Finite Group Databasebeta

← #28491761

Subgroups — #28491761

Order 15104.

400 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 93 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#21425068C212yesmaximal
2#21424915C211yesmaximal, characteristic
3#21421977C211yesmaximal, characteristic
4#21421977C212yesmaximal
5C59×#2844C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273650—21no—
8C2×#14273650—21no—
9C2×#14273638C2211yescharacteristic
10C2×#14273638C2212yes—
11#16837164—22no—
12#16837164—22no—
13C59×#400—21no—
14C59×#400—22no—
15C59×#395—21no—
16C59×#395—22no—
17C59×#382C2211yescharacteristic
18C59×#382C2212yes—
19C22×Dic118—21no—
20C22×C59⋊C8—21no—
21C22×C59⋊C8—22no—
22#14273727—22no—
23#14273727—21no—
24#14273727Q812yescomplemented
25C2×C4⋊Dic59—22no—
26#14273650—42no—
27#14273650—42no—
28#14273638—22no—
29#14273638—22no—
30#14273638—22no—
31C22×Q8×C59—21no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C4⋊C4×C59D411yescharacteristic
36C2×C4⋊C4×C59D412yes—
37C2×C4⋊C4×C59D412yes—
38C2×C42×C59C2311yescharacteristic
39#95×C59—22no—
40#95×C59—21no—
41#95×C59—21no—
42#95×C59—22no—
43#95×C59—22no—
44#95×C59—22no—
45C2×Q8×C59—81no—
46C2×Q8×C59—82no—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59—21no—
50C22×C4×C59—22no—
51C22×C4×C59C2×D412yes—
52C22×C4×C59C2×D411yescharacteristic
53C22×Dic59—41no—
54C2×Dic118—22no—
55C2×Dic118—41no—
56C2×Dic118—41no—
57C2×Dic118—41no—
58C2×Dic118—41no—
59C2×Dic118—41no—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—81no—
65C4⋊C4×C59—82no—
66C42×C59C2×Q811yescharacteristic
67C42×C59#4112yes—
68C42×C59#4111yescharacteristic
69C4⋊Dic59—24no—
70C4⋊Dic59—42no—
71C2×C59⋊C8—22no—
72C2×C59⋊C8—24no—
73C2×C59⋊C8—41no—
74C2×C59⋊C8—42no—
75C2×C59⋊C8—42no—
76C2×C59⋊C8—41no—
77C23×C59#12011yescharacteristic
78Q8×C59—84no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59C22⋊Q812yes—
94C2×C4×C59C22⋊Q811yescharacteristic
95C2×C4×C59#12412yes—
96C2×C4×C59#12411yescharacteristic
97C2×Dic59—42no—
98C2×Dic59—81no—
99C2×Dic59—81no—
100Dic118—42no—
101Dic118—42no—
102Dic118—81no—
103Dic118—81no—
104C59⋊C8—42no—
105C59⋊C8—44no—
106#10359—591nomaximal, Sylow
107C22×C59#39311yescharacteristic
108C22×C59#45311yescharacteristic
109C22×C59#45411yescharacteristic
110C22×C59#45512yes—
111C22×C59#45612yes—
112C4×C59—21no—
113C4×C59—21no—
114C4×C59—22no—
115C4×C59—22no—
116C4×C59—81no—
117C4×C59—82no—
118C4×C59—81no—
119C4×C59—82no—
120Dic59—82no—
121#2844D5911yescharacteristic
122#1993—592no—
123#1960—591no—
124#1295—591no—
125#1295—592no—
126C2×C59#165911yescharacteristic
127C2×C59#166011yescharacteristic
128C2×C59#166412yes—
129C2×C59#202711yescharacteristic
130C2×C59#202812yes—
131C2×C4⋊Q8—591no—
132C2×#103—1181no—
133C2×#103—1181no—
134C2×#97—591no—
135C2×#97—592no—
136#400—22no—
137#400—21no—
138#395—22no—
139#395—21no—
140#382C2×D5911yescharacteristic
141#382C2×D5912yes—
142#340—1182no—
143#340—1182no—
144C59#1035911yescharacteristic, complemented, Sylow
145C22×Q8—21no—
146C22×Q8—1181no—
147C22×C8—1182no—
148C22×C8—1181no—
149C4⋊Q8—592no—
150C4⋊Q8—1182no—
151C4⋊Q8—1181no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—1182no—
155C2×C4⋊C4C22⋊D5912yes—
156C2×C4⋊C4C22⋊D5912yes—
157C2×C4⋊C4C22⋊D5911yescharacteristic
158C2×C4⋊C4C22⋊D5911yescharacteristic
159C2×C42C22×D5911yescharacteristic
160#103—2362no—
161#103—2362no—
162#97—1182no—
163#97—1182no—
164#97—1182no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—81no—
172C2×Q8—82no—
173C2×Q8—1182no—
174C2×Q8—2361no—
175C2×Q8—2361no—
176C2×Q8—2361no—
177C2×Q8—2361no—
178C2×Q8—2361no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2361no—
184C22×C4C2×C22⋊D5912yes—
185C22×C4C2×C22⋊D5911yescharacteristic
186C2×C8—1182no—
187C2×C8—1184no—
188C2×C8—2361no—
189C2×C8—2362no—
190C2×C8—2362no—
191C2×C8—2361no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—1184no—
199C4⋊C4—2362no—
200C42Q8×D5911yescharacteristic
201C42#1174568912yes—
202C42#1174568911yescharacteristic
203C23#1427369611yescharacteristic
204Q8—84no—
205Q8—4721no—
206Q8—4721no—
207Q8—2362no—
208Q8—2362no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4721no—
224C2×C4—4721no—
225C2×C4—2362no—
226C2×C4#1427370312yes—
227C2×C4#1427370311yescharacteristic
228C2×C4#1427371612yes—
229C2×C4#1427371611yescharacteristic
230C8—2362no—
231C8—2364no—
232C22#1683731311yescharacteristic
233C22#1683757011yescharacteristic
234C22#1683757511yescharacteristic
235C22#1683758012yes—
236C22#1683758512yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—4722no—
246C2#2142340211yescharacteristic
247C2#2142340711yescharacteristic
248C2#2142343012yes—
249C2#2142524911yescharacteristic
250C2#2142525412yes—
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2844C2
socle, prime radicalC23×C59#120
hypercenter#2844D59
hypocenterC59#10359
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #14273727Q8semidirect
144. C59#10359semidirect

Maximal subgroups

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

subgroupindex
#214250682
#214249152
#214219772
C59×#28442
#1035959