Finite Group Databasebeta

← #28505522

Subgroups — #28505522

Order 15104.

492 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 95 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#21427208C211yesmaximal, characteristic
2C2×#16837199C211yesmaximal, characteristic
3#21424311C212yesmaximal, complemented
4#21422222C212yesmaximal, complemented
5C59×#1888C211yesmaximal, characteristic
6#16837445—22no—
7C2×#14273646C2212yes—
8#16837309C2212yes—
9#16837295—22no—
10#16837282—21no—
11C22⋊C4×Dic59—22no—
12#16837248—21no—
13#16837199C414yescomplemented
14#16837161—22no—
15C2×#124×C59C2211yescharacteristic
16C59×#519C2211yescharacteristic
17C59×#418—21no—
18C2×#101×C59C2211yescharacteristic
19C59×#339—21no—
20C23×Dic59—21no—
21C2×#11745664—22no—
22C2×C4⋊Dic59D412yes—
23C2×C22⋊Dic59—22no—
24C2×C22⋊Dic59—21no—
25C2×C4×Dic59—22no—
26C2×C4×Dic59D412yes—
27#14273660—42no—
28#14273658—42no—
29#14273646—22no—
30#14273646C2×C414yescomplemented
31C59⋊#95—21no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C22×Q8×C59C2311yescharacteristic
36C22×D4×C59D411yescharacteristic
37C2×M16×C59—21no—
38#124×C59—22no—
39#124×C59C2×C412yes—
40#124×C59Q812yes—
41C4×Q8×C59—42no—
42#117×C59—21no—
43#117×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C42×C59D411yescharacteristic
46#102×C59—42no—
47#101×C59—22no—
48C24×C59—21no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59C22×C411yescharacteristic
52C2×Q8×C59C2×Q811yescharacteristic
53C2×Q8×C59#4112yes—
54C2×D4×C59—21no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C4⋊C412yes—
58C22×C4×C59—21no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D412yes—
61C22×Dic59—22no—
62C22×Dic59—22no—
63C22×Dic59—44no—
64C22×Dic59—81no—
65M16×C59—41no—
66M16×C59—41no—
67C2×C8×C59—41no—
68C4⋊C4×C59—22no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—22no—
71C22⋊C4×C59—24no—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—41no—
74C42×C59—21no—
75C42×C59—42no—
76C42×C59—41no—
77C42×C59C4⋊C412yes—
78C22⋊Dic59—44no—
79C22⋊Dic59—81no—
80C4⋊Dic59—42no—
81C4×Dic59—44no—
82C4×Dic59—42no—
83#11745664—24no—
84#11745664—42no—
85C2×C59⋊C8—42no—
86C23×C59—22no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59#12011yescharacteristic
92Q8×C59—41no—
93Q8×C59—41no—
94Q8×C59—42no—
95D4×C59—42no—
96D4×C59—81no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59C2×C4⋊C411yescharacteristic
105C2×C4×C59C4×D412yes—
106C2×C4×C59C22⋊Q812yes—
107C2×C4×C59#12311yescharacteristic
108C2×Dic59—24no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—84no—
115C8×C59—81no—
116C59⋊C8—44no—
117#12972—591nomaximal, Sylow
118C22×C59—22no—
119C22×C59—44no—
120C22×C59—42no—
121C22×C59—81no—
122C22×C59—81no—
123C22×C59—82no—
124C22×C59#38511yescharacteristic
125C22×C59B(4,𝔽2)11yescharacteristic
126C22×C59#45811yescharacteristic
127C4×C59—22no—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—22no—
131C4×C59—42no—
132C4×C59—82no—
133Dic59—44no—
134Dic59—82no—
135#2372—591no—
136C2×#355—591no—
137#1888D5911yescharacteristic
138#1857—592no—
139#1394—592no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—82no—
143C2×C59#186711yescharacteristic
144C2×C59#220412yes—
145C2×#124C2×D5911yescharacteristic
146#519C2×D5911yescharacteristic
147#428—1182no—
148#418—21no—
149C2×#101—592no—
150C2×#101C2×D5911yescharacteristic
151#391—592no—
152#388—1182no—
153#386—1181no—
154C4×C22⋊C4—1182no—
155#376—1181no—
156#355—594no—
157#339—21no—
158#339—1182no—
159C59#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D5911yescharacteristic
161C22×D4C22⋊D5911yescharacteristic
162C23×C4—1181no—
163C2×M16—21no—
164C2×M16—1182no—
165#124—22no—
166#124Dic11812yes—
167#124C4×D5912yes—
168C4×Q8—42no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—592no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—1181no—
174C2×C22⋊C4—1182no—
175C2×C42—592no—
176C2×C42—1182no—
177C2×C42C22⋊D5911yescharacteristic
178#107—2362no—
179#106—2362no—
180#102—42no—
181#101—22no—
182#101—594no—
183#101—1182no—
184#95—1181no—
185#95—1181no—
186#95—1182no—
187#95—1182no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic11811yescharacteristic
192C2×Q8C2×C4×D5911yescharacteristic
193C2×Q8#1174568712yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4Dic59⋊C412yes—
198C22×C4—21no—
199C22×C4—1182no—
200C22×C4—1182no—
201C22×C4—4721no—
202C22×C4—2364no—
203C22×C4D4×D5912yes—
204C22×C4C2×C22⋊D5911yescharacteristic
205M16—41no—
206M16—41no—
207M16—1184no—
208M16—2362no—
209C2×C8—41no—
210C2×C8—2362no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—2362no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—41no—
218C22⋊C4—4721no—
219C22⋊C4—2364no—
220C42—21no—
221C42—42no—
222C42—41no—
223C42—2362no—
224C42—2364no—
225C42Dic59⋊C412yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1427369511yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—1184no—
245C2×C4—4724no—
246C2×C4—2362no—
247C2×C4—2362no—
248C2×C4—2362no—
249C2×C4—2362no—
250C2×C4—2362no—
251C2×C4C2×Dic59⋊C411yescharacteristic
252C2×C4(C22⋊D59)⋊C412yes—
253C2×C4#1427370412yes—
254C2×C4#1427370811yescharacteristic
255C8—81no—
256C8—2364no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1683727411yescharacteristic
264C22#1683758811yescharacteristic
265C22#1683759111yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—82no—
272C4—4722no—
273C4—2364no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2142437611yescharacteristic
278C2#2142612012yes—
subgroupG/H
centerC22#16837588
commutatorC2×Q8×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#1888C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
cosocleC22×Q8×C59C23
hypercenter#1888D59
hypocenterC59#12972
LCSG ▷ C2×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1888

Complements

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

NH ≅ G/Ntype
3. #21424311C2semidirect
4. #21422222C2semidirect
13. #16837199C4semidirect
30. #14273646C2×C4semidirect
159. C59#12972semidirect

Maximal subgroups

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

subgroupindex
#214272082
C2×#168371992
#214243112
#214222222
C59×#18882
#1297259