Finite Group Databasebeta

← #28500916

Subgroups — #28500916

Order 15104.

440 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425227C211yesmaximal, characteristic
2#21423115C211yesmaximal, characteristic
3#21423047C211yesmaximal, characteristic
4#21422120C211yesmaximal, characteristic, complemented
5#21422117C211yesmaximal, characteristic, complemented
6#21422026C211yesmaximal, characteristic, complemented
7C59×#2841C211yesmaximal, characteristic, complemented
8#16837964C2211yescharacteristic
9C2×#14273710—21no—
10#16837720—21no—
11#16837541—21no—
12C2×#14273641—21no—
13#16837169C2211yescharacteristic
14#16837135—21no—
15#16837135—21no—
16#16837126C2211yescharacteristic
17#16837126C2211yescharacteristic
18C59×#400—21no—
19C59×#399C2211yescharacteristic
20C59×#398—21no—
21C59×#398—21no—
22C59×#396—21no—
23C59×#395—21no—
24C59×#393C2211yescharacteristic
25C59×#381C2211yescharacteristic
26C59×#380—21no—
27C22×C22⋊D59—21no—
28C2×#11745671—41no—
29#14273711—41no—
30#14273710—42no—
31#14273710—42no—
32#14273701—21no—
33#14273701—21no—
34#14273697—41no—
35C2×Dic59⋊C4—41no—
36C2×C22⋊Dic59—21no—
37C2×(C2×D59)⋊C4—41no—
38#14273658—41no—
39#14273651—41no—
40#14273647—41no—
41#14273641—42no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C22×Q8×C59D411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59—21no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59C2311yescharacteristic
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59—21no—
63#95×C59D411yescharacteristic
64#95×C59D411yescharacteristic
65C24×C59C2×D411yescharacteristic
66C23×D59—21no—
67C2×Q8×C59—41no—
68C2×Q8×C59—81no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59SD1612yescomplemented
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C2×C22⋊D59—41no—
82C2×C22⋊D59—41no—
83C2×C22⋊D59—41no—
84C2×C22⋊D59—41no—
85C2×C22⋊D59—41no—
86C2×C22⋊D59—41no—
87C22×Dic59—21no—
88C22×Dic59—41no—
89C2×D236—21no—
90C2×D236—21no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—81no—
93C4⋊C4×C59—81no—
94C4⋊C4×C59—81no—
95C4⋊C4×C59—81no—
96C4⋊C4×C59—81no—
97C4⋊C4×C59—81no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—81no—
101C22⋊C4×C59—81no—
102C22⋊Dic59—22no—
103C22⋊Dic59—41no—
104#11745671—82no—
105(C2×D59)⋊C4—41no—
106(C2×D59)⋊C4—41no—
107(C2×D59)⋊C4—44no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110Dic59⋊C4—44no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59#12011yescharacteristic
118C23×C59C2×SD1611yescharacteristic
119C23×C59#13711yescharacteristic
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—81no—
124C22×D59—81no—
125Q8×C59—42no—
126Q8×C59—81no—
127C2×C4×C59—22no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—81no—
151C22⋊D59—81no—
152C22⋊D59—81no—
153C22⋊D59—81no—
154C22⋊D59—81no—
155C2×Dic59—41no—
156C2×Dic59—42no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—81no—
160C2×Dic59—81no—
161D236—81no—
162D236—81no—
163C59⋊C8—81no—
164C59⋊C8—81no—
165#12125—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—81no—
175C22×C59#44811yescharacteristic
176C22×C59#45311yescharacteristic
177C22×C59B(4,𝔽2)11yescharacteristic
178C2×D59—82no—
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C2×D59—161no—
183C4×C59—42no—
184C4×C59—41no—
185C4×C59—41no—
186C4×C59—81no—
187C4×C59—81no—
188C4×C59—81no—
189C4×C59—81no—
190C4×C59—81no—
191Dic59—82no—
192Dic59—81no—
193#2841D5911yescharacteristic, complemented
194#2022—591no—
195#1613—591no—
196#1601—591no—
197#1352—591no—
198#1351—591no—
199#1317—591no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—41no—
