Finite Group Databasebeta

← #28483688

Subgroups — #28483688

Order 15104.

800 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16837322C211yesmaximal, characteristic
2#21424030C212yesmaximal
3C2×#16837128C211yesmaximal, characteristic
4#21421974C212yesmaximal
5C59×#2316C211yesmaximal, characteristic
6C23×Dic118C2211yescharacteristic
7C22×Dic59⋊C4—21no—
8C2×#14273650—24no—
9C2×C22⋊(C59⋊C8)C2212yes—
10#16837322—22no—
11#16837322C414yescomplemented
12C2×C59⋊#95—21no—
13C2×C59⋊#95C2211yescharacteristic
14#16837128—22no—
15C2×#117×C59—21no—
16C22×C4⋊C4×C59C2211yescharacteristic
17C59×#389—24no—
18C59×#380—21no—
19C59×#380C2212yes—
20C22×Dic118—22no—
21C22×Dic118—22no—
22C22×Dic118—21no—
23C22×Dic118—44no—
24C22×Dic118C2×C412yes—
25C23×Dic59—21no—
26C23×Dic59—21no—
27C22×C59⋊C8—22no—
28C2×Dic59⋊C4—24no—
29C2×Dic59⋊C4—44no—
30#14273650—48no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—24no—
34C59⋊#95—21no—
35C59⋊#95C2×C412yes—
36C23×C4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59C2311yescharacteristic
39#117×C59—42no—
40#117×C59—42no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—22no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59C2×C412yes—
46C2×C4⋊C4×C59D412yes—
47C2×C4⋊C4×C59D414yes—
48C2×C22⋊C4×C59—22no—
49C2×C42×C59—22no—
50#95×C59—22no—
51#95×C59—22no—
52C24×C59C2×D411yescharacteristic
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
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⋊C412yes—
64C22×C4×C59SD1612yes—
65C22×C4×C59Q1612yes—
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D412yes—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4112yes—
70C22×Dic59—24no—
71C22×Dic59—24no—
72C22×Dic59—22no—
73C22×Dic59—22no—
74C22×Dic59—42no—
75C22×Dic59—42no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C2×Dic118—28no—
81C2×Dic118—24no—
82C2×Dic118—44no—
83C2×Dic118—48no—
84C2×Dic118—44no—
85C2×Dic118—44no—
86C2×Dic118—42no—
87C4⋊C4×C59—24no—
88C4⋊C4×C59—24no—
89C4⋊C4×C59—44no—
90C4⋊C4×C59—44no—
91C4⋊C4×C59—82no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—42no—
94C42×C59—44no—
95Dic59⋊C4—48no—
96C2×C59⋊C8—44no—
97C2×C59⋊C8—42no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59C2×C22⋊C411yescharacteristic
101C23×C59#12011yescharacteristic
102C23×C59#12311yescharacteristic
103C23×C59C2×SD1611yescharacteristic
104C23×C59C2×Q1611yescharacteristic
105C23×C59AGL(1,ℤ/8)11yescharacteristic
106C23×C59#13711yescharacteristic
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—44no—
121C2×C4×C59—44no—
122C2×C4×C59—44no—
123C2×C4×C59—81no—
124C2×C4×C59#10314yescomplemented
125C2×C4×C59C4×D412yes—
126C2×C4×C59#12112yes—
127C2×Dic59—24no—
128C2×Dic59—24no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—44no—
132C2×Dic59—44no—
133C2×Dic59—44no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—44no—
137Dic118—28no—
138Dic118—48no—
139Dic118—44no—
140C59⋊C8—82no—
141#9015—591nomaximal, Sylow
142C22×C59—22no—
143C22×C59—22no—
144C22×C59—22no—
145C22×C59—22no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59#38611yescharacteristic
151C22×C59C2×#10311yescharacteristic
152C22×C59#41711yescharacteristic
153C22×C59#45011yescharacteristic
154C22×C59#45111yescharacteristic
155C22×C59#45612yes—
156C22×C59B(4,𝔽2)12yes—
157C22×C59#48311yescharacteristic
158C22×C59#48411yescharacteristic
159C4×C59—24no—
160C4×C59—22no—
161C4×C59—22no—
162C4×C59—42no—
163C4×C59—84no—
164Dic59—44no—
165Dic59—44no—
166C2×#395—591no—
167#2316D5911yescharacteristic
168#1787—592no—
169C2×#324—591no—
170#1294—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59—21no—
