Finite Group Databasebeta

← #28483733

Subgroups — #28483733

Order 15104.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16837338C211yesmaximal, characteristic
2#21424184C212yesmaximal, complemented
3C2×#16837126C211yesmaximal, characteristic
4#21421974C212yesmaximal, complemented
5C59×#2312C211yesmaximal, characteristic
6C22×C4⋊Dic59C2211yescharacteristic
7C22×Dic59⋊C4—21no—
8C22×C22⋊Dic59—21no—
9C2×C22⋊(C59⋊C8)C2212yes—
10#16837338—22no—
11#16837338C414yescomplemented
12C2×C59⋊#95C2211yescharacteristic
13#16837164—24no—
14#16837126—22no—
15C22×C22⋊C4×C59C2211yescharacteristic
16C59×#388—24no—
17C59×#380C2212yes—
18C59×#379—21no—
19C4×C22⋊C4×C59—21no—
20C23×Dic59—21no—
21C23×Dic59—21no—
22C22×C59⋊C8—22no—
23C2×C4⋊Dic59—22no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—24no—
28C2×Dic59⋊C4—44no—
29C2×C22⋊Dic59—24no—
30C2×C22⋊Dic59—44no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—21no—
34C59⋊#95C2×C412yes—
35C25×C59D411yescharacteristic
36C23×C4×C59D411yescharacteristic
37C23×C4×C59C2311yescharacteristic
38C2×C4⋊C4×C59D414yes—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59C2×C412yes—
45C2×C22⋊C4×C59D412yes—
46C2×C42×C59—22no—
47#95×C59—22no—
48#95×C59—22no—
49C24×C59—21no—
50C24×C59—22no—
51C24×C59—21no—
52C24×C59—22no—
53C24×C59—42no—
54C24×C59—42no—
55C24×C59C22⋊C412yes—
56C24×C59C2×D411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—24no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C412yes—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D412yes—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59#4112yes—
67C22×Dic59—24no—
68C22×Dic59—22no—
69C22×Dic59—22no—
70C22×Dic59—42no—
71C22×Dic59—42no—
72C22×Dic59—42no—
73C22×Dic59—42no—
74C22×Dic59—41no—
75C22×Dic59—41no—
76C4⋊C4×C59—44no—
77C22⋊C4×C59—24no—
78C22⋊C4×C59—24no—
79C22⋊C4×C59—42no—
80C22⋊C4×C59—42no—
81C22⋊C4×C59—82no—
82C22⋊C4×C59—82no—
83C42×C59—82no—
84C22⋊Dic59—48no—
85C4⋊Dic59—24no—
86C4⋊Dic59—42no—
87Dic59⋊C4—48no—
88C2×C59⋊C8—44no—
89C2×C59⋊C8—42no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—42no—
96C23×C59—42no—
97C23×C59—48no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—81no—
103C23×C59—81no—
104C23×C59—84no—
105C23×C59C2×C22⋊C411yescharacteristic
106C23×C59#12011yescharacteristic
107C23×C59#12311yescharacteristic
108C23×C59#13512yes—
109C23×C59#13712yes—
110C2×C4×C59—24no—
111C2×C4×C59—22no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—44no—
118C2×C4×C59—44no—
119C2×C4×C59—44no—
120C2×C4×C59—81no—
121C2×C4×C59C4×D412yes—
122C2×C4×C59#12112yes—
123C2×Dic59—24no—
124C2×Dic59—48no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—82no—
131C2×Dic59—82no—
132C59⋊C8—82no—
133#9024—591nomaximal, Sylow
134C22×C59—24no—
135C22×C59—24no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—44no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59—82no—
145C22×C59—84no—
146C22×C59—84no—
147C22×C59#38611yescharacteristic
148C22×C59#41611yescharacteristic
149C22×C59#41911yescharacteristic
150C22×C59#45212yes—
151C22×C59#45312yes—
152C22×C59#45812yes—
153C22×C59#48512yes—
154C4×C59—22no—
155C4×C59—22no—
156C4×C59—42no—
157C4×C59—84no—
158Dic59—44no—
159Dic59—82no—
