Finite Group Databasebeta

← #28477409

Subgroups — #28477409

Order 15104.

604 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 87 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#21430468C211yesmaximal, characteristic, complemented
2#21425327C212yesmaximal, complemented
3#21421918C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic, complemented
5C59×#2011C211yesmaximal, characteristic
6C2×#14273704—22no—
7C2×#14273648—22no—
8C2×#14273648—22no—
9#16837325—22no—
10#16837322—21no—
11#16837300C2212yes—
12#16837282—21no—
13#16837157C2212yes—
14C22×D8×C59—21no—
15C2×#127×C59C2211yescharacteristic
16C2×#102×C59—21no—
17C59×#390C2211yescharacteristic
18C59×#336C2211yescharacteristic
19C22×Dic118—21no—
20C23×Dic59—21no—
21C22×C59⋊C8—22no—
22#14273704—44no—
23#14273704—44no—
24C2×C4⋊Dic59D412yes—
25C2×C4⋊Dic59D412yes—
26C2×Dic59⋊C4—22no—
27C2×Dic59⋊C4—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—21no—
30#14273648—44no—
31#14273648—44no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C22×D4×C59—22no—
35C22×D4×C59—21no—
36C22×D4×C59D411yescharacteristic
37C2×D8×C59—42no—
38C2×D8×C59—42no—
39C2×D8×C59—42no—
40C22×C8×C59—21no—
41#127×C59—42no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59—22no—
44C2×C42×C59C2311yescharacteristic
45#102×C59—42no—
46C24×C59—21no—
47C24×C59—41no—
48C2×D4×C59—22no—
49C2×D4×C59—24no—
50C2×D4×C59—21no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59—42no—
56C2×D4×C59—42no—
57C2×D4×C59—42no—
58C2×D4×C59—41no—
59C2×D4×C59—42no—
60C2×D4×C59—41no—
61C2×D4×C59—42no—
62C2×D4×C59—41no—
63C2×D4×C59SD1612yes—
64C22×C4×C59—21no—
65C22×C4×C59C2×D412yes—
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—44no—
70C22×Dic59—81no—
71C2×Dic118—44no—
72C2×Dic118—81no—
73D8×C59—48no—
74C2×C8×C59—22no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—42no—
80C42×C59—41no—
81C22⋊Dic59—44no—
82C22⋊Dic59—81no—
83C4⋊Dic59—22no—
84C4⋊Dic59—42no—
85C4⋊Dic59D814yescomplemented
86Dic59⋊C4—44no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89C2×C59⋊C8—42no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—42no—
92C23×C59—22no—
93C23×C59—42no—
94C23×C59—41no—
95C23×C59—42no—
96C23×C59—41no—
97C23×C59—42no—
98C23×C59—82no—
99C23×C59—81no—
100C23×C59—81no—
101C23×C59—82no—
102C23×C59#12011yescharacteristic
103D4×C59—42no—
104D4×C59—42no—
105D4×C59—42no—
106D4×C59—44no—
107D4×C59—44no—
108D4×C59—81no—
109D4×C59—81no—
110D4×C59—81no—
111D4×C59—82no—
112D4×C59—82no—
113C2×C4×C59—21no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59C2×D812yes—
120C2×C4×C59C2×SD1611yescharacteristic
121C2×C4×C59AGL(1,ℤ/8)12yes—
122C2×C4×C59#13711yescharacteristic
123C2×Dic59—24no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—84no—
130Dic118—84no—
131C8×C59—42no—
132C59⋊C8—82no—
133#7738—591nomaximal, Sylow
134C22×C59—42no—
135C22×C59—44no—
136C22×C59—82no—
137C22×C59—81no—
138C22×C59—81no—
139C22×C59—82no—
140C22×C59—84no—
141C22×C59—161no—
142C22×C59—162no—
143C22×C59—161no—
144C22×C59#44712yes—
145C22×C59#44811yescharacteristic
146C22×C59#45311yescharacteristic
147C22×C59#45412yes—
148C22×C59B(4,𝔽2)11yescharacteristic
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—22no—
153C4×C59—81no—
154Dic59—44no—
155Dic59—82no—
