Finite Group Databasebeta

← #28507333

Subgroups — #28507333

Order 15104.

484 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 107 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#21428413C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422215C214yesmaximal
4C4×#100×C59C211yesmaximal, characteristic
5C22×C4⋊Dic59C411yescharacteristic, complemented
6C2×#14273642C2212yes—
7#16837341—21no—
8#16837338—21no—
9#16837300—21no—
10#16837282C2212yes—
11#16837274C412yescomplemented
12#16837262—21no—
13#16837262—21no—
14#16837262C411yescharacteristic, complemented
15#16837169—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#100×C59C2212yes—
18C4×M16×C59—21no—
19C59×#343—21no—
20C23×Dic59C2×C412yes—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—41no—
25C2×C4⋊Dic59—41no—
26C2×C4⋊Dic59—41no—
27C2×Dic59⋊C4—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59C2×C412yes—
31#14273642—24no—
32C59⋊#95—22no—
33C59⋊#95—21no—
34C59⋊#95—22no—
35C59⋊#95—41no—
36C22×D4×C59C2311yescharacteristic
37C23×C4×C59C2×C411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C2×M16×C59—22no—
40C4×D4×C59—41no—
41C4×D4×C59—41no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C42×C59D411yescharacteristic
46#100×C59—22no—
47#100×C59Q814yes—
48#97×C59—41no—
49C4×C8×C59—41no—
50C24×C59C22×C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59C2×Q812yes—
55C2×D4×C59#4112yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C22×Dic59—82no—
72M16×C59—24no—
73M16×C59—42no—
74C2×C8×C59—42no—
75C4⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C42×C59—21no—
79C42×C59—21no—
80C22⋊Dic59—42no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59—82no—
84C4⋊Dic59—42no—
85C4⋊Dic59—42no—
86C4⋊Dic59—41no—
87C4⋊Dic59—41no—
88C4⋊Dic59—82no—
89Dic59⋊C4—82no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59C2×C22⋊C411yescharacteristic
99C23×C59C4×Q812yes—
100C23×C59#12411yescharacteristic
101C23×C59C4⋊Q811yescharacteristic
102D4×C59—42no—
103D4×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—81no—
117C2×C4×C59#9911yescharacteristic
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59C22⋊Q812yes—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—84no—
127C2×Dic59—82no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C8×C59—44no—
131#13292—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—22no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—81no—
146C22×C59#39111yescharacteristic
147C22×C59C2×#9911yescharacteristic
148C22×C59#41011yescharacteristic
149C4×C59—21no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59#35212yes—
155Dic59—82no—
156Dic59—82no—
157Dic59—162no—
158#2487—591no—
159#1799—591no—
160C4×#100D5911yescharacteristic
161#1390—594no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—41no—
165C2×C59—42no—
166C2×C59—41no—
167C2×C59#199611yescharacteristic
168C2×C59C2×#35211yescharacteristic
169C2×C59#212211yescharacteristic
170C2×C4×D4C2×D5911yescharacteristic
171C22×C4⋊C4—591no—
172C2×#100C2×D5912yes—
173C2×#99—592no—
174C4×M16—21no—
175#400—1181no—
176#399—1181no—
177#389—1181no—
178#386—592no—
179#385—592no—
180#380—591no—
181#380—1181no—
182#380—1181no—
183#343—21no—
184#342—1181no—
185C59#1329211yescharacteristic, complemented, Sylow
186C22×D4C22×D5911yescharacteristic
187C23×C4—592no—
188C23×C4C4×D5911yescharacteristic
189C23×C4D23611yescharacteristic
190C2×M16—22no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—1182no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—2361no—
198C2×C4⋊C4—2361no—
199C2×C4⋊C4—2361no—
200C2×C4⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4—592no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4—1182no—
204C2×C22⋊C4C4×D5911yescharacteristic
205C2×C22⋊C4D23611yescharacteristic
206C2×C42C22⋊D5911yescharacteristic
207#100—22no—
208#100Dic11814yes—
209#99—1184no—
210#97—41no—
211C4×C8—41no—
212#95—1181no—
213#95—1182no—
214#95—1182no—
215#95—2361no—
216C24C2×C4×D5911yescharacteristic
217C24C2×D23611yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×Dic11812yes—
221C2×D4#1174568512yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—1182no—
229C22×C4—1182no—
230C22×C4—4722no—
231C22×C4—2362no—
232C22×C4—2362no—
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4C2×C22⋊D5911yescharacteristic
238M16—24no—
239M16—42no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—4722no—
243C4⋊C4—4722no—
244C4⋊C4—2362no—
245C4⋊C4—2362no—
246C4⋊C4—2361no—
247C4⋊C4—2361no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—4722no—
251C22⋊C4—2362no—
252C22⋊C4—2362no—
253C22⋊C4—2362no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D59)⋊C411yescharacteristic
265C23C4×Dic11812yes—
266C23#1427371511yescharacteristic
267C23#1427372711yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—4722no—
284C2×C4—4722no—
285C2×C4—4722no—
286C2×C4—4724no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4#1427364111yescharacteristic
294C2×C4#1427364111yescharacteristic
295C2×C4#1427370612yes—
296C8—44no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#1683730811yescharacteristic
312C22C2×#1427364111yescharacteristic
313C22#1683737511yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—9442no—
320C4—4722no—
321C4—4722no—
322C4#1683719312yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2142508111yescharacteristic
329C2C2×#1683719311yescharacteristic
330C2#2142574711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#100×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#100D59
hypocenterC59#13292
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic59C4semidirect
11. #16837274C4semidirect
14. #16837262C4semidirect
185. C59#13292semidirect

Maximal subgroups

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

subgroupindex
#214284132
#214240842
#214222152
C4×#100×C592
#1329259