Finite Group Databasebeta

← #28491768

Subgroups — #28491768

Order 15104.

476 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 89 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#21432371C211yesmaximal, characteristic, complemented
2#21424920C212yesmaximal
3#21424919C211yesmaximal, characteristic
4#21421978C212yesmaximal, complemented
5C59×#1295C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7C2×#14273649—22no—
8C2×#14273649—22no—
9C2×#14273647—22no—
10C2×#14273638C2211yescharacteristic
11#16837336—22no—
12#16837335—22no—
13#16837334—22no—
14#16837268C2212yes—
15C2×#97×C59C2212yes—
16C59×#382C2211yescharacteristic
17C22×D236—21no—
18C22×D236—22no—
19C22×C59⋊C8—21no—
20#14273724—22no—
21#14273724—21no—
22#14273724Q812yes—
23C2×C4⋊Dic59—22no—
24C2×C4⋊Dic59D412yes—
25C2×C4⋊Dic59D412yes—
26C2×Dic59⋊C4—22no—
27C2×Dic59⋊C4—22no—
28#14273649—44no—
29#14273649—44no—
30#14273647—44no—
31#14273638—22no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C22×C8×C59—22no—
36C2×C4⋊C4×C59D412yes—
37C2×C42×C59C2311yescharacteristic
38#97×C59—22no—
39#97×C59—22no—
40#95×C59—22no—
41C23×D59—41no—
42C22×C4×C59—22no—
43C22×C4×C59C2×D411yescharacteristic
44C22×C4×C59C2×D412yes—
45C22×Dic59—22no—
46C22×Dic59—22no—
47C2×D236—24no—
48C2×D236—22no—
49C2×D236—42no—
50C2×D236—42no—
51C2×D236—42no—
52C2×D236—42no—
53C2×D236—42no—
54C2×D236—42no—
55C2×D236—42no—
56C2×D236—41no—
57C2×C8×C59—24no—
58C2×C8×C59—42no—
59C2×C8×C59—42no—
60C4⋊C4×C59—42no—
61C42×C59C2×Q811yescharacteristic
62C42×C59#4112yes—
63C42×C59#4111yescharacteristic
64C4⋊Dic59—42no—
65C4⋊Dic59—42no—
66C4⋊Dic59—82no—
67Dic59⋊C4—82no—
68Dic59⋊C4—82no—
69C2×C59⋊C8—22no—
70C2×C59⋊C8—41no—
71C2×C59⋊C8—41no—
72C23×C59#12011yescharacteristic
73C22×D59—42no—
74C22×D59—82no—
75C22×D59—81no—
76C22×D59—82no—
77C22×D59—81no—
78C2×C4×C59—21no—
79C2×C4×C59—22no—
80C2×C4×C59—21no—
81C2×C4×C59—22no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59C22⋊Q812yes—
86C2×C4×C59C22⋊Q811yescharacteristic
87C2×C4×C59#12412yes—
88C2×C4×C59#12411yescharacteristic
89C2×Dic59—42no—
90C2×Dic59—42no—
91C2×Dic59—42no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95D236—44no—
96D236—44no—
97D236—44no—
98D236—82no—
99D236—82no—
100D236—82no—
101C8×C59—44no—
102C59⋊C8—42no—
103#10360—591nomaximal, Sylow
104C22×C59#39311yescharacteristic
105C22×C59#45312yes—
106C22×C59#45311yescharacteristic
107C22×C59#45412yes—
108C22×C59#45411yescharacteristic
109C2×D59—84no—
110C2×D59—82no—
111C2×D59—162no—
112C2×D59—161no—
113C2×D59—161no—
114C4×C59—21no—
115C4×C59—22no—
116C4×C59—22no—
117C4×C59—21no—
118C4×C59—82no—
119Dic59—82no—
120Dic59—82no—
121#2845—591no—
122#1961—592no—
123#1961—591no—
124#1295—592no—
125#1295D5911yescharacteristic, complemented
126C2×C59#165612yes—
127C2×C59#165611yescharacteristic