156#2683—591no—
157#2041—592no—
158#2011D5911yescharacteristic
159#1271—592no—
160#1270—591no—
161C2×C59—82no—
162C2×C59—162no—
163C2×C59#159611yescharacteristic
164C2×C59#159812yes—
165C2×C59#161911yescharacteristic
166C2×C59#162712yes—
167C2×C59#165711yescharacteristic
168C22×D8—21no—
169C2×#127C2×D5911yescharacteristic
170C2×C22⋊Q8—1182no—
171C2×#102—21no—
172C2×#102—1182no—
173C2×#102—1182no—
174#396—1182no—
175#395—1181no—
176#390C2×D5911yescharacteristic
177#389—592no—
178#386—1181no—
179#336—592no—
180#336C2×D5911yescharacteristic
181C59#773811yescharacteristic, complemented, Sylow
182C22×Q8—1181no—
183C22×D4—22no—
184C22×D4—21no—
185C22×D4C22⋊D5911yescharacteristic
186C23×C4—1181no—
187C2×D8—42no—
188C2×D8—42no—
189C2×D8—42no—
190C22×C8—21no—
191C22×C8—1182no—
192#127—42no—
193C22⋊Q8—2364no—
194C22⋊Q8—2364no—
195C2×C4⋊C4—592no—
196C2×C4⋊C4—592no—
197C2×C4⋊C4—1182no—
198C2×C4⋊C4—1182no—
199C2×C4⋊C4C22⋊D5911yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—1181no—
202C2×C22⋊C4—1182no—
203C2×C42C22×D5911yescharacteristic
204#102—42no—
205#102—2364no—
206#102—2364no—
207#95—1181no—
208#95—1182no—
209C24—21no—
210C24—41no—
211C2×Q8—4721no—
212C2×Q8—2364no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4#1174567012yes—
229C22×C4—21no—
230C22×C4—1182no—
231C22×C4—1182no—
232C22×C4—4721no—
233C22×C4—2364no—
234C22×C4D4×D5912yes—
235C22×C4C2×C22⋊D5911yescharacteristic
236D8—48no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—2362no—
241C2×C8—2362no—
242C2×C8—2362no—
243C4⋊C4—41no—
244C4⋊C4—594no—
245C4⋊C4—1182no—
246C4⋊C4—2362no—
247C4⋊C4—2364no—
248C4⋊C4—2362no—
249C4⋊C4—2362no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—4721no—
253C22⋊C4—2364no—
254C42—41no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—82no—
262C23—81no—
263C23—81no—
264C23—82no—
265C23#1427369511yescharacteristic
266Q8—4724no—
267D4—42no—
268D4—42no—
269D4—42no—
270D4—44no—
271D4—44no—
272D4—81no—
273D4—81no—
274D4—81no—
275D4—82no—
276D4—82no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—1184no—
284C2×C4—4724no—
285C2×C4—2362no—
286C2×C4—2362no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4D8×D5912yes—
291C2×C4C2×#1174567011yescharacteristic
292C2×C4#1427375112yes—
293C2×C4#1427375711yescharacteristic
294C8—42no—
295C8—4722no—
296C22—44no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22—161no—
304C22—161no—
305C22—162no—
306C22#1683753412yes—
307C22#1683754011yescharacteristic
308C22#1683756911yescharacteristic
309C22#1683757312yes—
310C22#1683758811yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—81no—
316C4—4722no—
317C4—2364no—
318C2—82no—
319C2—162no—
320C2#2142301811yescharacteristic
321C2#2142302912yes—
322C2#2142314811yescharacteristic
323C2#2142319812yes—
324C2#2142339211yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#2011C2
socleC23×C59#120
hypercenter#2011D59
hypocenterC59#7738
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. #21430468C2semidirect
2. #21425327C2semidirect
3. #21421918C2semidirect
4. #21421917C2semidirect
85. C4⋊Dic59D8semidirect
181. C59#7738semidirect

Maximal subgroups

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

subgroupindex
#214304682
#214253272
#214219182
#214219172
C59×#20112
#773859