Finite Group Databasebeta

← #28492303

Subgroups — #28492303

Order 15104.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16837609C211yesmaximal, characteristic
2#21424089C211yesmaximal, characteristic
3#21422867C211yesmaximal, characteristic
4#21422768C211yesmaximal, characteristic
5#21422490C211yesmaximal, characteristic
6#21421991C211yesmaximal, characteristic
7C59×#1546C211yesmaximal, characteristic
8C22×#11745672—21no—
9C2×C4×Dic118C2211yescharacteristic
10#16837609—22no—
11#16837609—22no—
12#16837455C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273651—21no—
16C2×#14273645—21no—
17#16837170—21no—
18#16837153—22no—
19#16837148—22no—
20#16837132—22no—
21C2×C4⋊Q8×C59C2211yescharacteristic
22C59×#431C2211yescharacteristic
23C59×#423C2211yescharacteristic
24C59×#332—22no—
25C22×Dic118D411yescharacteristic
26C2×#11745672—24no—
27C2×#11745672—42no—
28C2×#11745672—42no—
29C2×#11745664—21no—
30C22×C59⋊C8—21no—
31C4×Dic118—21no—
32C4×Dic118—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C2×C4⋊Dic59D411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)D412yes—
43#14273651—24no—
44#14273645—42no—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C22×Q8×C59—21no—
48C2×M16×C59—21no—
49C4⋊Q8×C59—21no—
50C4⋊Q8×C59—21no—
51C4⋊Q8×C59—21no—
52C4⋊Q8×C59D412yes—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59D412yes—
59C22⋊C8×C59—21no—
60C4×C8×C59—21no—
61C2×Q8×C59—22no—
62C2×Q8×C59—41no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C2×Dic118—21no—
74C2×Dic118—21no—
75C2×Dic118—21no—
76C2×Dic118—21no—
77C2×Dic118—21no—
78C2×Dic118SD1612yes—
79M16×C59—41no—
80M16×C59—41no—
81C2×C8×C59—22no—
82C2×C8×C59—41no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C42×C59C2×D411yescharacteristic
88C42×C59C2×D411yescharacteristic
89C42×C59C2×D411yescharacteristic
90C42×C59#4111yescharacteristic
91#11745672—48no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94Dic59⋊C4—41no—
95Dic59⋊C4—41no—
96C4×Dic59—41no—
97C4×Dic59—41no—
98#11745664—41no—
99#11745664—41no—
100C2×C59⋊C8—22no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C23×C59#12311yescharacteristic
106Q8×C59—42no—
107Q8×C59—42no—
108Q8×C59—81no—
109Q8×C59—81no—
110C2×C4×C59—22no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—81no—
114C2×C4×C59—81no—
115C2×C4×C59#12011yescharacteristic
116C2×C4×C59#12111yescharacteristic
117C2×C4×C59#12111yescharacteristic
118C2×C4×C59#12111yescharacteristic
119C2×C4×C59#12311yescharacteristic
120C2×C4×C59#12411yescharacteristic
121C2×C4×C59C2×SD1611yescharacteristic
122C2×C4×C59#13511yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—81no—
130Dic118—22no—
131Dic118—22no—
132Dic118—41no—
133Dic118—41no—
134C8×C59—42no—
135C8×C59—81no—
136C59⋊C8—42no—
137C59⋊C8—81no—
138C59⋊C8—81no—
139#10447—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59#39411yescharacteristic
143C22×C59#48111yescharacteristic
144C22×C59#48211yescharacteristic
145C4×C59—21no—
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—82no—
149C4×C59#44911yescharacteristic
150C4×C59#45011yescharacteristic
151C4×C59#46511yescharacteristic
152C4×C59#46911yescharacteristic
153C4×C59#47312yes—
154Dic59—42no—
155Dic59—81no—
156C2×#462—591no—
157#1801—591no—
158#1570—591no—
159#1546—591no—
160#1546D5911yescharacteristic
161#1479—591no—
162#1301—591no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59#203411yescharacteristic
166C2×C59#204711yescharacteristic
167C2×C59#204911yescharacteristic
