Finite Group Databasebeta

← #28500217

Subgroups — #28500217

Order 15104.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434724C211yesmaximal, characteristic
2#21425862C211yesmaximal, characteristic, complemented
3#21422158C212yesmaximal, complemented
4#21422136C212yesmaximal, complemented
5C59×#2166C211yesmaximal, characteristic, complemented
6#16838014C411yescharacteristic
7#16838007C411yescharacteristic
8C2×#14273667—21no—
9#16837804C2211yescharacteristic
10#16837775C2211yescharacteristic
11#16837771C411yescharacteristic
12#16837767C411yescharacteristic
13#16837648—21no—
14#16837631—21no—
15#16837504—21no—
16C4×#11745660—21no—
17#16837223—21no—
18#16837223C2212yes—
19C59×#405C2211yescharacteristic
20C59×#364C2212yes—
21C59×#349—21no—
22C59×#347—21no—
23C2×#11745660—21no—
24C2×#11745660—21no—
25#14273727C2×C411yescharacteristic
26#14273724C2×C411yescharacteristic
27#14273720—41no—
28#14273719C2×C411yescharacteristic
29#14273719C2×C411yescharacteristic
30#14273717—41no—
31#14273712—41no—
32#14273706—21no—
33#14273701—21no—
34#14273700—41no—
35C4×Dic118—21no—
36C4×D236—21no—
37#14273667—22no—
38#14273667—41no—
39C2×C59⋊C16—41no—
40#14273661—21no—
41#14273659—21no—
42#14273652—21no—
43#14273652—21no—
44#14273649—21no—
45#14273649—21no—
46C2×M16×C59D411yescharacteristic
47C22×C8×C59D411yescharacteristic
48#117×C59C2×C411yescharacteristic
49M32×C59—22no—
50#97×C59C2×C411yescharacteristic
51#97×C59D411yescharacteristic
52C4×C8×C59D411yescharacteristic
53C4×C8×C59C2311yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C2×C22⋊D59—41no—
56C2×D236—21no—
57C2×D236—41no—
58C2×C4×D59—41no—
59C2×Dic118—21no—
60M16×C59—22no—
61C2×C8×C59—21no—
62C2×C8×C59—21no—
63C2×C8×C59C22⋊C411yescharacteristic
64C2×C8×C59C22⋊C411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C2×C8×C59C2×D411yescharacteristic
67C4⋊C4×C59—21no—
68C22⋊C4×C59—21no—
69C42×C59C22⋊C411yescharacteristic
70C42×C59C22×C411yescharacteristic
71C22⋊Dic59—41no—
72(C2×D59)⋊C4—41no—
73(C2×D59)⋊C4—41no—
74C4⋊Dic59—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59—41no—
77Dic59⋊C4—41no—
78Dic59⋊C4—41no—
79Dic59⋊C4—41no—
80Dic59⋊C4—41no—
81C4×Dic59—41no—
82#11745660—42no—
83#11745660—41no—
84#11745660—41no—
85C16×C59—42no—
86C59⋊C16—42no—
87C23×C59#10011yescharacteristic
88C22×D59—41no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59#10011yescharacteristic
94C2×C4×C59C2×C22⋊C411yescharacteristic
95C22⋊D59—161no—
96C2×Dic59—41no—
97C2×Dic59—41no—
98C2×Dic59—41no—
99D236—41no—
100D236—41no—
101D236—161no—
102C4×D59—161no—
103Dic118—41no—
104Dic118—41no—
105C8×C59—21no—
106C8×C59—21no—
107C8×C59#13512yes—
108C8×C59AGL(1,ℤ/8)11yescharacteristic
109C8×C59#13711yescharacteristic
110#11991—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—41no—
113C22×C59C2×#10011yescharacteristic
114C2×D59—81no—
115C2×D59—81no—
116C2×D59—81no—
117C4×C59—21no—
118C4×C59—41no—
119C4×C59—41no—
120C4×C59#41611yescharacteristic
121C4×C59#41711yescharacteristic
122Dic59—81no—
123Dic59—81no—
124Dic59—81no—
125#3174—591no—
126#2166D5911yescharacteristic, complemented
127#2153—591no—
