Finite Group Databasebeta

← #28483718

Subgroups — #28483718

Order 15104.

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

normal subgroups only

This group has 812 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×#16837338C211yesmaximal, characteristic
2#21424184C212yesmaximal, complemented
3C2×#16837126C211yesmaximal, characteristic
4#21421974C212yesmaximal, complemented
5C59×#2315C211yesmaximal, 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×C59—21no—
16C22×C22⋊C4×C59C2211yescharacteristic
17C59×#385—24no—
18C59×#380—21no—
19C59×#380C2212yes—
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×C412yescomplemented
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×C412yescomplemented
35C25×C59D411yescharacteristic
36C23×C4×C59—22no—
37C23×C4×C59D411yescharacteristic
38C23×C4×C59C2311yescharacteristic
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59D414yes—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—42no—
46C2×C22⋊C4×C59—42no—
47C2×C22⋊C4×C59C2×C412yes—
48C2×C22⋊C4×C59D412yes—
49#95×C59—22no—
50#95×C59—22no—
51C24×C59—21no—
52C24×C59—21no—
53C24×C59—21no—
54C24×C59—21no—
55C24×C59—42no—
56C24×C59—42no—
57C24×C59C22⋊C412yes—
58C24×C59SD1612yes—
59C24×C59Q1612yes—
60C24×C59C2×D411yescharacteristic
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—24no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59—44no—
67C22×C4×C59—82no—
68C22×C4×C59—82no—
69C22×C4×C59C22⋊C412yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D412yes—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59#4112yes—
74C22×Dic59—24no—
75C22×Dic59—22no—
76C22×Dic59—22no—
77C22×Dic59—42no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C22×Dic59—42no—
81C22×Dic59—42no—
82C22×Dic59—41no—
83C22×Dic59—41no—
84C4⋊C4×C59—44no—
85C4⋊C4×C59—82no—
86C22⋊C4×C59—24no—
87C22⋊C4×C59—24no—
88C22⋊C4×C59—44no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—42no—
91C22⋊Dic59—48no—
92C4⋊Dic59—24no—
93C4⋊Dic59—42no—
94Dic59⋊C4—48no—
95C2×C59⋊C8—44no—
96C2×C59⋊C8—42no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—22no—
103C23×C59—22no—
104C23×C59—22no—
105C23×C59—42no—
106C23×C59—42no—
107C23×C59—44no—
108C23×C59—44no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—82no—
112C23×C59—82no—
113C23×C59—81no—
114C23×C59—81no—
115C23×C59#10314yes—
116C23×C59C2×C22⋊C411yescharacteristic
117C23×C59#12011yescharacteristic
118C23×C59#12311yescharacteristic
119C23×C59C2×SD1611yescharacteristic
120C23×C59C2×Q1611yescharacteristic
121C23×C59AGL(1,ℤ/8)11yescharacteristic
122C23×C59#13711yescharacteristic
123C2×C4×C59—24no—
124C2×C4×C59—22no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—42no—
130C2×C4×C59—44no—
131C2×C4×C59—44no—
132C2×C4×C59—44no—
133C2×C4×C59—81no—
134C2×C4×C59—88no—
135C2×C4×C59C4×D412yes—
136C2×C4×C59#12112yes—
137C2×Dic59—24no—
138C2×Dic59—44no—
139C2×Dic59—44no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—42no—
143C2×Dic59—42no—
144C2×Dic59—42no—
145C2×Dic59—82no—
146C2×Dic59—82no—
147C59⋊C8—82no—
148#9021—591nomaximal, Sylow
149C22×C59—22no—
150C22×C59—22no—
151C22×C59—22no—
152C22×C59—22no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—24no—
158C22×C59—44no—
159C22×C59—44no—
160C22×C59—44no—
161C22×C59—42no—
162C22×C59—42no—
163C22×C59—82no—
164C22×C59—82no—
165C22×C59—82no—
166C22×C59#38611yescharacteristic
167C22×C59C2×#10311yescharacteristic
168C22×C59#41711yescharacteristic
169C22×C59#45011yescharacteristic
170C22×C59#45111yescharacteristic
171C22×C59#45312yes—
172C22×C59#45812yes—
173C22×C59#48311yescharacteristic
174C22×C59#48411yescharacteristic
175C4×C59—22no—
176C4×C59—22no—
177C4×C59—42no—
178C4×C59—84no—
179Dic59—44no—
180Dic59—82no—
181C2×#399—591no—
