Finite Group Databasebeta

← #28486561

Subgroups — #28486561

Order 15104.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#21429001C211yesmaximal, characteristic
3#21422520C212yesmaximal
4#21421956C212yesmaximal, complemented
5C59×#1801C211yesmaximal, characteristic, complemented
6C2×#14273703—21no—
7C2×C4×D236C2211yescharacteristic
8#16837533—24no—
9#16837459—22no—
10#16837455C2212yes—
11#16837332C412yes—
12#16837319C412yes—
13#16837293—22no—
14#16837268C2211yescharacteristic
15#16837259—21no—
16#16837129—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C59×#423C2212yes—
19C2×#101×C59—21no—
20C59×#343—21no—
21C22×D236C2×C411yescharacteristic
22C22×C4×D59—21no—
23#14273703—44no—
24C4×D236—22no—
25C4×D236—22no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—21no—
32C4⋊(C59⋊C8)—24no—
33C4⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35#14273638—24no—
36C4×C59⋊C8D414yes—
37C59⋊#95—22no—
38C59⋊#95—21no—
39C59⋊#95C2×C412yes—
40C22×Q8×C59D411yescharacteristic
41C2×M16×C59—22no—
42C4×Q8×C59—22no—
43C4×Q8×C59—22no—
44C2×C4⋊C4×C59C2×C412yes—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C42×C59D412yes—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49C4⋊C8×C59—22no—
50#101×C59—42no—
51C23×D59—21no—
52C2×Q8×C59—22no—
53C2×Q8×C59—21no—
54C2×Q8×C59—22no—
55C2×Q8×C59—21no—
56C22×C4×C59C22⋊C412yes—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×Dic59—22no—
62C22×Dic59—21no—
63C2×D236—22no—
64C2×D236—42no—
65C2×C4×D59—42no—
66C2×C4×D59—42no—
67C2×C4×D59—41no—
68C2×C4×D59—41no—
69M16×C59—42no—
70M16×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—24no—
73C4⋊C4×C59—22no—
74C42×C59—22no—
75C42×C59—22no—
76C42×C59C2×D412yes—
77C42×C59#4112yes—
78(C2×D59)⋊C4—44no—
79(C2×D59)⋊C4—44no—
80(C2×D59)⋊C4—41no—
81(C2×D59)⋊C4—41no—
82C4⋊Dic59—22no—
83C4⋊Dic59—42no—
84Dic59⋊C4—44no—
85C2×C59⋊C8—24no—
86C23×C59C2×C22⋊C411yescharacteristic
87C22×D59—22no—
88C22×D59—41no—
89C22×D59—41no—
90C22×D59—41no—
91C22×D59—41no—
92C22×D59—41no—
93C22×D59—41no—
94Q8×C59—42no—
95Q8×C59—42no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—42no—
103C2×C4×C59C4×D412yes—
104C2×C4×C59#12011yescharacteristic
105C2×C4×C59#12112yes—
106C2×C4×C59#12311yescharacteristic
107C2×C4×C59AGL(1,ℤ/8)12yes—
108C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C59#13711yescharacteristic
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—41no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116D236—24no—
117D236—42no—
118C4×D59—82no—
119C4×D59—82no—
120C8×C59—82no—
121C59⋊C8—44no—
122#9490—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59#38611yescharacteristic
126C22×C59#41711yescharacteristic
127C22×C59#41811yescharacteristic
128C2×D59—42no—
129C2×D59—42no—
130C2×D59—42no—
131C2×D59—81no—
132C2×D59—81no—
133C2×D59—81no—
134C2×D59—81no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—42no—
139C4×C59—42no—
140C4×C59#46812yes—
141C4×C59#46912yes—
142Dic59—82no—
143Dic59—81no—
144#2983—591no—
145#2536—591no—
146#1801D5911yescharacteristic, complemented
147#1486—592no—
148#1286—592no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59#189111yescharacteristic
152C2×C59#189511yescharacteristic
153C2×C59#191211yescharacteristic
154D59—82no—
155C2×C22⋊Q8—1181no—
156C2×C4×Q8C2×D5911yescharacteristic
157C2×C4×D4—591no—
158#446—1184no—
159#432—1182no—
160#431—592no—
161#423C2×D5912yes—
162C2×#101—21no—
163#398—592no—
164#394—592no—
165#388—1182no—
166#382—591no—
167#380—1181no—
168#343—21no—
169D4⋊C8—1184no—
170C59#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D5911yescharacteristic
172C22×D4—591no—
173C23×C4—1181no—
174C2×M16—22no—
175C22⋊Q8—2364no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1182no—
179C4×D4—1182no—
180C2×C4⋊C4—591no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4—1181no—
183C2×C4⋊C4C4×D5912yes—
184C2×C4⋊C4C22⋊D5911yescharacteristic
185C2×C22⋊C4—1182no—
186C2×C22⋊C4—1182no—
187C2×C22⋊C4—1181no—
188C2×C42D23612yes—
189C2×C42C22×D5911yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—1184no—
193C4⋊C8—1182no—
194#101—42no—
195C22⋊C8—1182no—
196#97—1184no—
197C4×C8—594no—
198#95—592no—
199#95—1182no—
200#95—1181no—
201C24—1181no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—1182no—
207C2×D4—2362no—
208C22×C4—1182no—
209C22×C4—1181no—
210C22×C4—2362no—
211C22×C4—2362no—
212C22×C4—2361no—
213C22×C4—2361no—
214C22×C4(C2×D59)⋊C412yes—
215C22×C4(C2×D59)⋊C412yes—
216C22×C4C2×C4×D5911yescharacteristic
217C22×C4C2×D23611yescharacteristic
218C22×C4C2×C22⋊D5911yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—1184no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—1182no—
226C4⋊C4—2364no—
227C4⋊C4—2362no—
228C22⋊C4—2364no—
229C22⋊C4—2364no—
230C22⋊C4—2361no—
231C22⋊C4—2361no—
232C42—22no—
233C42—22no—
234C42D4×D5912yes—
235C42#1174568712yes—
236C23—1182no—
237C23—2361no—
238C23—2361no—
239C23—2361no—
240C23—2361no—
241C23—2361no—
242C23—2361no—
243C23C2×(C2×D59)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—1184no—
247D4—2362no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—4722no—
256C2×C4—4722no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4—2361no—
260C2×C4—2362no—
261C2×C4—2361no—
262C2×C4—2361no—
263C2×C4(C22⋊D59)⋊C412yes—
264C2×C4#1427369711yescharacteristic
265C2×C4#1427369912yes—
266C2×C4#1427371111yescharacteristic
267C2×C4#1427375212yes—
268C2×C4#1427375311yescharacteristic
269C2×C4#1427375811yescharacteristic
270C8—82no—
271C8—2364no—
272C22—21no—
273C22—21no—
274C22—4721no—
275C22—4721no—
276C22—4721no—
277C22—4721no—
278C22—2362no—
279C22—2362no—
280C22—2362no—
281C22#1683727911yescharacteristic
282C22#1683740111yescharacteristic
283C22#1683740611yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—4722no—
290C4—4721no—
291C4#1683765012yes—
292C4#1683765312yes—
293C2—21no—
294C2—21no—
295C2—4722no—
296C2#2142451211yescharacteristic
297C2#2142453211yescharacteristic
298C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1801D59
hypocenterC59#9490
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
4. #21421956C2semidirect
5. C59×#1801C2semidirect
146. #1801D59semidirect
170. C59#9490semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214290012
#214225202
#214219562
C59×#18012
#949059