Finite Group Databasebeta

← #28476707

Subgroups — #28476707

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426579C211yesmaximal, characteristic
2C2×#16837157C211yesmaximal, characteristic
3#21421920C212yesmaximal, complemented
4#21421918C212yesmaximal, complemented
5C59×#1817C211yesmaximal, characteristic
6C23×C59⋊C8—21no—
7C2×#14273686C411yescharacteristic
8C2×#14273686C411yescharacteristic
9#16837300C411yescharacteristic
10#16837300C411yescharacteristic
11#16837300C2211yescharacteristic
12#16837300C2211yescharacteristic
13#16837275—21no—
14#16837275—21no—
15C4×Dic59⋊C4—21no—
16C4×Dic59⋊C4—21no—
17#16837157—21no—
18#16837157C2212yes—
19C22×C42×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#336—21no—
22C59×#336C2212yes—
23C22×C59⋊C8—24no—
24C22×C59⋊C8—22no—
25C22×C59⋊C8—42no—
26C22×C59⋊C8—42no—
27#14273686—22no—
28#14273686—42no—
29#14273686C814yescomplemented
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×Dic59⋊C4—21no—
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—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×C59—22no—
50C2×C42×C59—42no—
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59C2311yescharacteristic
55C22⋊C8×C59—42no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—41no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C2×C812yes—
70C22×C4×C59M1612yes—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C2×C8×C59—42no—
79C2×C8×C59—42no—
80C2×C8×C59—42no—
81C42×C59—41no—
82C42×C59—41no—
83C42×C59—41no—
84C42×C59—41no—
85C22⋊Dic59—24no—
86C22⋊Dic59—42no—
87C4⋊Dic59—21no—
88C4⋊Dic59—21no—
89C4⋊Dic59—41no—
90C4⋊Dic59—41no—
91C4⋊Dic59C2×C812yescomplemented
92C4⋊Dic59C2×C812yescomplemented
93Dic59⋊C4—42no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96Dic59⋊C4—41no—
97Dic59⋊C4—41no—
98C4×Dic59—24no—
99C4×Dic59—42no—
100C2×C59⋊C8—24no—
101C2×C59⋊C8—48no—
102C2×C59⋊C8—42no—
103C2×C59⋊C8—42no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59C22⋊C812yes—
109C23×C59#9912yes—
110C23×C59#10112yes—
111C23×C59C2×C22⋊C411yescharacteristic
112C2×C4×C59—24no—
113C2×C4×C59—24no—
114C2×C4×C59—24no—
115C2×C4×C59—24no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—44no—
130C2×C4×C59C22⋊C812yes—
131C2×C4×C59#10412yes—
132C2×C4×C59C22×C811yescharacteristic
133C2×C4×C59C2×M1611yescharacteristic
134C2×C4×C59#13512yes—
135C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C59#13711yescharacteristic
137C2×Dic59—22no—
138C2×Dic59—22no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C8×C59—82no—
150C59⋊C8—44no—
151#7579—591nomaximal, Sylow
152C22×C59—22no—
153C22×C59—22no—
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59—22no—
158C22×C59#32414yes—
159C22×C59C2×C22⋊C811yescharacteristic
160C22×C59C2×#9911yescharacteristic
161C22×C59C2×#10111yescharacteristic
162C22×C59#41611yescharacteristic
163C22×C59#41711yescharacteristic
164C22×C59C2×#10411yescharacteristic
165C22×C59#42111yescharacteristic
166C4×C59—22no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—22no—
171C4×C59—21no—
172C4×C59—22no—
173C4×C59—41no—
174Dic59—42no—
175Dic59—42no—
176Dic59—81no—
177Dic59—81no—
178#2296—591no—
179#1817D5911yescharacteristic
180C2×#336—591no—
181#1271—592no—
182#1271—592no—
183C2×C59—24no—
184C2×C59C2×#32411yescharacteristic
185C2×C59#147611yescharacteristic
186C2×C59#147711yescharacteristic
187C2×C59#150411yescharacteristic
188C2×C59#151611yescharacteristic