160C2×#399—591no—
161#2312D5911yescharacteristic
162#1824—592no—
163C2×#323—591no—
164#1294—592no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—22no—
168C2×C59—82no—
169C2×C59#161814yes—
170C2×C59#187311yescharacteristic
171C2×C59#188711yescharacteristic
172C2×C59#189011yescharacteristic
173C22×C4⋊C4—591no—
174C22×C4⋊C4—1181no—
175C22×C22⋊C4—1181no—
176C22×C22⋊C4C2×D5911yescharacteristic
177C2×C22⋊C8—592no—
178#399—594no—
179#399—1182no—
180#388—24no—
181#380C2×D5912yes—
182#379—21no—
183C4×C22⋊C4—21no—
184C2×#95—591no—
185#340—1184no—
186#323—1182no—
187C59#902411yescharacteristic, complemented, Sylow
188C25C22⋊D5911yescharacteristic
189C23×C4—1181no—
190C23×C4—1181no—
191C23×C4C22⋊D5911yescharacteristic
192C23×C4C22×D5911yescharacteristic
193C22×C8—1182no—
194C2×C4⋊C4—592no—
195C2×C4⋊C4—1182no—
196C2×C4⋊C4—1182no—
197C2×C4⋊C4—1184no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4—2364no—
200C2×C4⋊C4C22⋊D5914yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1184no—
207C2×C22⋊C4—2364no—
208C2×C22⋊C4C4×D5912yes—
209C2×C22⋊C4D23612yes—
210C2×C42—22no—
211C22⋊C8—1182no—
212C22⋊C8—1182no—
213#95—22no—
214#95—22no—
215#95—592no—
216#95—1181no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24(C2×D59)⋊C412yes—
224C24C2×C22⋊D5911yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1184no—
231C22×C4—1182no—
232C22×C4—1182no—
233C22×C4—2362no—
234C22×C4—2362no—
235C22×C4—2361no—
236C22×C4—2361no—
237C22×C4—2362no—
238C22×C4—2362no—
239C22×C4(C2×D59)⋊C412yes—
240C22×C4C2×C4×D5911yescharacteristic
241C22×C4C2×D23611yescharacteristic
242C22×C4#1174568512yes—
243C22×C4C2×C22⋊D5912yes—
244C2×C8—2364no—
245C2×C8—2362no—
246C4⋊C4—44no—
247C4⋊C4—1184no—
248C4⋊C4—2368no—
249C4⋊C4—2362no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—2368no—
257C42—82no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×(C2×D59)⋊C411yescharacteristic
274C23#1427369611yescharacteristic
275C23#1427371011yescharacteristic
276C23#1427374712yes—
277C23#1427375712yes—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—1184no—
290C2×C4—4722no—
291C2×C4—4722no—
292C2×C4—2362no—
293C2×C4—2362no—
294C2×C4—2362no—
295C2×C4—2368no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4C4×C22⋊D5912yes—
299C2×C4#1427370112yes—
300C8—4722no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1683728111yescharacteristic
315C22#1683739611yescharacteristic
316C22#1683741011yescharacteristic
317C22#1683756412yes—
318C22#1683757012yes—
319C22#1683759212yes—
320C22#1683774312yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—84no—
325C4—4722no—
326C4—2364no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#2142314414yes—
332C2#2142440811yescharacteristic
333C2#2142449411yescharacteristic
334C2#2142450911yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2312C2
socleC23×C59#120
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2312D59
hypocenterC59#9024
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21424184C2semidirect
4. #21421974C2semidirect
11. #16837338C4semidirect
187. C59#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#168373382
#214241842
C2×#168371262
#214219742
C59×#23122
#902459