Finite Group Databasebeta

← #28484000

Subgroups — #28484000

Order 15104.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#21434418C211yesmaximal, characteristic
2#21422688C211yesmaximal, characteristic
3#21422516C212yesmaximal, complemented
4#21421932C212yesmaximal, complemented
5C59×#2979C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7#16837756—22no—
8#16837752C412yes—
9#16837748C412yes—
10#16837398—22no—
11C4×#11745664C2212yes—
12C2×C4×C59⋊C8C2211yescharacteristic
13#16837142—22no—
14#16837132—22no—
15#16837132—22no—
16C59×#532C2211yescharacteristic
17C59×#436—22no—
18C8×D4×C59—22no—
19C59×#431C2212yes—
20C59×#408—22no—
21C2×#11745664—22no—
22C22×C59⋊C8—21no—
23#14273727C2×C412yes—
24#14273719—21no—
25#14273706—22no—
26C4×Dic118—22no—
27#14273686—21no—
28#14273651—24no—
29#14273648—24no—
30#14273638—22no—
31C4×C59⋊C8—21no—
32C4×C59⋊C8—22no—
33C4×C59⋊C8C2×C412yes—
34C2×#41×C59D412yes—
35C2×M16×C59—22no—
36C22×C8×C59—22no—
37C4⋊Q8×C59C2×C411yescharacteristic
38#127×C59C2×C411yescharacteristic
39#124×C59D412yes—
40#121×C59—22no—
41C4×D4×C59—22no—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59D412yes—
44C4⋊C8×C59D412yes—
45C22⋊C8×C59—24no—
46C22⋊C8×C59—22no—
47C4×C8×C59—22no—
48#41×C59—42no—
49#41×C59—42no—
50C2×Q8×C59C22⋊C412yes—
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59C22⋊C412yes—
54C22×C4×C59—22no—
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59#4112yes—
57C2×Dic118—22no—
58M16×C59—82no—
59C2×C8×C59—22no—
60C2×C8×C59—22no—
61C2×C8×C59—82no—
62C4⋊C4×C59—22no—
63C22⋊C4×C59—24no—
64C42×C59C22×C411yescharacteristic
65C42×C59C2×D411yescharacteristic
66C42×C59C2×D412yes—
67C22⋊Dic59—41no—
68C4⋊Dic59—22no—
69C4⋊Dic59—22no—
70Dic59⋊C4—42no—
71C4×Dic59—41no—
72#11745664—24no—
73#11745664—42no—
74C2×C59⋊C8—22no—
75C2×C59⋊C8—41no—
76C2×C59⋊C8—41no—
77C2×C59⋊C8—42no—
78C23×C59—22no—
79C23×C59#12311yescharacteristic
80Q8×C59—42no—
81D4×C59—42no—
82D4×C59—42no—
83D4×C59—42no—
84D4×C59—82no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59C4×D412yes—
94C2×C4×C59#12011yescharacteristic
95C2×C4×C59#12112yes—
96C2×Dic59—42no—
97Dic118—42no—
98Dic118—42no—
99C8×C59—42no—
100C8×C59—42no—
101C59⋊C8—42no—
102C59⋊C8—44no—
103#9067—591nomaximal, Sylow
104C22×C59—22no—
105C22×C59—21no—
106C22×C59—44no—
107C22×C59—42no—
108C22×C59#38611yescharacteristic
109C4×C59—22no—
110C4×C59—21no—
111C4×C59—42no—
112C4×C59#44312yes—
113C4×C59#44412yes—
114C4×C59#45511yescharacteristic
115C4×C59#45511yescharacteristic
116Dic59—82no—
117#3131—591no—
118#2979D5911yescharacteristic
119#1525—591no—
120#1485—592no—
121#1278—592no—
122C2×C59—41no—
123C2×C59—82no—
124C2×C59#187612yes—
125C2×C59#188511yescharacteristic
126#533—591no—
127#532C2×D5911yescharacteristic
128#488—1182no—
129#487—592no—
130#486—592no—
131#436—22no—
132C8×D4—22no—
133#431C2×D5912yes—
134#416—1182no—
135#408—22no—
136C4×M16—592no—
137C2×C4×C8—591no—
138#329—1182no—
139Q8⋊C8—1182no—
140Q8⋊C8—1182no—
141C59#906711yescharacteristic, complemented, Sylow
142C2×#41C22⋊D5912yes—
143C2×M16—22no—
144C2×M16—1182no—
145C22×C8—22no—
146C22×C8—1181no—
147C4⋊Q8—592no—
148C4⋊Q8C4×D5911yescharacteristic
149#127C4×D5911yescharacteristic
150#125—1181no—
151#124D23612yes—
152C22⋊Q8—1182no—
153#121—22no—
154C4×Q8—1182no—
155C4×D4—22no—
156#117—1181no—
157C2×C42C22×D5911yescharacteristic
158C4⋊C8C22⋊D5912yes—
159C4⋊C8C22⋊D5912yes—
160#103—1184no—
161#102—1184no—
162C22⋊C8—24no—
163C22⋊C8—22no—
164#97—1182no—
165C4×C8—22no—
166C4×C8—592no—
167C4×C8—1181no—
168C4×C8—1182no—
169#41—42no—
170#41—42no—
171C2×Q8—1182no—
172C2×Q8(C2×D59)⋊C412yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4(C2×D59)⋊C412yes—
176C22×C4—22no—
177C22×C4#1174568512yes—
178C22×C4C2×C22⋊D5911yescharacteristic
179M16—82no—
180M16—1184no—
181M16—2362no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—82no—
185C2×C8—1182no—
186C2×C8—2361no—
187C2×C8—2361no—
188C2×C8—2362no—
189C4⋊C4—22no—
190C4⋊C4—1182no—
191C4⋊C4—1182no—
192C4⋊C4—2362no—
193C22⋊C4—24no—
194C22⋊C4—2361no—
195C42—2361no—
196C42C2×C4×D5911yescharacteristic
197C42C2×D23611yescharacteristic
198C42C2×C22⋊D5912yes—
199C23—22no—
200C23#1427371011yescharacteristic
201Q8—42no—
202Q8—2362no—
203Q8—2362no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—2362no—
216C2×C4C2×(C2×D59)⋊C411yescharacteristic
217C2×C4C4×C22⋊D5912yes—
218C2×C4#1427369611yescharacteristic
219C2×C4#1427370112yes—
220C8—42no—
221C8—42no—
222C8—2362no—
223C8—2364no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1683728111yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—4722no—
233C4#1683751712yes—
234C4#1683752212yes—
235C4#1683758011yescharacteristic
236C4#1683758011yescharacteristic
237C2—41no—
238C2—82no—
239C2#2142442312yes—
240C2#2142448411yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2979C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#2979D59
hypocenterC59#9067
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #21422516C2semidirect
4. #21421932C2semidirect
141. C59#9067semidirect

Maximal subgroups

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

subgroupindex
#214344182
#214226882
#214225162
#214219322
C59×#29792
#906759