168C22×Q16—1181no—
169C2×C4⋊Q8C2×D5911yescharacteristic
170C2×C4×Q8—591no—
171#462—1182no—
172#462—1182no—
173#431—591no—
174#431C2×D5911yescharacteristic
175#423—591no—
176#423C2×D5911yescharacteristic
177C2×C4⋊C8—591no—
178C2×#103—1181no—
179C2×#101—1181no—
180#343—1181no—
181#334—1182no—
182#332—22no—
183#332—1182no—
184Q8⋊C8—1182no—
185C59#1044711yescharacteristic, complemented, Sylow
186C22×Q8—21no—
187C22×Q8—591no—
188C2×Q16—1184no—
189C2×Q16—2362no—
190C2×Q16—2362no—
191C2×M16—21no—
192C2×M16—1181no—
193C22×C8—1181no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8D23612yes—
198C4×Q8—1181no—
199C4×Q8—1181no—
200C4×Q8—1181no—
201C4×Q8—1181no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—1181no—
206C2×C42—1181no—
207C2×C42C22×D5911yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—592no—
211C4⋊C8—1181no—
212C4⋊C8—1181no—
213C4⋊C8—1181no—
214C4⋊C8—1181no—
215C4⋊C8C22⋊D5912yes—
216#103—1184no—
217#101—2362no—
218C22⋊C8—21no—
219C22⋊C8—1181no—
220C4×C8—21no—
221C4×C8—1181no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—592no—
229C2×Q8—1181no—
230C2×Q8—1181no—
231C2×Q8—1181no—
232C2×Q8—1181no—
233C2×Q8—1181no—
234C22×C4—41no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4#1174568511yescharacteristic
238C22×C4D4×D5911yescharacteristic
239C22×C4#1174568711yescharacteristic
240Q16—2368no—
241M16—41no—
242M16—41no—
243M16—2361no—
244M16—2361no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—1182no—
248C2×C8—2361no—
249C2×C8—2361no—
250C2×C8—2361no—
251C2×C8—2361no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—1181no—
257C4⋊C4—1181no—
258C4⋊C4—2361no—
259C4⋊C4—2361no—
260C42—2361no—
261C42—2361no—
262C42C2×D23611yescharacteristic
263C42D4×D5911yescharacteristic
264C42#1174568711yescharacteristic
265C42C2×C22⋊D5911yescharacteristic
266C23#1427370911yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—81no—
270Q8—81no—
271Q8—1182no—
272Q8—1182no—
273Q8—2361no—
274Q8—2361no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—1182no—
281C2×C4—4721no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4#1427369711yescharacteristic
286C2×C4#1427369811yescharacteristic
287C2×C4#1427369911yescharacteristic
288C2×C4#1427370111yescharacteristic
289C2×C4#1427371111yescharacteristic
290C2×C4#1427371311yescharacteristic
291C2×C4SD16×D5911yescharacteristic
292C2×C4#1427374811yescharacteristic
293C2×C4#1427375111yescharacteristic
294C2×C4#1427375111yescharacteristic
295C8—42no—
296C8—81no—
297C8—4721no—
298C8—4721no—
299C8—2362no—
300C22—21no—
301C22—21no—
302C22#1683731811yescharacteristic
303C22#1683771911yescharacteristic
304C22#1683772511yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—82no—
309C4—4721no—
310C4—2362no—
311C4#1683754811yescharacteristic
312C4#1683755411yescharacteristic
313C4#1683762811yescharacteristic
314C4#1683765211yescharacteristic
315C4#1683767512yes—
316C2—21no—
317C2—21no—
318C2#2142528411yescharacteristic
319C2#2142536311yescharacteristic
320C2#2142537511yescharacteristic
subgroupG/H
centerC22#16837318
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1546C2
socleC22×C59#394
prime radicalC23×C59#123
hypercenter#1546D59
hypocenterC59#10447
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
185. C59#10447semidirect

Maximal subgroups

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

subgroupindex
C2×#168376092
#214240892
#214228672
#214227682
#214224902
#214219912
C59×#15462
#1044759