128#1366—592no—
129#1359—592no—
130C2×C59—21no—
131C2×C59—41no—
132C2×C59#155111yescharacteristic
133D59—161no—
134#541—591no—
135#540—591no—
136C2×M32—1181no—
137#499—591no—
138#492—591no—
139#491—591no—
140#490—591no—
141#468—1181no—
142#465—1181no—
143#440—1181no—
144C4×SD16—1181no—
145#405C2×D5911yescharacteristic
146#364—592no—
147#364—1181no—
148#364C2×D5912yes—
149#349—21no—
150#347—21no—
151C59#1199111yescharacteristic, complemented, Sylow
152C2×SD16—1181no—
153C2×SD16—1181no—
154C2×M16C22⋊D5911yescharacteristic
155C22×C8D23611yescharacteristic
156C4⋊Q8—591no—
157#127—591no—
158#126—2361no—
159#125—591no—
160#125—591no—
161#125—2361no—
162#123—2361no—
163C22⋊Q8—1181no—
164#121—1181no—
165#121—2361no—
166C4×Q8—1181no—
167C4×D4—1181no—
168#117C4×D5911yescharacteristic
169M32—22no—
170M32—1182no—
171M32—2361no—
172C2×C16—2361no—
173#107—1181no—
174#106—1181no—
175#103—1181no—
176#103—1181no—
177#102—1181no—
178#102—1181no—
179#97C4×D5911yescharacteristic
180#97C22⋊D5911yescharacteristic
181C4×C8D23611yescharacteristic
182C4×C8C22×D5911yescharacteristic
183C2×Q8—1181no—
184C2×D4—1181no—
185C2×D4—2361no—
186C2×D4—2361no—
187C22×C4—2361no—
188C22×C4(C2×D59)⋊C411yescharacteristic
189SD16—2362no—
190SD16—2361no—
191SD16—2361no—
192M16—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8(C2×D59)⋊C411yescharacteristic
196C2×C8(C2×D59)⋊C411yescharacteristic
197C2×C8C2×D23611yescharacteristic
198C2×C8C2×C22⋊D5911yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—1181no—
201C4⋊C4—1181no—
202C4⋊C4—2361no—
203C4⋊C4—2361no—
204C4⋊C4—2361no—
205C4⋊C4—2361no—
206C4⋊C4—2361no—
207C22⋊C4—21no—
208C22⋊C4—2361no—
209C22⋊C4—2361no—
210C22⋊C4—2361no—
211C42—2361no—
212C42(C2×D59)⋊C411yescharacteristic
213C42C2×C4×D5911yescharacteristic
214C16—42no—
215C16—2362no—
216C23—2361no—
217C23#1427364311yescharacteristic
218Q8—2361no—
219Q8—2361no—
220D4—9441no—
221D4—9441no—
222D4—2361no—
223D4—2361no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—9441no—
229C2×C4—2361no—
230C2×C4—2361no—
231C2×C4—2361no—
232C2×C4#1427364311yescharacteristic
233C2×C4C2×(C2×D59)⋊C411yescharacteristic
234C8—21no—
235C8—21no—
236C8#1427374712yes—
237C8#1427375511yescharacteristic
238C8#1427376011yescharacteristic
239C22—21no—
240C22—41no—
241C22—4721no—
242C22—4721no—
243C22—4721no—
244C22C2×#1427364311yescharacteristic
245C4—21no—
246C4—41no—
247C4—41no—
248C4—4721no—
249C4—4721no—
250C4—4721no—
251C4#1683739611yescharacteristic
252C4#1683740411yescharacteristic
253C2—21no—
254C2—41no—
255C2—9441no—
256C2#2142279211yescharacteristic
subgroupG/H
centerC2#21422792
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#2166C2
socleC2×C59#1551
prime radical#16838007C4
cosocleC4×C8×C59C23
hypercenter#2166D59
hypocenterC59#11991
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #21425862C2semidirect
3. #21422158C2semidirect
4. #21422136C2semidirect
5. C59×#2166C2semidirect
126. #2166D59semidirect
151. C59#11991semidirect

Maximal subgroups

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

subgroupindex
#214347242
#214258622
#214221582
#214221362
C59×#21662
#1199159