Finite Group Databasebeta

← #28484675

Subgroups — #28484675

Order 15104.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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#21433181C211yesmaximal, characteristic, complemented
2#21429956C211yesmaximal, characteristic
3#21422693C212yesmaximal
4#21421961C212yesmaximal, complemented
5C4×#101×C59C211yesmaximal, characteristic, complemented
6C2×#14273715C2211yescharacteristic
7#16837533—24no—
8#16837459—21no—
9#16837459C2212yes—
10#16837455—21no—
11#16837337—21no—
12#16837322C411yescharacteristic
13#16837322C411yescharacteristic
14#16837319C411yescharacteristic
15#16837319C411yescharacteristic
16#16837313—21no—
17#16837293—21no—
18#16837293—21no—
19#16837270C2211yescharacteristic
20#16837141—24no—
21C2×C4×Q8×C59C2211yescharacteristic
22C2×#101×C59—21no—
23C4×M16×C59C2212yes—
24C59×#343—21no—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27#14273715—22no—
28#14273715—22no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—22no—
39C4⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—22no—
41#14273638—22no—
42#14273638D414yes—
43C4×C59⋊C8—22no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95C2×C411yescharacteristic
49C59⋊#95C2×C411yescharacteristic
50C22×Q8×C59D411yescharacteristic
51C2×M16×C59—22no—
52C4×Q8×C59—22no—
53C4×Q8×C59—22no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60#101×C59—24no—
61#97×C59—22no—
62C4×C8×C59—22no—
63C23×D59—21no—
64C2×Q8×C59—22no—
65C2×Q8×C59—22no—
66C2×Q8×C59—41no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×D236—22no—
78C2×D236—81no—
79C2×Dic118—22no—
80C2×Dic118—81no—
81M16×C59—24no—
82M16×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—22no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59C2×D412yes—
92C42×C59#4112yes—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—41no—
98(C2×D59)⋊C4—81no—
99(C2×D59)⋊C4—81no—
100Dic59⋊C4—81no—
101Dic59⋊C4—81no—
102C2×C59⋊C8—24no—
103C23×C59C2×C22⋊C411yescharacteristic
104C22×D59—41no—
105C22×D59—41no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—41no—
111Q8×C59—44no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59#10412yes—
126C2×C4×C59C4×D412yes—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59#12112yes—
129C2×C4×C59#12311yescharacteristic
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139D236—82no—
140Dic118—82no—
141C8×C59—44no—
142C59⋊C8—44no—
143#9188—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59#38611yescharacteristic
147C22×C59C2×#10411yescharacteristic
148C22×C59#42111yescharacteristic
149C2×D59—81no—
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—81no—
153C2×D59—81no—
154C2×D59—81no—
155C2×D59—81no—
156C4×C59—22no—
157C4×C59—22no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—22no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—41no—
164Dic59—81no—
165Dic59—81no—
166Dic59—81no—
167#2983—591no—
168#2632—591no—
169C4×#101D5911yescharacteristic, complemented
170#1527—592no—
171#1288—592no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#187111yescharacteristic
175C2×C59#191211yescharacteristic
176C2×C59#191411yescharacteristic
177D59—161no—
178C2×#124—591no—
179C2×C4×Q8C2×D5911yescharacteristic
180#446—1184no—
181#432—592no—
182#432—1181no—
183#431—1181no—
184C2×#101—21no—
185C4×M16C2×D5912yes—
186#399—1181no—
187#395—591no—
188#395—591no—
189#394—591no—
190#394—591no—
191#393—1181no—
192#388—1181no—
193#388—1181no—
194#383—591no—
195#343—21no—
196#329—1184no—
197C59#918811yescharacteristic, complemented, Sylow
198C22×Q8—591no—
199C22×Q8C22⋊D5911yescharacteristic
200C22×D4—591no—
201C2×M16—22no—
202#124—1182no—
203#124—1182no—
204C4×Q8—22no—
205C4×Q8—22no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4C4×D5911yescharacteristic
209C2×C4⋊C4C4×D5911yescharacteristic
210C2×C4⋊C4C22⋊D5911yescharacteristic
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C42D23611yescharacteristic
218C2×C42D23611yescharacteristic
219C2×C42C22×D5911yescharacteristic
220C4⋊C8—1182no—
221C4⋊C8—1182no—
222C4⋊C8—1182no—
223#101—24no—
224C22⋊C8—1182no—
225#97—22no—
226#97—594no—
227#97—1182no—
228C4×C8—22no—
229C4×C8—1182no—
230#95—591no—
231#95—591no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235#95—1181no—
236C24—1181no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—1182no—
241C2×Q8—4721no—
242C2×D4—1182no—
243C2×D4—4721no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4(C2×D59)⋊C411yescharacteristic
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254M16—24no—
255M16—42no—
256C2×C8—42no—
257C2×C8—1184no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—4721no—
262C4⋊C4—4721no—
263C22⋊C4—4721no—
264C22⋊C4—4721no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C22⋊C4—2361no—
268C22⋊C4—2362no—
269C22⋊C4—2361no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42D4×D5912yes—
275C42#1174568712yes—
276C23—2361no—
277C23—2361no—
278C23—2361no—
279C23—2361no—
280C23—2361no—
281C23—2361no—
282C23—2361no—
283C23C2×(C2×D59)⋊C411yescharacteristic
284Q8—44no—
285Q8—4722no—
286D4—4722no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4#1427365512yes—
310C2×C4(C22⋊D59)⋊C412yes—
311C2×C4#1427369711yescharacteristic
312C2×C4#1427369912yes—
313C2×C4#1427371111yescharacteristic
314C8—44no—
315C8—2364no—
316C22—21no—
317C22—21no—
318C22—4721no—
319C22—4721no—
320C22—4721no—
321C22—4721no—
322C22—4721no—
323C22—4721no—
324C22—4721no—
325C22#1683727911yescharacteristic
326C22C2×#1427365511yescharacteristic
327C22#1683742011yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—4721no—
337C4—4721no—
338C4—4721no—
339C2—21no—
340C2—21no—
341C2—9441no—
342C2#2142439611yescharacteristic
343C2#2142463111yescharacteristic
344C2#2142464111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC4×#101×C59C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenterC4×#101D59
hypocenterC59#9188
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
4. #21421961C2semidirect
5. C4×#101×C59C2semidirect
169. C4×#101D59semidirect
197. C59#9188semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214299562
#214226932
#214219612
C4×#101×C592
#918859