Finite Group Databasebeta

← #28499183

Subgroups — #28499183

Order 15104.

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

normal subgroups only

This group has 484 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#21429662C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic, complemented
3#21422122C212yesmaximal, complemented
4#21422098C212yesmaximal, complemented
5C59×#1742C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic
7#16837459—21no—
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837319C2211yescharacteristic
11#16837311C411yescharacteristic
12#16837293—21no—
13#16837281—21no—
14#16837278—21no—
15#16837273—21no—
16#16837270C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C22×M16×C59—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#343—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C4×D59—21no—
26C4×D236—41no—
27C4×D236—41no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37#14273638—41no—
38C59⋊#95—21no—
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—42no—
79C2×C4×D59—42no—
80C2×C4×D59—81no—
81C2×C4×D59—81no—
82M16×C59—24no—
83M16×C59—44no—
84M16×C59—42no—
85C2×C8×C59—44no—
86C2×C8×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95C22⋊Dic59—81no—
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—
102C4⋊Dic59—41no—
103Dic59⋊C4—81no—
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—22no—
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—41no—
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—42no—
149D236—42no—
150C4×D59—84no—
151C8×C59—44no—
152C59⋊C8—82no—
153#11761—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—22no—
157C22×C59—42no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—81no—
161C22×C59C2×#9911yescharacteristic
162C22×C59#41711yescharacteristic
163C22×C59C2×#10411yescharacteristic
164C2×D59—42no—
165C2×D59—42no—
166C2×D59—42no—
167C2×D59—81no—
168C2×D59—81no—
169C2×D59—81no—
170C2×D59—81no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—41no—
179Dic59—81no—
180Dic59—81no—
181Dic59—81no—
182#2598—591no—
183#1742D5911yescharacteristic, complemented
184#1464—591no—
185#1353—592no—
186#1345—592no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—42no—
190C2×C59#150411yescharacteristic
191C2×C59#212011yescharacteristic
192C2×C59#213411yescharacteristic
193D59—82no—
194C22×M16—21no—
195C2×C4×D4—591no—
196C2×#117C2×D5911yescharacteristic
197#432—1181no—
198#400—591no—
199#399—591no—
200#394—591no—
201#392—591no—
202#388—1181no—
203#386—1181no—
204#386—1181no—
205#385—1181no—
206#383—591no—
207#343—21no—
208#343C2×D5912yes—
209#324—592no—
210#323—1181no—
211C59#1176111yescharacteristic, complemented, Sylow
212C22×D4—591no—
213C22×D4—1181no—
214C23×C4—1181no—
215C23×C4C22⋊D5911yescharacteristic
216C2×M16—24no—
217C2×M16—22no—
218C2×M16—41no—
219C2×M16—42no—
220C22×C8—41no—
221C4×D4—2361no—
222C4×D4—2361no—
223#117—41no—
224#117—41no—
225C2×C4⋊C4—591no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4C22⋊D5911yescharacteristic
228C2×C4⋊C4C22×D5911yescharacteristic
229C2×C22⋊C4—1181no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4—1181no—
232C2×C22⋊C4—1181no—
233C2×C22⋊C4—1181no—
234C2×C22⋊C4C4×D5911yescharacteristic
235C2×C22⋊C4D23611yescharacteristic
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—
246#95—1181no—
247C24—1181no—
248C24(C2×D59)⋊C411yescharacteristic
249C2×D4—4721no—
250C2×D4—2361no—
251C2×D4—2362no—
252C2×D4—2362no—
253C2×D4—2362no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—4721no—
263C22×C4—4721no—
264C22×C4—2362no—
265C22×C4(C2×D59)⋊C411yescharacteristic
266C22×C4(C2×D59)⋊C411yescharacteristic
267C22×C4(C2×D59)⋊C411yescharacteristic
268C22×C4C2×C4×D5911yescharacteristic
269C22×C4C2×D23611yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271M16—24no—
272M16—44no—
273M16—42no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—2362no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—4721no—
280C4⋊C4—2361no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—4721no—
284C22⋊C4—4721no—
285C22⋊C4—2362no—
286C22⋊C4—2361no—
287C22⋊C4—2361no—
288C22⋊C4—2361no—
289C22⋊C4—2361no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—1182no—
299C23—2361no—
300C23—2361no—
301C23—2361no—
302C23—2361no—
303C23—2361no—
304C23—2361no—
305C23#1427365512yes—
306C23C2×(C2×D59)⋊C411yescharacteristic
307D4—4724no—
308D4—2362no—
309D4—2362no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—4724no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4#1427364111yescharacteristic
337C2×C4#1427364111yescharacteristic
338C2×C4#1427375311yescharacteristic
339C2×C4#1427375811yescharacteristic
340C8—44no—
341C8—4722no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22—4721no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22—2362no—
354C22—2362no—
355C22—2362no—
356C22C2×#1427364111yescharacteristic
357C22#1683740111yescharacteristic
358C22C2×#1427365511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—4721no—
368C4—4721no—
369C4—4721no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—4722no—
374C2#2142259811yescharacteristic
375C2#2142573811yescharacteristic
376C2#2142578711yescharacteristic
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#11761
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. #21422122C2semidirect
4. #21422098C2semidirect
5. C59×#1742C2semidirect
183. #1742D59semidirect
211. C59#11761semidirect

Maximal subgroups

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

subgroupindex
#214296622
#214224302
#214221222
#214220982
C59×#17422
#1176159