Finite Group Databasebeta

← #28477498

Subgroups — #28477498

Order 15104.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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#21430468C211yesmaximal, characteristic
2#21425426C212yesmaximal, complemented
3#21421918C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#2013C211yesmaximal, characteristic
6C2×#14273704—22no—
7C2×#14273658—22no—
8#16837325—22no—
9#16837322—21no—
10#16837300C2212yes—
11#16837282—21no—
12#16837164—22no—
13#16837157C2212yes—
14C22×SD16×C59—21no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C2×#103×C59—21no—
17C59×#390C2211yescharacteristic
18C59×#336C2211yescharacteristic
19C22×Dic118—21no—
20C23×Dic59—21no—
21C22×C59⋊C8—22no—
22#14273704—44no—
23#14273704—44no—
24C2×C4⋊Dic59D412yes—
25C2×C4⋊Dic59D412yes—
26C2×Dic59⋊C4—22no—
27C2×Dic59⋊C4—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—21no—
30#14273658—44no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C22×Q8×C59—21no—
34C22×D4×C59D411yescharacteristic
35C2×SD16×C59—42no—
36C2×SD16×C59—42no—
37C2×SD16×C59—42no—
38C22×C8×C59—21no—
39C4⋊Q8×C59—42no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C42×C59C2311yescharacteristic
44#103×C59—42no—
45C24×C59—21no—
46C2×Q8×C59—22no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59—41no—
52C2×D4×C59—21no—
53C2×D4×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59SD1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—41no—
58C22×C4×C59C2×D412yes—
59C22×C4×C59C2×D411yescharacteristic
60C22×Dic59—22no—
61C22×Dic59—22no—
62C22×Dic59—44no—
63C22×Dic59—81no—
64C2×Dic118—44no—
65C2×Dic118—81no—
66SD16×C59—48no—
67C2×C8×C59—22no—
68C2×C8×C59—41no—
69C2×C8×C59—41no—
70C4⋊C4×C59—24no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—41no—
76C22⋊Dic59—44no—
77C22⋊Dic59—81no—
78C4⋊Dic59—22no—
79C4⋊Dic59—42no—
80C4⋊Dic59SD1614yescomplemented
81Dic59⋊C4—44no—
82Dic59⋊C4—42no—
83Dic59⋊C4—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—42no—
86C2×C59⋊C8—42no—
87C23×C59—22no—
88C23×C59—42no—
89C23×C59—41no—
90C23×C59—42no—
91C23×C59—41no—
92C23×C59#12011yescharacteristic
93Q8×C59—42no—
94Q8×C59—42no—
95Q8×C59—81no—
96Q8×C59—81no—
97D4×C59—42no—
98D4×C59—81no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—42no—
106C2×C4×C59—81no—
107C2×C4×C59—81no—
108C2×C4×C59C2×SD1612yes—
109C2×C4×C59C2×SD1611yescharacteristic
110C2×C4×C59AGL(1,ℤ/8)12yes—
111C2×C4×C59#13711yescharacteristic
112C2×Dic59—24no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—84no—
119Dic118—84no—
120C8×C59—42no—
121C59⋊C8—44no—
122#7754—591nomaximal, Sylow
123C22×C59—42no—
124C22×C59—44no—
125C22×C59—82no—
126C22×C59—81no—
127C22×C59—81no—
128C22×C59#44811yescharacteristic
129C22×C59#45012yes—
130C22×C59#45311yescharacteristic
131C22×C59#45612yes—
132C22×C59B(4,𝔽2)11yescharacteristic
133C4×C59—21no—
134C4×C59—21no—
135C4×C59—22no—
136C4×C59—22no—
137C4×C59—82no—
138C4×C59—81no—
139Dic59—44no—
140Dic59—82no—
141#2683—591no—
142#2058—592no—
143#2013D5911yescharacteristic
144#1271—592no—
145#1270—591no—
146C2×C59—82no—
147C2×C59#159611yescharacteristic
148C2×C59#160412yes—
149C2×C59#162911yescharacteristic
150C2×C59#165112yes—
151C2×C59#166011yescharacteristic
152C22×SD16—21no—
153C2×C4⋊Q8C2×D5911yescharacteristic
154C2×C22⋊Q8—1182no—
155C2×#106—1182no—
156C2×#103—21no—
157#396—1182no—
158#395—1181no—
159#390C2×D5911yescharacteristic
160#389—592no—
161#386—1181no—
162#340—1182no—
163#336—592no—
164#336C2×D5911yescharacteristic
165C59#775411yescharacteristic, complemented, Sylow
166C22×Q8—21no—
167C22×Q8—1181no—
168C22×D4C22⋊D5911yescharacteristic
169C23×C4—1181no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×SD16—42no—
173C22×C8—21no—
174C22×C8—1182no—
175C4⋊Q8—42no—
176C22⋊Q8—2364no—
177C22⋊Q8—2364no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—592no—
180C2×C4⋊C4—592no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4—1182no—
183C2×C4⋊C4C22⋊D5911yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—1181no—
186C2×C22⋊C4—1182no—
187C2×C42C22×D5911yescharacteristic
188#106—2364no—
189#103—42no—
190#95—1181no—
191#95—1182no—
192C24—21no—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—4721no—
200C2×Q8—2364no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4#1174567012yes—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—1182no—
208C22×C4—1182no—
209C22×C4—4721no—
210C22×C4—2364no—
211C22×C4D4×D5912yes—
212C22×C4C2×C22⋊D5911yescharacteristic
213SD16—48no—
214C2×C8—22no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—1184no—
218C2×C8—2362no—
219C2×C8—2362no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—594no—
224C4⋊C4—1182no—
225C4⋊C4—2362no—
226C4⋊C4—2364no—
227C4⋊C4—2362no—
228C4⋊C4—2362no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—4721no—
232C22⋊C4—2364no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23#1427369511yescharacteristic
240Q8—42no—
241Q8—42no—
242Q8—81no—
243Q8—81no—
244Q8—4724no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—1184no—
257C2×C4—4724no—
258C2×C4—2362no—
259C2×C4—2362no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4SD16×D5912yes—
264C2×C4C2×#1174567011yescharacteristic
265C2×C4#1427375412yes—
266C2×C4#1427375711yescharacteristic
267C8—42no—
268C8—2364no—
269C22—42no—
270C22—44no—
271C22—81no—
272C22—81no—
273C22—82no—
274C22#1683754011yescharacteristic
275C22#1683755112yes—
276C22#1683756911yescharacteristic
277C22#1683758312yes—
278C22#1683758811yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—81no—
284C4—82no—
285C4—4722no—
286C4—2364no—
287C2—82no—
288C2#2142301811yescharacteristic
289C2#2142306312yes—
290C2#2142321011yescharacteristic
291C2#2142335712yes—
292C2#2142340511yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2013C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2013D59
hypocenterC59#7754
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #21425426C2semidirect
3. #21421918C2semidirect
80. C4⋊Dic59SD16semidirect
165. C59#7754semidirect

Maximal subgroups

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

subgroupindex
#214304682
#214254262
#214219182
#214219172
C59×#20132
#775459