Finite Group Databasebeta

← #28500946

Subgroups — #28500946

Order 15104.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425142C211yesmaximal, characteristic
2#21423064C212yesmaximal
3#21422126C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#2811C211yesmaximal, characteristic, complemented
6C2×#14273758—21no—
7#16837983C2211yescharacteristic
8#16837541—22no—
9C2×#14273655—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C59×#395—22no—
14C59×#395C2212yes—
15C59×#391—21no—
16C59×#391—21no—
17C59×#391—22no—
18C59×#391C2211yescharacteristic
19C2×#11745685—21no—
20#14273758—42no—
21#14273758—42no—
22C2×#11745672—41no—
23C2×#11745671—41no—
24C2×#11745671—42no—
25C2×#11745664—21no—
26#14273707—41no—
27#14273702—41no—
28#14273701—22no—
29#14273655—42no—
30#14273651—42no—
31C22⋊(C59⋊C8)—22no—
32C22×Q8×C59—21no—
33C22×Q8×C59D411yescharacteristic
34C22×Q8×C59D412yes—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59—21no—
40C2×C42×C59D411yescharacteristic
41#95×C59—22no—
42#95×C59—22no—
43#95×C59—22no—
44#95×C59—22no—
45#95×C59D412yes—
46C2×Q8×C59—24no—
47C2×Q8×C59—21no—
48C2×Q8×C59—42no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59—82no—
52C2×Q8×C59—81no—
53C2×Q8×C59—81no—
54C2×Q8×C59SD1612yescomplemented
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D412yes—
61C2×C22⋊D59—41no—
62#11745685—41no—
63#11745685—41no—
64#11745685—41no—
65#11745685—41no—
66C2×D236—21no—
67C2×D236—22no—
68C2×C4×D59—41no—
69C2×Dic118—21no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—82no—
72C4⋊C4×C59—81no—
73C4⋊C4×C59—81no—
74C42×C59—41no—
75C42×C59—81no—
76#11745672—44no—
77#11745671—44no—
78#11745671—84no—
79(C2×D59)⋊C4—42no—
80C4⋊Dic59—22no—
81#11745664—22no—
82#11745664—41no—
83C2×C59⋊C8—41no—
84C2×C59⋊C8—42no—
85C23×C59#12011yescharacteristic
86C22×D59—41no—
87Q8×C59—42no—
88Q8×C59—44no—
89Q8×C59—81no—
90Q8×C59—84no—
91Q8×C59—82no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—24no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59C2×SD1611yescharacteristic
109C2×C4×C59#13711yescharacteristic
110C22⋊D59—81no—
111C22⋊D59—81no—
112C2×Dic59—41no—
113D236—41no—
114D236—41no—
115D236—82no—
116C4×D59—81no—
117C4×D59—81no—
118Dic118—41no—
119Dic118—41no—
120C59⋊C8—42no—
121C59⋊C8—82no—
122#12130—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59#44811yescharacteristic
126C22×C59#45311yescharacteristic
127C22×C59B(4,𝔽2)11yescharacteristic
128C2×D59—81no—
129C2×D59—81no—
130C2×D59—81no—
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—44no—
135C4×C59—42no—
136C4×C59—42no—
137C4×C59—82no—
138Dic59—81no—
139#2811D5911yescharacteristic, complemented
140#2007—591no—
141#1604—592no—
142#1354—592no—
143#1319—591no—
144C2×C59—21no—
145C2×C59—21no—
146C2×C59#159611yescharacteristic
147C2×C59#219911yescharacteristic
148C2×C59#220511yescharacteristic
149D59—161no—
150C2×#137—1181no—
151#537—591no—
152#448—1182no—
153C2×#104—1181no—
154#395—22no—
155#395C2×D5912yes—
156#391—21no—
157#391—21no—
158#391—22no—
159#391C2×D5911yescharacteristic
160#343—591no—
161#327—1182no—
162#324—592no—
163C59#1213011yescharacteristic, complemented, Sylow
164C2×#41—1181no—
165C22×Q8—21no—
166C22×Q8C22⋊D5911yescharacteristic
167C22×Q8C22⋊D5912yes—
168#137—2362no—
169#137—2362no—
170C2×Q16—2361no—
171C2×SD16—2362no—
172C2×SD16—2361no—
173C2×M16—1181no—
174C22⋊Q8—2361no—
175#121—1182no—
176#121—2361no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22×D5911yescharacteristic
181C2×C42—21no—
182C2×C42C22⋊D5911yescharacteristic
183#104—2362no—
184#103—2362no—
185C22⋊C8—1182no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95C22⋊D5912yes—
191#41—2361no—
192#41—2361no—
193#41—2361no—
194#41—2361no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8—42no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—81no—
202C2×Q8—81no—
203C2×Q8—1181no—
204C2×Q8#1174567112yescomplemented
205C2×D4—1182no—
206C2×D4—1181no—
207C2×D4—2361no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—2361no—
213C22×C4C2×C22⋊D5911yescharacteristic
214C22×C4C2×C22⋊D5912yes—
215Q16—2364no—
216SD16—4724no—
217SD16—2364no—
218M16—1182no—
219M16—2361no—
220C2×C8—2362no—
221C2×C8—2361no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—1182no—
227C22⋊C4—2362no—
228C42—41no—
229C42—81no—
230C23—2361no—
231C23#1427369611yescharacteristic
232Q8—42no—
233Q8—44no—
234Q8—81no—
235Q8—84no—
236Q8—82no—
237Q8—2361no—
238Q8—2361no—
239D4—4722no—
240D4—4721no—
241D4—4721no—
242D4—2361no—
243D4—2361no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—4721no—
261C2×C4—4721no—
262C2×C4—2361no—
263C2×C4C2×#1174567111yescharacteristic
264C2×C4#1427375811yescharacteristic
265C8—4722no—
266C8—2362no—
267C22—21no—
268C22—21no—
269C22—4721no—
270C22—4721no—
271C22—4721no—
272C22#1683754111yescharacteristic
273C22#1683757011yescharacteristic
274C22#1683758911yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—44no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—4721no—
283C2—21no—
284C2—21no—
285C2—9441no—
286C2#2142301911yescharacteristic
287C2#2142609711yescharacteristic
288C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2811C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2811D59
hypocenterC59#12130
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
3. #21422126C2semidirect
4. #21422030C2semidirect
5. C59×#2811C2semidirect
54. C2×Q8×C59SD16semidirect
139. #2811D59semidirect
163. C59#12130semidirect
204. C2×Q8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214251422
#214230642
#214221262
#214220302
C59×#28112
#1213059