364no—
190C2×C8—2364no—
191C4⋊C4—28no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—41no—
201C42—42no—
202C42C4×Dic5912yes—
203C23C2×C22⋊Dic5911yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4#1427364212yes—
229C2×C4C2×C4×Dic5911yescharacteristic
230C2×C4#1427368611yescharacteristic
231C2×C4C4×C22⋊D5914yes—
232C8—2368no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊Dic5911yescharacteristic
236C22C2×#1427364211yescharacteristic
237C22#1683737611yescharacteristic
238C4—24no—
239C4—21no—
240C4—21no—
241C4—22no—
242C4—42no—
243C4—42no—
244C4—41no—
245C4—42no—
246C4—81no—
247C4—81no—
248C2—21no—
249C2—21no—
250C2C4×#1427364211yescharacteristic
251C2#2142578511yescharacteristic
252C2#2142579111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC2×C4×C59C2×C42
FrattiniC22×Q8C22×D59
FittingC59×#2327C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC22×Q8×C59C23
hypercenter#2327D59
hypocenterC59#12631
LCSG ▷ C2×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
7. #16837201C4semidirect
144. C59#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#168372012
C4×#142736462
#214222232
C59×#23272
#1263159