Finite Group Databasebeta

← #28493084

Subgroups — #28493084

Order 15104.

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

normal subgroups only

This group has 490 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#21425082C212yesmaximal
2#21424520C212yesmaximal
3#21424273C211yesmaximal, characteristic
4#21421964C211yesmaximal, characteristic
5C59×#3189C211yesmaximal, characteristic
6C23×C59⋊C8C2211yescharacteristic
7C2×Dic59⋊C8—21no—
8C2×#14273651—22no—
9#16837262C2212yes—
10C4×C22⋊Dic59C2211yescharacteristic
11#16837169—22no—
12#16837169—22no—
13#16837164—22no—
14#16837156—21no—
15C2×C22⋊Q8×C59C2212yes—
16C59×#484—22no—
17C59×#467—21no—
18C59×#467—21no—
19C59×#451—22no—
20C2×C22⋊C8×C59C2211yescharacteristic
21C22×C59⋊C8—22no—
22C22×C59⋊C8—21no—
23C22×C59⋊C8—42no—
24C2×C4⋊Dic59—22no—
25C2×C22⋊Dic59—22no—
26C2×C22⋊Dic59D411yescharacteristic
27C2×C22⋊Dic59D411yescharacteristic
28C2×C4×Dic59—21no—
29Dic59⋊C8—42no—
30#14273651—44no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C22×Q8×C59D412yes—
34C23×C4×C59C2311yescharacteristic
35C2×Q16×C59—21no—
36C2×Q16×C59—21no—
37C22×C8×C59D411yescharacteristic
38C22×C8×C59D411yescharacteristic
39C22⋊Q8×C59—22no—
40C22⋊Q8×C59—22no—
41C22⋊Q8×C59—22no—
42C22⋊Q8×C59—22no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59D412yes—
45C2×C22⋊C4×C59—22no—
46#107×C59—21no—
47#107×C59—21no—
48#103×C59—22no—
49#103×C59—22no—
50C22⋊C8×C59—22no—
51C24×C59C2×D411yescharacteristic
52C2×Q8×C59—22no—
53C2×Q8×C59—22no—
54C2×Q8×C59—42no—
55C2×Q8×C59—42no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—41no—
59C22×C4×C59—41no—
60C22×C4×C59C2×D412yes—
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59#4111yescharacteristic
63C22×C4×C59#4112yes—
64C22×Dic59—21no—
65C22×Dic59—21no—
66C22×Dic59—42no—
67Q16×C59—81no—
68Q16×C59—81no—
69C2×C8×C59—21no—
70C2×C8×C59—21no—
71C2×C8×C59—41no—
72C2×C8×C59—41no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—22no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—42no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—42no—
79C22⋊Dic59—21no—
80C22⋊Dic59—22no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59Q1612yescomplemented
84C4⋊Dic59—42no—
85C4⋊Dic59—42no—
86C4×Dic59—41no—
87C4×Dic59—41no—
88C2×C59⋊C8—24no—
89C2×C59⋊C8—22no—
90C2×C59⋊C8—41no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59#12111yescharacteristic
96C23×C59#12112yes—
97C23×C59C2×Q1611yescharacteristic
98C23×C59#13511yescharacteristic
99C23×C59#13712yes—
100Q8×C59—44no—
101Q8×C59—42no—
102Q8×C59—42no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—41no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—82no—
112C2×C4×C59—82no—
113C2×C4×C59#12011yescharacteristic
114C2×C4×C59#12312yes—
115C2×C4×C59#12411yescharacteristic
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—82no—
123C2×Dic59—82no—
124C2×Dic59—82no—
125C8×C59—41no—
126C8×C59—41no—
127C59⋊C8—42no—
128#10566—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—42no—
136C22×C59#39411yescharacteristic
137C22×C59#46212yes—
138C22×C59#46412yes—
139C22×C59#46711yescharacteristic
140C22×C59#47011yescharacteristic
141C4×C59—22no—
142C4×C59—21no—
143C4×C59—82no—
144C4×C59—82no—
145C4×C59B(4,𝔽2)11yescharacteristic
146C4×C59#45811yescharacteristic