128C2×C59#165811yescharacteristic
129C2×C59#202712yes—
130C2×C59#202711yescharacteristic
131D59—162no—
132C2×#127—591no—
133C2×#102—1182no—
134C2×#102—1182no—
135C2×#102—1182no—
136C2×#97—591no—
137C2×#97C2×D5912yes—
138#398—1182no—
139#398—1182no—
140#398—1182no—
141#382—592no—
142#382C2×D5911yescharacteristic
143C59#1036011yescharacteristic, complemented, Sylow
144C22×D4—1181no—
145C22×D4—1182no—
146C22×C8—22no—
147C22×C8—1181no—
148#127—592no—
149#127—1182no—
150#127—1181no—
151C2×C4⋊C4—592no—
152C2×C4⋊C4—592no—
153C2×C4⋊C4—1182no—
154C2×C4⋊C4—1182no—
155C2×C4⋊C4—1182no—
156C2×C4⋊C4D23612yes—
157C2×C42C22×D5911yescharacteristic
158#102—2364no—
159#102—2364no—
160#102—2364no—
161#97—22no—
162#97—22no—
163#97—1182no—
164#95—22no—
165#95—1182no—
166#95—1182no—
167#95—1182no—
168C24—2361no—
169C2×D4—1184no—
170C2×D4—1182no—
171C2×D4—2362no—
172C2×D4—2362no—
173C2×D4—2362no—
174C2×D4—2362no—
175C2×D4—2362no—
176C2×D4—2362no—
177C2×D4—2362no—
178C2×D4—2361no—
179C22×C4—22no—
180C22×C4—1182no—
181C22×C4—1182no—
182C22×C4C2×D23611yescharacteristic
183C22×C4D4×D5912yes—
184C2×C8—24no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—1182no—
188C2×C8—2361no—
189C2×C8—2361no—
190C4⋊C4—42no—
191C4⋊C4—4722no—
192C4⋊C4—4722no—
193C4⋊C4—4722no—
194C4⋊C4—2362no—
195C4⋊C4—2362no—
196C42#1174568512yes—
197C42#1174568711yescharacteristic
198C42Q8×D5911yescharacteristic
199C23—4722no—
200C23—4721no—
201C23—4722no—
202C23—4721no—
203C23—2362no—
204C23#1427369711yescharacteristic
205D4—4722no—
206D4—4722no—
207D4—4722no—
208D4—2364no—
209D4—2364no—
210D4—2364no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—2362no—
219C2×C4—2362no—
220C2×C4—2362no—
221C2×C4—2362no—
222C2×C4—2362no—
223C2×C4—2362no—
224C2×C4#1427370512yes—
225C2×C4#1427370711yescharacteristic
226C2×C4#1427371312yes—
227C2×C4#1427371511yescharacteristic
228C8—44no—
229C8—2362no—
230C22—9442no—
231C22—9441no—
232C22—9441no—
233C22—4724no—
234C22—4722no—
235C22#1683731511yescharacteristic
236C22#1683757112yes—
237C22#1683757211yescharacteristic
238C22#1683757612yes—
239C22#1683757711yescharacteristic
240C4—22no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—82no—
245C4—4722no—
246C4—4722no—
247C2—9442no—
248C2#2142339012yes—
249C2#2142339111yescharacteristic
250C2#2142339911yescharacteristic
251C2#2142525112yes—
252C2#2142525311yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1295C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenter#1295D59
hypocenterC59#10360
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
1. #21432371C2semidirect
4. #21421978C2semidirect
5. C59×#1295C2semidirect
125. #1295D59semidirect
143. C59#10360semidirect

Maximal subgroups

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

subgroupindex
#214323712
#214249202
#214249192
#214219782
C59×#12952
#1036059