Finite Group Databasebeta

← #28499203

Subgroups — #28499203

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#21429783C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic, complemented
3#21422125C212yesmaximal, complemented
4#21422098C212yesmaximal, complemented
5C59×#1742C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7#16837459—21no—
8#16837322C2211yescharacteristic
9#16837319C2211yescharacteristic
10#16837314C411yescharacteristic
11#16837311C411yescharacteristic
12#16837302—21no—
13#16837293—21no—
14#16837292—21no—
15#16837281—21no—
16#16837248C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C22×M16×C59—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26#14273715—41no—
27#14273715—41no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×C4×Dic59—21no—
36C4⋊(C59⋊C8)—41no—
37C22⋊(C59⋊C8)—22no—
38#14273638—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C2×M16×C59—24no—
46C2×M16×C59—22no—
47C2×M16×C59—41no—
48C2×M16×C59—42no—
49C22×C8×C59—41no—
50#117×C59—41no—
51#117×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C23×D59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C2×C22⋊D59—42no—
72C2×C22⋊D59—42no—
73C2×C22⋊D59—81no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×D236—41no—
78C2×D236—81no—
79C2×Dic118—41no—
80C2×Dic118—81no—
81M16×C59—24no—
82M16×C59—44no—
83M16×C59—42no—
84C2×C8×C59—44no—
85C2×C8×C59—42no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C22⋊Dic59—81no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—41no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—41no—
100(C2×D59)⋊C4—41no—
101(C2×D59)⋊C4—81no—
102Dic59⋊C4—81no—
103C4×Dic59—42no—
104C2×C59⋊C8—42no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59#10412yes—
110C23×C59C2×C22⋊C411yescharacteristic
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
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—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59#9911yescharacteristic
135C2×C4×C59#9911yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C22⋊D59—84no—
139C2×Dic59—22no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148D236—82no—
149Dic118—82no—
150C8×C59—44no—
151C59⋊C8—82no—
152#11765—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—22no—
156C22×C59—42no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—81no—
160C22×C59C2×#9911yescharacteristic
161C22×C59#41711yescharacteristic
162C22×C59C2×#10411yescharacteristic
163C2×D59—81no—
164C2×D59—81no—
165C2×D59—81no—
166C2×D59—81no—
167C2×D59—81no—
168C2×D59—81no—
169C2×D59—81no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—41no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178Dic59—42no—
179Dic59—81no—
180Dic59—81no—
181#2610—591no—
182#1742D5911yescharacteristic, complemented
183#1464—591no—
184#1354—592no—
185#1345—592no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—42no—
189C2×C59#150411yescharacteristic
190C2×C59#212211yescharacteristic
191C2×C59#213211yescharacteristic
192D59—161no—
193C22×M16—21no—
194C2×#124—591no—
195C2×#117C2×D5911yescharacteristic
196#432—1181no—
197#395—591no—
198#394—591no—
199#393—591no—
200#392—591no—
201#390—1181no—
202#388—1181no—
203#388—1181no—
204#386—1181no—
205#376—591no—
206#343—21no—
207#343C2×D5912yes—
208#324—592no—
209#323—1181no—
210C59#1176511yescharacteristic, complemented, Sylow
211C22×Q8—591no—
212C22×D4—591no—
213C22×D4—1181no—
214C23×C4C22⋊D5911yescharacteristic
215C2×M16—24no—
216C2×M16—22no—
217C2×M16—41no—
218C2×M16—42no—
219C22×C8—41no—
220#124—2361no—
221#124—2361no—
222#117—41no—
223#117—41no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4C22⋊D5911yescharacteristic
226C2×C4⋊C4C22×D5911yescharacteristic
227C2×C22⋊C4—1181no—
228C2×C22⋊C4—1181no—
229C2×C22⋊C4—1181no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4—1181no—
232C2×C22⋊C4—1181no—
233C2×C22⋊C4C4×D5911yescharacteristic
234C2×C22⋊C4D23611yescharacteristic
235C2×C42—1181no—
236C2×C42C4×D5911yescharacteristic
237C2×C42D23611yescharacteristic
238C4⋊C8—2361no—
239C22⋊C8—1182no—
240#97—2361no—
241#95—591no—
242#95—591no—
243#95—1181no—
244#95—1181no—
245#95—1181no—
246C24—1181no—
247C24(C2×D59)⋊C411yescharacteristic
248C2×Q8—4721no—
249C2×Q8—2361no—
250C2×D4—4721no—
251C2×D4—4721no—
252C2×D4—2361no—
253C2×D4—2362no—
254C2×D4—2362no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4(C2×D59)⋊C411yescharacteristic
264C22×C4(C2×D59)⋊C411yescharacteristic
265C22×C4(C2×D59)⋊C411yescharacteristic
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4C2×D23611yescharacteristic
268C22×C4C2×C22⋊D5911yescharacteristic
269M16—24no—
270M16—44no—
271M16—42no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—2362no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—4721no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—4721no—
281C22⋊C4—4721no—
282C22⋊C4—2362no—
283C22⋊C4—2362no—
284C22⋊C4—2361no—
285C22⋊C4—2361no—
286C22⋊C4—2361no—
287C22⋊C4—2361no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—2362no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—2361no—
298C23—2361no—
299C23—2361no—
300C23—2361no—
301C23—2361no—
302C23—2361no—
303C23—2361no—
304C23#1427365512yes—
305C23C2×(C2×D59)⋊C411yescharacteristic
306Q8—4722no—
307D4—4722no—
308D4—4724no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—1182no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4#1427364111yescharacteristic
335C2×C4#1427364111yescharacteristic
336C2×C4#1427375311yescharacteristic
337C2×C4#1427375811yescharacteristic
338C8—44no—
339C8—4722no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—4721no—
348C22—4721no—
349C22—4721no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22—4721no—
354C22C2×#1427364111yescharacteristic
355C22#1683740111yescharacteristic
356C22C2×#1427365511yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—4721no—
366C4—4721no—
367C4—2362no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—9441no—
372C2#2142259811yescharacteristic
373C2#2142574711yescharacteristic
374C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1742C2
socleC22×C59C2×#99
prime radical#16837311C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1742D59
hypocenterC59#11765
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. #21422430C2semidirect
3. #21422125C2semidirect
4. #21422098C2semidirect
5. C59×#1742C2semidirect
182. #1742D59semidirect
210. C59#11765semidirect

Maximal subgroups

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

subgroupindex
#214297832
#214224302
#214221252
#214220982
C59×#17422
#1176559