Finite Group Databasebeta

← #28485332

Subgroups — #28485332

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435221C211yesmaximal, characteristic
2#21425069C211yesmaximal, characteristic, complemented
3#21422778C211yesmaximal, characteristic, complemented
4#21422656C211yesmaximal, characteristic
5#21421956C211yesmaximal, characteristic, complemented
6#21421927C211yesmaximal, characteristic, complemented
7C59×#2932C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9#16837798C412yescomplemented
10#16837748C412yescomplemented
11#16837718—22no—
12#16837692—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837268C2211yescharacteristic
16#16837165—21no—
17#16837165—21no—
18#16837150—22no—
19#16837136—22no—
20C2×#131×C59—21no—
21C59×#532C2211yescharacteristic
22C59×#435—22no—
23C59×#435—22no—
24C59×#423C2211yescharacteristic
25C59×#408—21no—
26C2×#97×C59C2211yescharacteristic
27C22×Dic118—21no—
28#14273727—21no—
29#14273727—21no—
30#14273727—21no—
31#14273727C2×C412yes—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35#14273658—24no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38#14273648—24no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C4×C59⋊C8—21no—
42C4×C59⋊C8C2×C412yes—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C2×#41×C59—21no—
46#131×C59—42no—
47#131×C59—42no—
48C2×M16×C59—22no—
49C2×M16×C59—21no—
50C22×C8×C59—22no—
51C22×C8×C59D411yescharacteristic
52C22×C8×C59D411yescharacteristic
53C4⋊Q8×C59C2×C411yescharacteristic
54#127×C59C2×C411yescharacteristic
55#124×C59—21no—
56#121×C59—21no—
57#121×C59D412yes—
58C4×D4×C59—21no—
59C4×D4×C59D412yes—
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—21no—
62C4⋊C8×C59D412yes—
63C22⋊C8×C59—24no—
64#97×C59—21no—
65#97×C59—21no—
66#41×C59—41no—
67#41×C59—41no—
68#41×C59—41no—
69#41×C59—41no—
70C2×Q8×C59—21no—
71C2×D4×C59—22no—
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59C22⋊C412yes—
75C22×C4×C59—22no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×Dic59—41no—
80C22×Dic59—41no—
81C2×Dic118—22no—
82C2×Dic118—41no—
83C2×Dic118—82no—
84M16×C59—44no—
85M16×C59—41no—
86M16×C59—41no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—21no—
90C2×C8×C59—22no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59—44no—
94C2×C8×C59SD1612yes—
95C4⋊C4×C59—21no—
96C4⋊C4×C59C22⋊C412yes—
97C22⋊C4×C59—22no—
98C22⋊C4×C59—22no—
99C42×C59C22×C411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59#4111yescharacteristic
103C4⋊Dic59—22no—
104C4⋊Dic59—22no—
105C4⋊Dic59—41no—
106C4⋊Dic59—41no—
107C4⋊Dic59—41no—
108C4⋊Dic59—41no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—41no—
111C23×C59—22no—
112C23×C59#12111yescharacteristic
113Q8×C59—41no—
114Q8×C59—41no—
115D4×C59—42no—
116D4×C59—42no—
117D4×C59—42no—
118D4×C59—41no—
119D4×C59—41no—
120D4×C59—81no—
121C2×C4×C59—22no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59C2×C22⋊C411yescharacteristic
127C2×C4×C59C4×D411yescharacteristic
128C2×C4×C59C4×D411yescharacteristic
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59#12111yescharacteristic
131C2×C4×C59#12311yescharacteristic
132C2×C4×C59C2×SD1611yescharacteristic
133C2×C4×C59#13511yescharacteristic
134C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C59#13711yescharacteristic
136C2×Dic59—42no—
137C2×Dic59—82no—
138C2×Dic59—81no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141Dic118—82no—
142Dic118—82no—
143C8×C59—22no—
144C8×C59—42no—
145C8×C59—41no—
146C59⋊C8—42no—
147C59⋊C8—81no—
148#9287—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59#38611yescharacteristic
155C22×C59#46111yescharacteristic
156C22×C59#46311yescharacteristic
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—42no—
162C4×C59C4×SD1611yescharacteristic
163C4×C59#44011yescharacteristic
164C4×C59#45011yescharacteristic
165C4×C59#45211yescharacteristic
166Dic59—82no—
167Dic59—161no—
168#3227—591no—