175C2×C59#160412yes—
176C2×C59#160512yes—
177C2×C59#187211yescharacteristic
178C2×C59#188611yescharacteristic
179C2×C59#188911yescharacteristic
180C23×Q8—591no—
181C2×#117—21no—
182C22×C4⋊C4—1181no—
183C22×C4⋊C4C2×D5911yescharacteristic
184C2×#103—1184no—
185C2×C22⋊C8—592no—
186#395—594no—
187#395—1182no—
188#389—24no—
189#380—21no—
190#380C2×D5912yes—
191C2×#95—591no—
192C2×#95—1181no—
193#324—1182no—
194C59#901511yescharacteristic, complemented, Sylow
195C22×Q8—592no—
196C22×Q8—1181no—
197C22×Q8—1182no—
198C22×Q8—1182no—
199C22×Q8—2364no—
200C23×C4—1181no—
201C23×C4—1181no—
202C23×C4C22⋊D5911yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C23×C4C22×D5911yescharacteristic
205C22×C8—1182no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—1184no—
213C2×C4⋊C4—2364no—
214C2×C4⋊C4C4×D5912yes—
215C2×C4⋊C4D23612yes—
216C2×C4⋊C4C22⋊D5914yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219#103—2368no—
220C22⋊C8—1182no—
221C22⋊C8—1182no—
222#95—22no—
223#95—22no—
224#95—592no—
225#95—1184no—
226#95—1181no—
227C24C2×C22⋊D5911yescharacteristic
228C2×Q8—1188no—
229C2×Q8—1184no—
230C2×Q8—2364no—
231C2×Q8—2364no—
232C2×Q8—2364no—
233C2×Q8—2368no—
234C2×Q8—2362no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1184no—
245C22×C4—1184no—
246C22×C4—1182no—
247C22×C4—1182no—
248C22×C4—2362no—
249C22×C4—2362no—
250C22×C4—2361no—
251C22×C4—2361no—
252C22×C4—2361no—
253C22×C4—2361no—
254C22×C4(C2×D59)⋊C412yes—
255C22×C4(C2×D59)⋊C412yes—
256C22×C4#1174567012yes—
257C22×C4#1174567212yes—
258C22×C4C2×C4×D5911yescharacteristic
259C22×C4C2×D23611yescharacteristic
260C22×C4#1174568512yes—
261C22×C4C2×C22⋊D5912yes—
262C2×C8—2364no—
263C2×C8—2362no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—82no—
269C4⋊C4—2368no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—44no—
273C23—22no—
274C23—22no—
275C23C2×(C2×D59)⋊C411yescharacteristic
276C23#1427369611yescharacteristic
277C23#1427371011yescharacteristic
278C23C2×#1174567011yescharacteristic
279C23C2×#1174567211yescharacteristic
280C23#1427375311yescharacteristic
281C23#1427375811yescharacteristic
282Q8—1188no—
283Q8—2368no—
284Q8—2364no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4—1184no—
303C2×C4—1184no—
304C2×C4—2362no—
305C2×C4—2364no—
306C2×C4—2364no—
307C2×C4—2362no—
308C2×C4—2364no—
309C2×C4—2362no—
310C2×C4—2362no—
311C2×C4—2364no—
312C2×C4#1427365014yescomplemented
313C2×C4C4×C22⋊D5912yes—
314C2×C4#1427370112yes—
315C8—4722no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1683728111yescharacteristic
325C22C2×#1427365011yescharacteristic
326C22#1683740111yescharacteristic
327C22#1683755311yescharacteristic
328C22#1683755811yescharacteristic
329C22#1683758512yes—
330C22#1683758912yes—
331C22#1683773211yescharacteristic
332C22#1683773711yescharacteristic
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—84no—
338C4—2364no—
339C4—2364no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2142306512yes—
345C2#2142307012yes—
346C2#2142440311yescharacteristic
347C2#2142448911yescharacteristic
348C2#2142450411yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2316C2
socleC23×C59#120
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2316D59
hypocenterC59#9015
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
11. #16837322C4semidirect
124. C2×C4×C59#103semidirect
194. C59#9015semidirect
312. C2×C4#14273650semidirect

Maximal subgroups

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

subgroupindex
C2×#168373222
#214240302
C2×#168371282
#214219742
C59×#23162
#901559