Finite Group Databasebeta

← #28505540

Subgroups — #28505540

Order 15104.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#21433163C211yesmaximal, characteristic
2#21424480C211yesmaximal, characteristic
3#21424318C212yesmaximal, complemented
4#21422226C212yesmaximal, complemented
5C59×#2132C211yesmaximal, characteristic
6#16838169C2211yescharacteristic
7#16837480—22no—
8D4×C59⋊C8—22no—
9#16837455—21no—
10#16837451—22no—
11#16837419—21no—
12#16837376C2211yescharacteristic
13#16837369—21no—
14#16837369C2212yes—
15#16837166—22no—
16#16837162—21no—
17C59×#532C2211yescharacteristic
18C59×#413C2212yes—
19C59×#356C412yescomplemented
20C59×#355C412yescomplemented
21#14273734—42no—
22C2×#11745664—21no—
23C2×#11745664—21no—
24C2×#11745664—22no—
25C2×#11745664D412yes—
26C22×C59⋊C8—21no—
27C22×C59⋊C8D412yes—
28#14273686—21no—
29#14273663—44no—
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)—21no—
32#14273654—42no—
33#14273642—22no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C4×C59⋊C8—22no—
38C2×#41×C59D411yescharacteristic
39C2×#41×C59C2311yescharacteristic
40C2×M16×C59—22no—
41C4⋊Q8×C59C2×C411yescharacteristic
42#127×C59Q811yescharacteristic
43#124×C59C2×C411yescharacteristic
44#124×C59Q811yescharacteristic
45#121×C59—22no—
46C4×D4×C59—22no—
47C2×C42×C59D411yescharacteristic
48#101×C59C2×C414yes—
49#100×C59—22no—
50#41×C59—41no—
51#41×C59—41no—
52#41×C59—41no—
53#41×C59—41no—
54C2×Q8×C59C4⋊C411yescharacteristic
55C2×Q8×C59C22×C411yescharacteristic
56C2×D4×C59—22no—
57C2×D4×C59—21no—
58C2×D4×C59C4⋊C411yescharacteristic
59C2×D4×C59C2×Q811yescharacteristic
60C2×D4×C59#4112yes—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65M16×C59—24no—
66M16×C59—42no—
67C2×C8×C59—42no—
68C4⋊C4×C59—22no—
69C22⋊C4×C59—22no—
70C22⋊C4×C59—22no—
71C42×C59—21no—
72C42×C59C4⋊C411yescharacteristic
73C42×C59C4⋊C411yescharacteristic
74C22⋊Dic59—41no—
75C4⋊Dic59—41no—
76C4×Dic59—41no—
77#11745664—22no—
78#11745664—24no—
79#11745664—41no—
80#11745664—42no—
81#11745664—81no—
82C2×C59⋊C8—22no—
83C2×C59⋊C8—22no—
84C2×C59⋊C8—42no—
85C2×C59⋊C8—41no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—81no—
88C23×C59—21no—
89C23×C59C22⋊Q812yes—
90C23×C59#12311yescharacteristic
91Q8×C59—41no—
92Q8×C59—41no—
93D4×C59—41no—
94D4×C59—42no—
95D4×C59—41no—
96D4×C59—44no—
97D4×C59—81no—
98C2×C4×C59—21no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—42no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59C2×C4⋊C411yescharacteristic
109C2×C4×C59C4×D412yes—
110C2×C4×C59#12011yescharacteristic
111C2×Dic59—81no—
112C8×C59—44no—
113C59⋊C8—42no—
114C59⋊C8—42no—
115C59⋊C8—42no—
116#12975—591nomaximal, Sylow
117C22×C59—21no—
118C22×C59—22no—
119C22×C59—41no—
120C22×C59—42no—
121C22×C59—42no—
122C22×C59—41no—
123C22×C59#38511yescharacteristic
124C4×C59—22no—
125C4×C59—22no—
126C4×C59—21no—
127C4×C59—41no—
128C4×C59—41no—
129C4×C59B(4,𝔽2)11yescharacteristic
130C4×C59#45811yescharacteristic
131Dic59—161no—
132#2979—591no—
133#2132D5911yescharacteristic
134#1884—591no—
135#1858—592no—
136#1395—592no—
137C2×C59—21no—
138C2×C59—41no—
139C2×C59—42no—
140C2×C59—81no—
141C2×C59#186711yescharacteristic
142#568—591no—
143#532C2×D5911yescharacteristic
144#436—1182no—
145C8×D4—1182no—
146#431—1181no—
147#430—1182no—
148#421—1181no—
149#413C2×D5912yes—
150#410—591no—
151#408—592no—
152#408—1181no—
153#356Dic5912yescomplemented
154#355Dic5912yescomplemented
155#341—1182no—
156#339—1181no—
157C59#1297511yescharacteristic, complemented, Sylow
158C2×#41D23611yescharacteristic
159C2×#41C22×D5911yescharacteristic
160#131—2362no—
161C2×M16—22no—
162C2×M16—592no—
163C2×M16—1181no—
164C2×M16—1181no—
165C2×M16—1182no—
166C22×C8—592no—
167C22×C8—1181no—
168C4⋊Q8C2×Dic5911yescharacteristic
169#127Dic11811yescharacteristic
170#124Dic11811yescharacteristic
171#124C2×Dic5911yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1181no—
175C2×C42D23611yescharacteristic
176#108—2364no—
177C4⋊C8—1181no—
178C4⋊C8—1181no—
179#104—2362no—
180#101C2×Dic5914yes—
181#100—22no—
182#99—1182no—
183C22⋊C8—1182no—
184C22⋊C8—1182no—
185C22⋊C8—1182no—
186C4×C8—1182no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊Dic5911yescharacteristic
192C2×Q8C22×Dic5911yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊Dic5911yescharacteristic
196C2×D4C2×Dic11811yescharacteristic
197C2×D4#1174568712yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×D23611yescharacteristic
201C22×C4D4×D5912yes—
202M16—24no—
203M16—42no—
204M16—1182no—
205M16—1184no—
206M16—4721no—
207M16—2361no—
208M16—2362no—
209C2×C8—42no—
210C2×C8—1182no—
211C2×C8—1182no—
212C2×C8—4721no—
213C2×C8—2361no—
214C2×C8—2362no—
215C2×C8—2362no—
216C4⋊C4—22no—
217C4⋊C4—2361no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—2361no—
221C42—21no—
222C42—2361no—
223C42C4⋊Dic5911yescharacteristic
224C42C4⋊Dic5911yescharacteristic
225C23—21no—
226C23#1427370412yes—
227C23#1427371111yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—44no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—4721no—
246C2×C4C2×C4⋊Dic5911yescharacteristic
247C2×C4D4×Dic5912yes—
248C2×C4#1427369711yescharacteristic
249C8—44no—
250C8—2362no—
251C8—2362no—
252C8—2362no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1683727711yescharacteristic
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—9441no—
266C4#1683759011yescharacteristic
267C4#1683759311yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2142437911yescharacteristic
subgroupG/H
centerC4#16837590
commutatorC2×Q8×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#2132C2
socleC2×C59#1867
prime radicalC59×#532C22
cosocleC2×#41×C59C23
hypercenter#2132D59
hypocenterC59#12975
LCSG ▷ C2×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #21424318C2semidirect
4. #21422226C2semidirect
19. C59×#356C4semidirect
20. C59×#355C4semidirect
153. #356Dic59semidirect
154. #355Dic59semidirect
157. C59#12975semidirect

Maximal subgroups

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

subgroupindex
#214331632
#214244802
#214243182
#214222262
C59×#21322
#1297559