Finite Group Databasebeta

← #28499074

Subgroups — #28499074

Order 15104.

470 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429980C211yesmaximal, characteristic
2C4×#14273644C211yesmaximal, characteristic, complemented
3#21422124C212yesmaximal, complemented
4#21422100C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic
7C2×#14273644—21no—
8C4×#11745664—21no—
9#16837341C2211yescharacteristic
10#16837338C411yescharacteristic
11#16837319C2211yescharacteristic
12#16837317—21no—
13#16837311C411yescharacteristic
14#16837310—21no—
15#16837293—21no—
16#16837281—21no—
17#16837265C411yescharacteristic
18#16837171C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C2×#11745664—22no—
27#14273701—41no—
28#14273701—41no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37#14273644—24no—
38#14273638—41no—
39C4×C59⋊C8—41no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×D4×C59D411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C22×C8×C59—41no—
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C22⋊C8×C59—22no—
56C22⋊C8×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C23×D59—21no—
60C2×D4×C59—22no—
61C2×D4×C59—22no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—81no—
68C22×C4×C59—81no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C2×C22⋊D59—42no—
75C2×C22⋊D59—42no—
76C2×C22⋊D59—81no—
77C2×C22⋊D59—81no—
78C2×C22⋊D59—81no—
79C2×C22⋊D59—81no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C2×D236—41no—
84C2×D236—81no—
85C2×C8×C59—42no—
86C2×C8×C59—82no—
87C4⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C42×C59—21no—
91C42×C59—21no—
92C22⋊Dic59—81no—
93C22⋊Dic59—81no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—81no—
98(C2×D59)⋊C4—81no—
99C4⋊Dic59—41no—
100Dic59⋊C4—81no—
101#11745664—24no—
102#11745664—42no—
103C2×C59⋊C8—42no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59#10412yes—
114C23×C59C2×C22⋊C411yescharacteristic
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—41no—
119C22×D59—41no—
120C22×D59—41no—
121C22×D59—41no—
122D4×C59—44no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59#9911yescharacteristic
137C2×C4×C59#9911yescharacteristic
138C2×C4×C59#13512yes—
139C22⋊D59—84no—
140C22⋊D59—84no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150D236—82no—
151C8×C59—82no—
152C59⋊C8—44no—
153#11737—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—22no—
157C22×C59—42no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59—81no—
165C22×C59—81no—
166C22×C59C2×#9911yescharacteristic
167C22×C59#41611yescharacteristic
168C22×C59C2×#10411yescharacteristic
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—81no—
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—22no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—41no—
182Dic59—81no—
183Dic59—81no—
184Dic59—81no—
185#2635—591no—
186C4×#100—591no—
187#1459D5911yescharacteristic, complemented
188#1353—592no—
189#1345—592no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—42no—
193C2×C59—81no—
194C2×C59#150111yescharacteristic
195C2×C59#212111yescharacteristic
196C2×C59#213411yescharacteristic
197D59—161no—
198C2×#121—591no—
199C2×C4×D4C2×D5911yescharacteristic
200C2×#100—1181no—
201C2×C22⋊C8—21no—
202C4×M16—1181no—
203#400—591no—
204#399—591no—
205#394—591no—
206#394—1181no—
207#392—591no—
208#392—1181no—
209#388—1181no—
210#386—1181no—
211#381—591no—
212#343—592no—
213#324C2×D5912yes—
214#323—21no—
215C59#1173711yescharacteristic, complemented, Sylow
216C22×D4—591no—
217C22×D4—1181no—
218C22×D4—1181no—
219C22×D4D23611yescharacteristic
220C23×C4C4×D5911yescharacteristic
221C23×C4C22⋊D5911yescharacteristic
222C2×M16—1182no—
223C22×C8—41no—
224#121—2361no—
225#121—2361no—
226C4×D4—41no—
227C4×D4—41no—
228C2×C4⋊C4—591no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4C22×D5911yescharacteristic
231C2×C22⋊C4—1181no—
232C2×C22⋊C4—1181no—
233C2×C22⋊C4—1181no—
234C2×C22⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4C4×D5911yescharacteristic
238C2×C22⋊C4C22⋊D5911yescharacteristic
239C2×C42D23611yescharacteristic
240#100—1184no—
241C22⋊C8—22no—
242C22⋊C8—41no—
243#97—2361no—
244C4×C8—2361no—
245#95—591no—
246#95—591no—
247#95—1181no—
248#95—1181no—
249#95—1181no—
250C24—1181no—
251C24(C2×D59)⋊C411yescharacteristic
252C24(C2×D59)⋊C411yescharacteristic
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—4721no—
257C2×D4—4721no—
258C2×D4—4721no—
259C2×D4—4721no—
260C2×D4—4721no—
261C2×D4—2361no—
262C2×D4—2362no—
263C2×D4—2362no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4(C2×D59)⋊C411yescharacteristic
274C22×C4(C2×D59)⋊C411yescharacteristic
275C22×C4C2×C4×D5911yescharacteristic
276C22×C4C2×D23611yescharacteristic
277C22×C4C2×C22⋊D5911yescharacteristic
278M16—1184no—
279M16—2362no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—2362no—
283C4⋊C4—41no—
284C4⋊C4—4721no—
285C4⋊C4—2361no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—4721no—
289C22⋊C4—4721no—
290C22⋊C4—4721no—
291C22⋊C4—4721no—
292C22⋊C4—2361no—
293C22⋊C4—2361no—
294C22⋊C4—2362no—
295C42—21no—
296C42—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—2361no—
307C23—2361no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23—2361no—
312C23—2361no—
313C23#1427365312yes—
314C23C2×(C2×D59)⋊C411yescharacteristic
315D4—44no—
316D4—4722no—
317D4—4724no—
318D4—4724no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4#1427364111yescharacteristic
342C2×C4#1427364111yescharacteristic
343C2×C4#1427374912yes—
344C8—82no—
345C8—2364no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—4721no—
359C22—4721no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22—4721no—
364C22—4721no—
365C22C2×#1427364111yescharacteristic
366C22#1683739911yescharacteristic
367C22C2×#1427365311yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—4721no—
375C4—4721no—
376C4—4721no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—81no—
381C2—9441no—
382C2#2142258811yescharacteristic
383C2#2142574011yescharacteristic
384C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radical#21429980C2
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11737
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. C4×#14273644C2semidirect
3. #21422124C2semidirect
4. #21422100C2semidirect
5. C59×#1459C2semidirect
187. #1459D59semidirect
215. C59#11737semidirect

Maximal subgroups

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

subgroupindex
#214299802
C4×#142736442
#214221242
#214221002
C59×#14592
#1173759