203C2×C59—41no—
204C2×C59#159611yescharacteristic
205C2×C59#220011yescharacteristic
206C2×C59#220111yescharacteristic
207D59—161no—
208#534—591no—
209C2×#123—1181no—
210#481—1181no—
211#448—1181no—
212C2×#99—1181no—
213#400—21no—
214#399C2×D5911yescharacteristic
215#398—21no—
216#398—21no—
217#396—21no—
218#395—21no—
219#393C2×D5911yescharacteristic
220#381C2×D5911yescharacteristic
221#380—21no—
222#342—591no—
223#327—1181no—
224#327—1181no—
225#323—591no—
226#323—591no—
227C59#1212511yescharacteristic, complemented, Sylow
228C22×Q8C22⋊D5911yescharacteristic
229C22×D4—1181no—
230C23×C4C22⋊D5911yescharacteristic
231C2×SD16—2361no—
232#123—2361no—
233#123—2362no—
234#123—2362no—
235#121—1181no—
236#121—1181no—
237#120—2361no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—2361no—
245C2×C4⋊C4C22⋊D5911yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—2361no—
250C2×C22⋊C4C22⋊D5911yescharacteristic
251C2×C22⋊C4C22×D5911yescharacteristic
252#106—2361no—
253#103—2361no—
254#102—2361no—
255#99—2362no—
256C22⋊C8—1181no—
257C22⋊C8—1181no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95C22⋊D5911yescharacteristic
265#95C22⋊D5911yescharacteristic
266C24—1181no—
267C24C2×C22⋊D5911yescharacteristic
268C2×Q8—41no—
269C2×Q8—81no—
270C2×D4—1181no—
271C2×D4—1181no—
272C2×D4—2361no—
273C2×D4—2361no—
274C2×D4—2361no—
275C2×D4—2361no—
276C2×D4—2361no—
277C2×D4—2361no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—1181no—
288C22×C4—2361no—
289C22×C4#1174567112yescomplemented
290C22×C4C2×C22⋊D5911yescharacteristic
291C22×C4C2×C22⋊D5911yescharacteristic
292SD16—4722no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—1181no—
303C4⋊C4—1181no—
304C4⋊C4—2364no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—1182no—
310C22⋊C4—2361no—
311C22⋊C4—2361no—
312C22⋊C4—2364no—
313C22⋊C4—2361no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—4721no—
319C23—4721no—
320C23—2361no—
321C23—2361no—
322C23—2361no—
323C23#1427369611yescharacteristic
324C23C2×#1174567111yescharacteristic
325C23#1427375811yescharacteristic
326Q8—42no—
327Q8—81no—
328D4—4721no—
329D4—4721no—
330D4—4721no—
331D4—4721no—
332D4—4721no—
333D4—4721no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—4721no—
359C2×C4—4721no—
360C2×C4—2362no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2361no—
364C8—4721no—
365C8—4721no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—9441no—
376C22—4722no—
377C22—4721no—
378C22—4721no—
379C22—4721no—
380C22#1683754111yescharacteristic
381C22#1683757011yescharacteristic
382C22#1683758911yescharacteristic
383C4—42no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—4722no—
392C4—4721no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—9441no—
398C2#2142301911yescharacteristic
399C2#2142610311yescharacteristic
400C2#2142610911yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2841C2
socleC22×C59B(4,𝔽2)
prime radical#16837964C22
hypercenter#2841D59
hypocenterC59#12125
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2841

Complements

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

NH ≅ G/Ntype
4. #21422120C2semidirect
5. #21422117C2semidirect
6. #21422026C2semidirect
7. C59×#2841C2semidirect
78. C22×C4×C59SD16semidirect
193. #2841D59semidirect
227. C59#12125semidirect
289. C22×C4#11745671semidirect

Maximal subgroups

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

subgroupindex
#214252272
#214231152
#214230472
#214221202
#214221172
#214220262
C59×#28412
#1212559