Finite Group Databasebeta

← #28499211

Subgroups — #28499211

Order 15104.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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#21429576C211yesmaximal, characteristic
2C2×#16837171C211yesmaximal, characteristic, complemented
3#21422124C212yesmaximal, complemented
4#21422100C212yesmaximal, complemented
5C59×#1463C211yesmaximal, characteristic, complemented
6C22×#11745664—21no—
7C2×#14273701C411yescharacteristic
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837319C411yescharacteristic
11#16837319C2211yescharacteristic
12#16837292—21no—
13#16837292—21no—
14#16837281—21no—
15#16837281—21no—
16C4×C22⋊Dic59C411yescharacteristic
17#16837171—21no—
18#16837171C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C2×#11745664—24no—
26C2×#11745664—22no—
27C2×#11745664—41no—
28C2×#11745664—42no—
29C22×C59⋊C8—41no—
30#14273701—41no—
31#14273701—41no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×C22⋊Dic59—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×C4×Dic59—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C2×M16×C59—41no—
50C2×C4⋊C4×C59—41no—
51C2×C4⋊C4×C59—41no—
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C4⋊C4×C59C2311yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C23×D59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—81no—
70C22×C4×C59—81no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—41no—
80C2×C22⋊D59—41no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C2×D236—41no—
85C2×D236—81no—
86M16×C59—82no—
87C2×C8×C59—42no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C22⋊Dic59—42no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—41no—
97(C2×D59)⋊C4—81no—
98(C2×D59)⋊C4—81no—
99C4⋊Dic59—41no—
100C4×Dic59—81no—
101#11745664—24no—
102#11745664—44no—
103#11745664—42no—
104C2×C59⋊C8—44no—
105C2×C59⋊C8—42no—
106C23×C59—22no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59#9911yescharacteristic
111C23×C59#9911yescharacteristic
112C23×C59C2×C22⋊C411yescharacteristic
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—41no—
119C22×D59—41no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—42no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59#10412yes—
141C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C59#13711yescharacteristic
143C22⋊D59—82no—
144C22⋊D59—82no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154D236—82no—
155C8×C59—82no—
156C59⋊C8—44no—
157#11766—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59#35512yes—
168C22×C59C2×#9911yescharacteristic
169C22×C59#41711yescharacteristic
170C22×C59C2×#10411yescharacteristic
171C2×D59—81no—
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C4×C59—21no—
179C4×C59—22no—
180C4×C59—21no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—41no—
185Dic59—81no—
186Dic59—81no—
187Dic59—81no—
188#2591—591no—
189C2×#343—591no—
190#1463D5911yescharacteristic, complemented
191#1353—592no—
192#1345—592no—
193C2×C59—21no—
194C2×C59—21no—
195C2×C59—22no—
196C2×C59—41no—
197C2×C59#150411yescharacteristic
198C2×C59#212111yescharacteristic
199C2×C59C2×#35511yescharacteristic
200D59—161no—
201C22×M16—1181no—
202C2×#121—591no—
203C22×C4⋊C4C2×D5911yescharacteristic
204#407—21no—
205#400—591no—
206#400—591no—
207#394—591no—
208#394—591no—
209#388—1181no—
210#388—1181no—
211#386—1181no—
212#386—1181no—
213C4×C22⋊C4—591no—
214#343—592no—
215#343—1181no—
216#324—21no—
217#324C2×D5912yes—
218C59#1176611yescharacteristic, complemented, Sylow
219C22×D4—591no—
220C22×D4—1181no—
221C23×C4C4×D5911yescharacteristic
222C23×C4D23611yescharacteristic
223C23×C4C22⋊D5911yescharacteristic
224C2×M16—41no—
225C2×M16—1184no—
226C2×M16—1182no—
227C2×M16—2361no—
228C2×M16—2362no—
229C22×C8—2361no—
230#121—2361no—
231#121—2361no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4C4×D5911yescharacteristic
236C2×C4⋊C4D23611yescharacteristic
237C2×C4⋊C4C22⋊D5911yescharacteristic
238C2×C4⋊C4C22×D5911yescharacteristic
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C22⋊C4—1181no—
245C2×C42—1181no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248#95—591no—
249#95—591no—
250#95—1181no—
251#95—1181no—
252#95—1181no—
253#95—1181no—
254C24—1181no—
255C24(C2×D59)⋊C411yescharacteristic
256C2×D4—4721no—
257C2×D4—2361no—
258C2×D4—2362no—
259C2×D4—2362no—
260C2×D4—2361no—
261C2×D4—2361no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—1181no—
274C22×C4—1181no—
275C22×C4—1181no—
276C22×C4(C2×D59)⋊C411yescharacteristic
277C22×C4(C2×D59)⋊C411yescharacteristic
278C22×C4(C2×D59)⋊C411yescharacteristic
279C22×C4C2×C4×D5911yescharacteristic
280C22×C4C2×D23611yescharacteristic
281C22×C4C2×C22⋊D5911yescharacteristic
282M16—82no—
283M16—1184no—
284M16—2364no—
285M16—2362no—
286C2×C8—42no—
287C2×C8—2364no—
288C2×C8—2362no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—2361no—
294C22⋊C4—4721no—
295C22⋊C4—4721no—
296C22⋊C4—2362no—
297C22⋊C4—2362no—
298C22⋊C4—2362no—
299C22⋊C4—2361no—
300C22⋊C4—2361no—
301C42—4721no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—41no—
306C23—2361no—
307C23—2361no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23—2361no—
312C23—2361no—
313C23#1427364111yescharacteristic
314C23#1427364111yescharacteristic
315C23C2×(C2×D59)⋊C411yescharacteristic
316D4—4722no—
317D4—4722no—
318D4—4722no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4#1427365312yes—
349C2×C4#1427375511yescharacteristic
350C2×C4#1427376011yescharacteristic
351C8—82no—
352C8—2364no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—4721no—
363C22—4721no—
364C22—4721no—
365C22—4721no—
366C22—4721no—
367C22—4721no—
368C22—4721no—
369C22#1683719812yes—
370C22C2×#1427364111yescharacteristic
371C22#1683740411yescharacteristic
372C22C2×#1427365311yescharacteristic
373C4—21no—
374C4—22no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—4721no—
381C4—4721no—
382C4—4721no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—9441no—
388C2#2142260111yescharacteristic
389C2#2142574011yescharacteristic
390C2C2×#1683719811yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1463C2
socleC22×C59C2×#99
prime radicalC2×#14273701C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1463D59
hypocenterC59#11766
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. C2×#16837171C2semidirect
3. #21422124C2semidirect
4. #21422100C2semidirect
5. C59×#1463C2semidirect
190. #1463D59semidirect
218. C59#11766semidirect

Maximal subgroups

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

subgroupindex
#214295762
C2×#168371712
#214221242
#214221002
C59×#14632
#1176659