189C2×C59#153111yescharacteristic
190C2×C59#155211yescharacteristic
191C23×C8—1181no—
192C2×#117—591no—
193C2×#117—591no—
194C22×C42C2×D5911yescharacteristic
195C2×C22⋊C8—21no—
196#389—591no—
197#389—591no—
198#389—591no—
199#389—591no—
200#385—1181no—
201#385—1181no—
202C4×C4⋊C4—1181no—
203C4×C4⋊C4—1181no—
204#336—21no—
205#336—592no—
206#336—1181no—
207#336C2×D5912yes—
208C59#757911yescharacteristic, complemented, Sylow
209C23×C4C4×D5911yescharacteristic
210C23×C4D23611yescharacteristic
211C23×C4C22⋊D5911yescharacteristic
212C22×C8—22no—
213C22×C8—1184no—
214C22×C8—1182no—
215C22×C8—2362no—
216C22×C8—2362no—
217#117—594no—
218#117—1182no—
219#117—2362no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—591no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C22⋊C4—1181no—
229C2×C22⋊C4—1181no—
230C2×C42—42no—
231C2×C42—1181no—
232C2×C42—1181no—
233C2×C42C4×D5911yescharacteristic
234C2×C42D23611yescharacteristic
235C2×C42C22⋊D5911yescharacteristic
236C2×C42C22×D5911yescharacteristic
237C22⋊C8—42no—
238#95—1181no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242C24(C2×D59)⋊C411yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4C8×D5912yes—
257C22×C4#1174565912yes—
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4C2×C4×D5911yescharacteristic
262C22×C4C2×D23611yescharacteristic
263C22×C4C2×C22⋊D5911yescharacteristic
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—1184no—
268C2×C8—2368no—
269C2×C8—2362no—
270C2×C8—2362no—
271C4⋊C4—592no—
272C4⋊C4—592no—
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2362no—
278C4⋊C4—2362no—
279C4⋊C4—2362no—
280C4⋊C4—2361no—
281C4⋊C4—2361no—
282C22⋊C4—1184no—
283C22⋊C4—2362no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—1184no—
289C42—2362no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23(C2×D59)⋊C812yes—
295C23#1427364112yes—
296C23#1427364512yes—
297C23C2×(C2×D59)⋊C411yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—44no—
316C2×C4—1182no—
317C2×C4—1182no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4(C2×D59)⋊C812yes—
329C2×C4#1427365312yes—
330C2×C4C2×C8×D5911yescharacteristic
331C2×C4C2×#1174565911yescharacteristic
332C2×C4#1427374712yes—
333C2×C4#1427375511yescharacteristic
334C2×C4#1427376011yescharacteristic
335C8—82no—
336C8—2364no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1683712714yes—
344C22C2×(C2×D59)⋊C811yescharacteristic
345C22C2×#1427364111yescharacteristic
346C22C2×#1427364511yescharacteristic
347C22#1683739611yescharacteristic
348C22#1683740411yescharacteristic
349C22C2×#1427365311yescharacteristic
350C22#1683742011yescharacteristic
351C4—21no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—4721no—
360C4—4721no—
361C4—2362no—
362C4—2362no—
363C2—24no—
364C2C2×#1683712711yescharacteristic
365C2#2142247611yescharacteristic
366C2#2142248411yescharacteristic
367C2#2142260111yescharacteristic
368C2#2142265011yescharacteristic
369C2#2142271211yescharacteristic
370C2#2142279711yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1817D59
hypocenterC59#7579
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #21421920C2semidirect
4. #21421918C2semidirect
29. #14273686C8semidirect
91. C4⋊Dic59C2×C8semidirect
92. C4⋊Dic59C2×C8semidirect
208. C59#7579semidirect

Maximal subgroups

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

subgroupindex
#214265792
C2×#168371572
#214219202
#214219182
C59×#18172
#757959