Finite Group Databasebeta

← #28501254

Subgroups — #28501254

Order 15104.

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

normal subgroups only

This group has 412 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#21425347C211yesmaximal, characteristic
2#21423115C211yesmaximal, characteristic
3#21423115C211yesmaximal, characteristic
4#21422120C211yesmaximal, characteristic, complemented
5#21422120C211yesmaximal, characteristic, complemented
6#21422044C211yesmaximal, characteristic, complemented
7C59×#2807C211yesmaximal, characteristic, complemented
8#16837964C2211yescharacteristic
9#16837720—21no—
10#16837720—21no—
11C2×#14273641—21no—
12#16837281—21no—
13#16837169C2211yescharacteristic
14#16837135—21no—
15#16837135—21no—
16#16837126C2211yescharacteristic
17#16837126C2211yescharacteristic
18C59×#400—21no—
19C59×#399C2211yescharacteristic
20C59×#399C2211yescharacteristic
21C59×#398—21no—
22C59×#389—21no—
23C59×#389—21no—
24C59×#388—21no—
25C59×#385—21no—
26C59×#385C2211yescharacteristic
27C22×C22⋊D59—21no—
28#14273711—41no—
29#14273701—21no—
30#14273701—21no—
31#14273697—41no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—41no—
34#14273658—41no—
35#14273658—41no—
36#14273647—41no—
37#14273647—41no—
38#14273641—42no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C59⋊#95—41no—
42C23×C4×C59D411yescharacteristic
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59D411yescharacteristic
62#95×C59D411yescharacteristic
63C24×C59C2×D411yescharacteristic
64C23×D59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—41no—
72C22×C4×C59—81no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—41no—
76C2×C22⋊D59—41no—
77C2×C22⋊D59—41no—
78C2×C22⋊D59—41no—
79C2×C22⋊D59—41no—
80C2×C22⋊D59—41no—
81C22×Dic59—21no—
82C22×Dic59—41no—
83C2×D236—21no—
84C2×D236—21no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—81no—
90C4⋊C4×C59—81no—
91C4⋊C4×C59—81no—
92C4⋊C4×C59—81no—
93C22⋊C4×C59—21no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—42no—
96C22⋊C4×C59—81no—
97C22⋊C4×C59SD1612yes—
98C42×C59—81no—
99C22⋊Dic59—41no—
100C22⋊Dic59—41no—
101(C2×D59)⋊C4—41no—
102(C2×D59)⋊C4—41no—
103(C2×D59)⋊C4—82no—
104C4⋊Dic59—21no—
105C4⋊Dic59—21no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59#12011yescharacteristic
113C23×C59C2×SD1611yescharacteristic
114C23×C59#13711yescharacteristic
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—81no—
119C22×D59—81no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—81no—
144C22⋊D59—81no—
145C22⋊D59—81no—
146C22⋊D59—81no—
147C22⋊D59—81no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153C2×Dic59—81no—
154D236—81no—
155D236—81no—
156C59⋊C8—81no—
157C59⋊C8—81no—
158#12179—591nomaximal, Sylow
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—81no—
168C22×C59#44811yescharacteristic
169C22×C59#45411yescharacteristic
170C22×C59B(4,𝔽2)11yescharacteristic
171C2×D59—82no—
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—161no—
176C4×C59—42no—
177C4×C59—42no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—81no—
181C4×C59—81no—
182C4×C59—81no—
183C4×C59—81no—
184Dic59—81no—
185Dic59—161no—
186#2807D5911yescharacteristic, complemented
187#2044—591no—
188#1613—591no—
189#1613—591no—
190#1352—591no—
191#1352—591no—
192#1326—591no—
193C2×C59—21no—
194C2×C59—21no—
195C2×C59—41no—
196C2×C59—41no—
197C2×C59#159911yescharacteristic
198C2×C59#219811yescharacteristic
199C2×C59#220011yescharacteristic
200D59—161no—
201#534—591no—
202#481—1181no—
203#481—1181no—
204C2×#99—1181no—
205#400—21no—
206#399C2×D5911yescharacteristic
207#399C2×D5911yescharacteristic
208#398—21no—
209#389—21no—
210#389—21no—
211#388—21no—
212#386—1181no—
213#385—21no—
214#385C2×D5911yescharacteristic
215#342—591no—
216#327—1181no—
217#327—1181no—
218#323—591no—
219#323—591no—
220C59#1217911yescharacteristic, complemented, Sylow
221C22×D4—1181no—
222C23×C4C22⋊D5911yescharacteristic
223#123—2361no—
224#121—1181no—
225#121—1181no—
226#120—2361no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22⋊D5911yescharacteristic
234C2×C4⋊C4C22⋊D5911yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4—2361no—
239C2×C22⋊C4C22⋊D5911yescharacteristic
240C2×C22⋊C4C22×D5911yescharacteristic
241C2×C42—21no—
242#106—2361no—
243#106—2361no—
244#102—2361no—
245#102—2361no—
246#99—2362no—
247C22⋊C8—1181no—
248C22⋊C8—1181no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—2361no—
255#95C22⋊D5911yescharacteristic
256#95C22⋊D5911yescharacteristic
257C24—1181no—
258C24C2×C22⋊D5911yescharacteristic
259C2×D4—1181no—
260C2×D4—1181no—
261C2×D4—2361no—
262C2×D4—2361no—
263C2×D4—2361no—
264C2×D4—2361no—
265C2×D4—2361no—
266C2×D4—2361no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—81no—
275C22×C4—1181no—
276C22×C4—2361no—
277C22×C4C2×C22⋊D5911yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279C2×C8—2361no—
280C2×C8—2361no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—1181no—
290C4⋊C4—1181no—
291C22⋊C4—21no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—4722no—
296C22⋊C4—2361no—
297C22⋊C4—2361no—
298C22⋊C4—2361no—
299C22⋊C4—2361no—
300C22⋊C4#1174567112yes—
301C42—81no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—4721no—
307C23—4721no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23#1427369611yescharacteristic
312C23C2×#1174567111yescharacteristic
313C23#1427375811yescharacteristic
314D4—4721no—
315D4—4721no—
316D4—4721no—
317D4—4721no—
318D4—4721no—
319D4—4721no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4—4721no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C8—4721no—
351C8—4721no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—9441no—
362C22—4722no—
363C22—4721no—
364C22—4721no—
365C22—4721no—
366C22#1683754111yescharacteristic
367C22#1683757511yescharacteristic
368C22#1683758911yescharacteristic
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—9441no—
378C4—4721no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—9441no—
384C2#2142303611yescharacteristic
385C2#2142609211yescharacteristic
386C2#2142610311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2807C2
socleC22×C59B(4,𝔽2)
prime radical#16837964C22
hypercenter#2807D59
hypocenterC59#12179
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2807

Complements

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

NH ≅ G/Ntype
4. #21422120C2semidirect
5. #21422120C2semidirect
6. #21422044C2semidirect
7. C59×#2807C2semidirect
186. #2807D59semidirect
220. C59#12179semidirect

Maximal subgroups

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

subgroupindex
#214253472
#214231152
#214221202
#214220442
C59×#28072
#1217959