182#2315D5911yescharacteristic
183#1824—592no—
184C2×#323—591no—
185#1294—592no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—21no—
190C2×C59—44no—
191C2×C59#161312yes—
192C2×C59#161412yes—
193C2×C59#187211yescharacteristic
194C2×C59#188711yescharacteristic
195C2×C59#189011yescharacteristic
196C22×C4⋊C4—591no—
197C22×C4⋊C4—1181no—
198C22×C22⋊C4—21no—
199C22×C22⋊C4—1181no—
200C22×C22⋊C4C2×D5911yescharacteristic
201C2×C22⋊C8—592no—
202#399—594no—
203#399—1182no—
204#385—24no—
205#380—21no—
206#380C2×D5912yes—
207C2×#95—591no—
208#340—1184no—
209#323—1182no—
210C59#902111yescharacteristic, complemented, Sylow
211C25C22⋊D5911yescharacteristic
212C23×C4—22no—
213C23×C4—1181no—
214C23×C4—1181no—
215C23×C4C22⋊D5911yescharacteristic
216C23×C4C22×D5911yescharacteristic
217C22×C8—1182no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—592no—
220C2×C4⋊C4—1182no—
221C2×C4⋊C4—1182no—
222C2×C4⋊C4—1184no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—2364no—
225C2×C4⋊C4C22⋊D5914yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—1184no—
233C2×C22⋊C4—2364no—
234C2×C22⋊C4C4×D5912yes—
235C2×C22⋊C4D23612yes—
236C22⋊C8—1182no—
237C22⋊C8—1182no—
238#95—22no—
239#95—22no—
240#95—592no—
241#95—1181no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24(C2×D59)⋊C412yes—
249C24#1174567012yes—
250C24#1174567212yes—
251C24C2×C22⋊D5911yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—1184no—
261C22×C4—1182no—
262C22×C4—1182no—
263C22×C4—2362no—
264C22×C4—2362no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4—2362no—
268C22×C4—2361no—
269C22×C4—2361no—
270C22×C4(C2×D59)⋊C412yes—
271C22×C4C2×C4×D5911yescharacteristic
272C22×C4C2×D23611yescharacteristic
273C22×C4#1174568512yes—
274C22×C4C2×C22⋊D5912yes—
275C2×C8—2364no—
276C2×C8—2362no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—1184no—
280C4⋊C4—2368no—
281C4⋊C4—2362no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—2368no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—82no—
303C23—82no—
304C23—81no—
305C23—81no—
306C23#1427365014yes—
307C23C2×(C2×D59)⋊C411yescharacteristic
308C23#1427369611yescharacteristic
309C23#1427371011yescharacteristic
310C23C2×#1174567011yescharacteristic
311C23C2×#1174567211yescharacteristic
312C23#1427375311yescharacteristic
313C23#1427375811yescharacteristic
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4—88no—
326C2×C4—1184no—
327C2×C4—4722no—
328C2×C4—4722no—
329C2×C4—2362no—
330C2×C4—2362no—
331C2×C4—2362no—
332C2×C4—2364no—
333C2×C4—2364no—
334C2×C4—2362no—
335C2×C4—2362no—
336C2×C4C4×C22⋊D5912yes—
337C2×C4#1427370112yes—
338C8—4722no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—24no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22#1683728111yescharacteristic
357C22C2×#1427365011yescharacteristic
358C22#1683740111yescharacteristic
359C22#1683755311yescharacteristic
360C22#1683755811yescharacteristic
361C22#1683757012yes—
362C22#1683759212yes—
363C22#1683773211yescharacteristic
364C22#1683773711yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—84no—
369C4—4722no—
370C4—2364no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#2142311612yes—
377C2#2142312112yes—
378C2#2142440311yescharacteristic
379C2#2142449411yescharacteristic
380C2#2142450911yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2315C2
socleC23×C59#120
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2315D59
hypocenterC59#9021
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #21424184C2semidirect
4. #21421974C2semidirect
11. #16837338C4semidirect
26. C2×C4⋊Dic59C2×C4semidirect
34. C59⋊#95C2×C4semidirect
210. C59#9021semidirect

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×#23152
#902159