Finite Group Databasebeta

← #28498243

Subgroups — #28498243

Order 15104.

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

normal subgroups only

This group has 496 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#21428563C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic, complemented
3#21422041C212yesmaximal, complemented
4#21422027C212yesmaximal, complemented
5C2×C59×#324C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273710—21no—
8C2×#14273701C411yescharacteristic
9#16837366—21no—
10#16837337—21no—
11#16837337—21no—
12#16837303C411yescharacteristic
13#16837303C2211yescharacteristic
14#16837271C411yescharacteristic
15#16837271C2211yescharacteristic
16C4×C22⋊Dic59C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C2×#11745664—41no—
26#14273710—44no—
27#14273710—44no—
28#14273701—41no—
29#14273701—41no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×C4×Dic59C2×C411yescharacteristic
42C2×C4×Dic59C2×C411yescharacteristic
43C22⋊(C59⋊C8)—22no—
44C22⋊(C59⋊C8)—41no—
45C59⋊#95—21no—
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×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—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—22no—
67C22×C4×C59—41no—
68C22×C4×C59—42no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—41no—
79C2×C22⋊D59—41no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C2×D236—22no—
84C2×D236—41no—
85C2×D236—41no—
86C2×C8×C59—42no—
87C2×C8×C59—82no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C22⋊Dic59—24no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—42no—
98C4⋊Dic59—41no—
99Dic59⋊C4—42no—
100Dic59⋊C4—42no—
101Dic59⋊C4—42no—
102Dic59⋊C4—42no—
103C4×Dic59—22no—
104C4×Dic59—41no—
105#11745664—82no—
106C2×C59⋊C8—42no—
107C23×C59—22no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—41no—
111C23×C59#9911yescharacteristic
112C23×C59#9911yescharacteristic
113C23×C59C2×C22⋊C411yescharacteristic
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—41no—
119C22×D59—41no—
120C22×D59—41no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
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—42no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59#13112yes—
141C22⋊D59—82no—
142C22⋊D59—82no—
143C2×Dic59—22no—
144C2×Dic59—22no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152D236—82no—
153C8×C59—82no—
154C59⋊C8—82no—
155#11549—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59#35312yes—
166C22×C59#40811yescharacteristic
167C22×C59C2×#9911yescharacteristic
168C22×C59#41311yescharacteristic
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—81no—
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C4×C59—22no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—41no—
182Dic59—42no—
183Dic59—42no—
184Dic59—81no—
185#2501—591no—
186#1463—591no—
187C2×#324D5911yescharacteristic, complemented
188#1325—592no—
189#1318—592no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—22no—
193C2×C59—41no—
194C2×C59#146811yescharacteristic
195C2×C59C2×#35311yescharacteristic
196C2×C59#213311yescharacteristic
197D59—161no—
198C2×#123—1181no—
199C2×#123—1181no—
200C2×#121—591no—
201C22×C4⋊C4C2×D5911yescharacteristic
202#407—1181no—
203C2×C22⋊C8—21no—
204#399—1181no—
205#399—1181no—
206#390—591no—
207#390—591no—
208#384—591no—
209#384—591no—
210C4×C22⋊C4—591no—
211#324—21no—
212#324—592no—
213#324—1181no—
214#324C2×D5912yes—
215C59#1154911yescharacteristic, complemented, Sylow
216C22×D4—591no—
217C22×D4—1181no—
218C23×C4C4×D5911yescharacteristic
219C23×C4D23611yescharacteristic
220C23×C4C22⋊D5911yescharacteristic
221C2×M16—2361no—
222C22×C8—41no—
223#123—2364no—
224#123—2364no—
225#121—2361no—
226#121—2361no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—591no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4—1181no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4C4×D5911yescharacteristic
235C2×C4⋊C4D23611yescharacteristic
236C2×C4⋊C4C22⋊D5911yescharacteristic
237C2×C4⋊C4C22×D5911yescharacteristic
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C42—591no—
245C2×C42—591no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—1182no—
249C22⋊C8—2361no—
250#95—1181no—
251C24—1181no—
252C24(C2×D59)⋊C411yescharacteristic
253C2×D4—1182no—
254C2×D4—2361no—
255C2×D4—2361no—
256C2×D4—2362no—
257C2×D4—2362no—
258C2×D4—2361no—
259C2×D4—2361no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4(C2×D59)⋊C411yescharacteristic
274C22×C4(C2×D59)⋊C411yescharacteristic
275C22×C4(C2×D59)⋊C411yescharacteristic
276C22×C4C2×C4×D5911yescharacteristic
277C22×C4C2×D23611yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279M16—4722no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—2362no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2361no—
288C4⋊C4—2362no—
289C4⋊C4—2362no—
290C4⋊C4—2362no—
291C4⋊C4—2362no—
292C22⋊C4—1184no—
293C22⋊C4—2362no—
294C22⋊C4—2362no—
295C22⋊C4—2362no—
296C22⋊C4—2362no—
297C22⋊C4—2362no—
298C42—1182no—
299C42—2361no—
300C23—21no—
301C23—22no—
302C23—21no—
303C23—41no—
304C23—2361no—
305C23—2361no—
306C23—2361no—
307C23—2361no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23#1427364111yescharacteristic
312C23#1427364111yescharacteristic
313C23C2×(C2×D59)⋊C411yescharacteristic
314D4—4722no—
315D4—4722no—
316D4—4722no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—1182no—
337C2×C4—1182no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4#1427373312yes—
346C8—82no—
347C8—4722no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—4721no—
358C22—4721no—
359C22—4721no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22—4721no—
364C22#1683719412yes—
365C22#1683736811yescharacteristic
366C22C2×#1427364111yescharacteristic
367C22#1683738411yescharacteristic
368C4—22no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—4721no—
375C4—2362no—
376C4—2362no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—9441no—
382C2#2142244211yescharacteristic
383C2C2×#1683719411yescharacteristic
384C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC2×C59×#324C2
socleC22×C59C2×#99
prime radicalC2×#14273701C4
cosocleC2×C4⋊C4×C59C23
hypercenterC2×#324D59
hypocenterC59#11549
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #21422426C2semidirect
3. #21422041C2semidirect
4. #21422027C2semidirect
5. C2×C59×#324C2semidirect
187. C2×#324D59semidirect
215. C59#11549semidirect

Maximal subgroups

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

subgroupindex
#214285632
#214224262
#214220412
#214220272
C2×C59×#3242
#1154959