169#2932D5911yescharacteristic
170#1993—591no—
171#1548—591no—
172#1517—591no—
173#1286—591no—
174#1275—591no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—82no—
178C2×C59#187711yescharacteristic
179C2×C59#190311yescharacteristic
180C2×C59#190911yescharacteristic
181C2×#131—21no—
182#532C2×D5911yescharacteristic
183C2×C4⋊Q8—591no—
184#498—592no—
185#486—592no—
186#481—1182no—
187#476—1182no—
188#435—22no—
189#435—22no—
190#431—591no—
191#431—591no—
192#423C2×D5911yescharacteristic
193#408—21no—
194C2×#97C2×D5911yescharacteristic
195#382—591no—
196#340—1181no—
197#340—1181no—
198#333—1182no—
199#327—1182no—
200C59#928711yescharacteristic, complemented, Sylow
201C2×#41—21no—
202C22×Q8—1181no—
203#131—42no—
204#131—42no—
205C2×M16—22no—
206C2×M16—21no—
207C22×C8—22no—
208C22×C8C22⋊D5911yescharacteristic
209C22×C8C22⋊D5911yescharacteristic
210C4⋊Q8—592no—
211C4⋊Q8—1181no—
212C4⋊Q8—1181no—
213C4⋊Q8—1181no—
214C4⋊Q8C4×D5911yescharacteristic
215#127C4×D5911yescharacteristic
216#124—21no—
217#121—21no—
218#121D23612yes—
219C4×D4—21no—
220C4×D4C22⋊D5912yes—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C42C22×D5911yescharacteristic
225#106—1184no—
226C4⋊C8—21no—
227C4⋊C8—1181no—
228C4⋊C8—1181no—
229C4⋊C8C22⋊D5912yes—
230#102—1184no—
231C22⋊C8—24no—
232C22⋊C8—1181no—
233C22⋊C8—1181no—
234#97—21no—
235#97—21no—
236C4×C8—592no—
237C4×C8—1181no—
238#95—1181no—
239#95—1181no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—1182no—
246C2×Q8—4722no—
247C2×Q8—2361no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4(C2×D59)⋊C412yes—
252C22×C4—22no—
253C22×C4—2361no—
254C22×C4—2361no—
255C22×C4C2×D23611yescharacteristic
256C22×C4#1174568511yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258M16—44no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—44no—
268C2×C8—1182no—
269C2×C8—2361no—
270C2×C8#1174567012yes—
271C4⋊C4—21no—
272C4⋊C4—1182no—
273C4⋊C4—1182no—
274C4⋊C4—2361no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2361no—
278C4⋊C4(C2×D59)⋊C412yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C2×C4×D5911yescharacteristic
282C42#1174568511yescharacteristic
283C42C2×C22⋊D5911yescharacteristic
284C42C2×C22⋊D5911yescharacteristic
285C23—22no—
286C23#1427370111yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—4722no—
290Q8—4722no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—4721no—
303C2×C4—4721no—
304C2×C4—4721no—
305C2×C4—4722no—
306C2×C4—2362no—
307C2×C4C2×(C2×D59)⋊C411yescharacteristic
308C2×C4C4×C22⋊D5911yescharacteristic
309C2×C4C4×C22⋊D5911yescharacteristic
310C2×C4#1427369611yescharacteristic
311C2×C4#1427370111yescharacteristic
312C2×C4#1427371011yescharacteristic
313C2×C4C2×#1174567011yescharacteristic
314C2×C4#1427374711yescharacteristic
315C2×C4#1427375311yescharacteristic
316C2×C4#1427375711yescharacteristic
317C8—22no—
318C8—42no—
319C8—41no—
320C8—4721no—
321C8—2362no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#1683728111yescharacteristic
328C22#1683760711yescharacteristic
329C22#1683761711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—9441no—
336C4—4722no—
337C4C4×#1174567011yescharacteristic
338C4#1683750211yescharacteristic
339C4#1683755311yescharacteristic
340C4#1683756411yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2142443011yescharacteristic
345C2#2142457611yescharacteristic
346C2#2142461811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2932C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#2932D59
hypocenterC59#9287
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #21425069C2semidirect
3. #21422778C2semidirect
5. #21421956C2semidirect
6. #21421927C2semidirect
9. #16837798C4semidirect
10. #16837748C4semidirect
200. C59#9287semidirect

Maximal subgroups

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

subgroupindex
#214352212
#214250692
#214227782
#214226562
#214219562
#214219272
C59×#29322
#928759