Finite Group Databasebeta

← #28483776

Subgroups — #28483776

Order 15104.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432660C211yesmaximal, characteristic
2#21427364C211yesmaximal, characteristic
3#21422526C212yesmaximal, complemented
4#21421998C212yesmaximal, complemented
5C59×#1458C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×C4×Dic118C2211yescharacteristic
8#16837429—24no—
9#16837426—21no—
10#16837426C2212yes—
11#16837341C411yescharacteristic
12#16837338C411yescharacteristic
13#16837335—21no—
14#16837325—21no—
15#16837322C411yescharacteristic
16#16837314C411yescharacteristic
17C4×Dic59⋊C4—21no—
18C4×C4⋊Dic59C2211yescharacteristic
19C4×C22⋊Dic59—21no—
20#16837134—24no—
21C2×#117×C59C2211yescharacteristic
22C59×#431C2212yes—
23C59×#324—21no—
24C59×#323—21no—
25C22×Dic118C2×C411yescharacteristic
26C2×#11745664—22no—
27C2×#11745664—22no—
28C2×#11745664—42no—
29C2×#11745664—42no—
30C22×C59⋊C8—22no—
31C4×Dic118—22no—
32C4×Dic118—22no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59—21no—
42C4⋊(C59⋊C8)—24no—
43C4⋊(C59⋊C8)—22no—
44C4⋊(C59⋊C8)D414yes—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95C2×C411yescharacteristic
50C59⋊#95C2×C411yescharacteristic
51C23×C4×C59D411yescharacteristic
52#117×C59—22no—
53#117×C59—22no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59C2×C411yescharacteristic
57C2×C22⋊C4×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60C4⋊C8×C59—22no—
61C22⋊C8×C59—22no—
62C4×C8×C59—22no—
63C24×C59C22⋊C411yescharacteristic
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59SD1612yes—
72C22×C4×C59Q1612yes—
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80C2×Dic118—22no—
81C2×Dic118—42no—
82C2×C8×C59—42no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—22no—
85C22⋊C4×C59—22no—
86C22⋊C4×C59—22no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59C2×D412yes—
90C42×C59#4112yes—
91C22⋊Dic59—42no—
92C22⋊Dic59—42no—
93C4⋊Dic59—22no—
94C4⋊Dic59—42no—
95Dic59⋊C4—42no—
96Dic59⋊C4—42no—
97Dic59⋊C4—42no—
98C4×Dic59—41no—
99C4×Dic59—41no—
100C4×Dic59—41no—
101C4×Dic59—41no—
102#11745664—44no—
103#11745664—42no—
104#11745664—42no—
105C2×C59⋊C8—24no—
106C2×C59⋊C8—44no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59#10312yes—
112C23×C59C2×C22⋊C411yescharacteristic
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
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#10312yes—
126C2×C4×C59C4×D412yes—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59#12112yes—
129C2×C4×C59#12311yescharacteristic
130C2×C4×C59C2×SD1611yescharacteristic
131C2×C4×C59C2×Q1611yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C59#13711yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146Dic118—24no—
147Dic118—42no—
148C8×C59—82no—
149C59⋊C8—44no—
150#9033—591nomaximal, Sylow
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—22no—
154C22×C59—42no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—81no—
158C22×C59#38611yescharacteristic
159C22×C59C2×#10311yescharacteristic
160C22×C59#41711yescharacteristic
161C4×C59—22no—
162C4×C59—22no—
163C4×C59—22no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59#45011yescharacteristic
167C4×C59#45111yescharacteristic
168C4×C59#48311yescharacteristic
169C4×C59#48411yescharacteristic
170Dic59—42no—
171Dic59—42no—
172Dic59—81no—
173Dic59—81no—
174#2905—591no—
175#2396—591no—
176#1487—592no—
177#1458D5911yescharacteristic
178#1303—592no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—42no—
182C2×C59#187211yescharacteristic
183C2×C59#188411yescharacteristic
184C2×C59#189111yescharacteristic
185C22×M16—1181no—
186C2×C4×Q8—591no—
187C2×#117C2×D5911yescharacteristic
188#431C2×D5912yes—
189#424—1184no—
190#423—592no—
191#423—1181no—
192#400—591no—
193#399—591no—
194#398—1181no—
195#396—1181no—
196#395—591no—
197#393—591no—
198C4×C4⋊C4—591no—
199C4×C4⋊C4—1181no—
200C4×C22⋊C4—1181no—
201Q8⋊C8—1184no—
202#324—21no—
203#323—21no—
204C59#903311yescharacteristic, complemented, Sylow
205C22×Q8—591no—
206C23×C4D23611yescharacteristic
207C2×M16—1182no—
208C2×M16—1182no—
209C2×M16—2362no—
210C2×M16—2362no—
211C22×C8—1182no—
212C4×Q8—1182no—
213C4×Q8—1182no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—591no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4C4×D5911yescharacteristic
222C2×C4⋊C4C22⋊D5911yescharacteristic
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4C4×D5911yescharacteristic
226C2×C22⋊C4C22⋊D5911yescharacteristic
227C2×C42—1181no—
228C2×C42—1181no—
229C2×C42D23611yescharacteristic
230C2×C42C22×D5911yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—594no—
233C4⋊C8—1184no—
234C4⋊C8—1182no—
235C22⋊C8—22no—
236C4×C8—22no—
237#95—591no—
238#95—591no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242#95—1181no—
243C24(C2×D59)⋊C411yescharacteristic
244C2×Q8—1182no—
245C2×Q8—2362no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1181no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4#1174566012yes—
255C22×C4Dic23612yes—
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4C2×C4×D5911yescharacteristic
260C22×C4C2×D23611yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262M16—2364no—
263M16—2362no—
264M16—2362no—
265C2×C8—42no—
266C2×C8—1184no—
267C2×C8—2364no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—1182no—
271C4⋊C4—2362no—
272C4⋊C4—2362no—
273C4⋊C4—2362no—
274C4⋊C4—2362no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—2362no—
278C22⋊C4—2362no—
279C42—21no—
280C42—21no—
281C42—2361no—
282C42—2361no—
283C42—2361no—
284C42—2361no—
285C42D4×D5912yes—
286C42#1174568712yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1427365212yes—
292C23C2×(C2×D59)⋊C411yescharacteristic
293Q8—1184no—
294Q8—2362no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—1182no—
308C2×C4—1182no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4#1427365212yes—
320C2×C4(C22⋊D59)⋊C412yes—
321C2×C4#1427369711yescharacteristic
322C2×C4#1427369912yes—
323C2×C4#1427371111yescharacteristic
324C2×C4C2×#1174566011yescharacteristic
325C2×C4C2×Dic23611yescharacteristic
326C2×C4#1427375511yescharacteristic
327C2×C4#1427376011yescharacteristic
328C8—82no—
329C8—2364no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1683727911yescharacteristic
338C22C2×#1427365211yescharacteristic
339C22#1683740411yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—4721no—
346C4—4721no—
347C4—2362no—
348C4—2362no—
349C4#1683755511yescharacteristic
350C4#1683756011yescharacteristic
351C4#1683773311yescharacteristic
352C4#1683773811yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2142440111yescharacteristic
357C2#2142447711yescharacteristic
358C2#2142451211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1458C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1458D59
hypocenterC59#9033
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #21422526C2semidirect
4. #21421998C2semidirect
204. C59#9033semidirect

Maximal subgroups

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

subgroupindex
#214326602
#214273642
#214225262
#214219982
C59×#14582
#903359