147Dic59—42no—
148Dic59—81no—
149Dic59—162no—
150#3189D5911yescharacteristic
151#1996—592no—
152#1892—592no—
153#1850—591no—
154#1289—591no—
155C2×C59—21no—
156C2×C59—21no—
157C2×C59—22no—
158C2×C59—22no—
159C2×C59#204411yescharacteristic
160C2×C59#205311yescharacteristic
161C2×C59#205611yescharacteristic
162C23×C8—591no—
163C2×C22⋊Q8C2×D5912yes—
164#484—22no—
165#467—21no—
166#467—21no—
167#451—22no—
168C2×C4⋊C8—1181no—
169C2×#103—1182no—
170C2×C22⋊C8C2×D5911yescharacteristic
171#380—592no—
172C4×C22⋊C4—591no—
173#342—1182no—
174#342—1182no—
175#340—1182no—
176#336—1181no—
177C59#1056611yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5912yes—
179C23×C4C22×D5911yescharacteristic
180C2×Q16—21no—
181C2×Q16—21no—
182C22×C8—1181no—
183C22×C8—1182no—
184C22×C8—2362no—
185C22×C8C22⋊D5911yescharacteristic
186C22×C8C22⋊D5911yescharacteristic
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—1182no—
193C2×C4⋊C4C22⋊D5912yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—591no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—1182no—
198C2×C42—1181no—
199#107—21no—
200#107—21no—
201C4⋊C8—2362no—
202#103—22no—
203#103—22no—
204#103—2364no—
205C22⋊C8—22no—
206#95—1181no—
207#95—1182no—
208C24D4×D5911yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4—2362no—
220C22×C4#1174568711yescharacteristic
221C22×C4#1174568712yes—
222C22×C4C2×C22⋊D5912yes—
223C22×C4C2×C22⋊D5911yescharacteristic
224Q16—81no—
225Q16—81no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—1184no—
231C2×C8—1182no—
232C2×C8—2361no—
233C2×C8—2362no—
234C2×C8—2361no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—2362no—
240C4⋊C4—2362no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—592no—
244C22⋊C4—1181no—
245C22⋊C4—1182no—
246C22⋊C4—2362no—
247C22⋊C4—2362no—
248C42—2361no—
249C42—2361no—
250C23—41no—
251C23—41no—
252C23#1427369811yescharacteristic
253C23#1427369912yes—
254C23Q16×D5911yescharacteristic
255C23#1427375011yescharacteristic
256C23#1427375912yes—
257Q8—44no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—1182no—
271C2×C4—4722no—
272C2×C4—4722no—
273C2×C4—4722no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4#1427369611yescharacteristic
280C2×C4#1427370812yes—
281C2×C4#1427371411yescharacteristic
282C8—41no—
283C8—41no—
284C8—2362no—
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1683731611yescharacteristic
293C22#1683761012yes—
294C22#1683762612yes—
295C22#1683764011yescharacteristic
296C22#1683765711yescharacteristic
297C4—22no—
298C4—21no—
299C4—82no—
300C4—82no—
301C4—9442no—
302C4—4721no—
303C4—2362no—
304C4#1683758911yescharacteristic
305C4#1683759211yescharacteristic
306C2—22no—
307C2—21no—
308C2—21no—
309C2—22no—
310C2#2142534411yescharacteristic
311C2#2142539911yescharacteristic
312C2#2142541611yescharacteristic
subgroupG/H
centerC22#16837316
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#3189C2
socleC22×C59#394
prime radicalC24×C59C2×D4
hypercenter#3189D59
hypocenterC59#10566
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
83. C22⋊Dic59Q16semidirect
177. C59#10566semidirect

Maximal subgroups

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

subgroupindex
#214250822
#214245202
#214242732
#214219642
C59×#31892
#1056659