Finite Group Databasebeta

← #28483008

Subgroups — #28483008

Order 15104.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432762C211yesmaximal, characteristic
2#21422872C211yesmaximal, characteristic, complemented
3#21422851C211yesmaximal, characteristic, complemented
4#21422478C211yesmaximal, characteristic
5#21421999C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1806C211yesmaximal, characteristic
8#16837463—22no—
9#16837459—21no—
10#16837455—21no—
11#16837455C2211yescharacteristic
12#16837426C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14#16837369—21no—
15#16837358—21no—
16C2×C4×C59⋊C8C2211yescharacteristic
17#16837154—21no—
18#16837154—21no—
19#16837152—22no—
20Q8⋊(C59⋊C8)C411yescharacteristic, complemented
21Q8⋊(C59⋊C8)C411yescharacteristic, complemented
22D4⋊(C59⋊C8)C411yescharacteristic, complemented
23D4⋊(C59⋊C8)C411yescharacteristic, complemented
24C4×#41×C59C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×#104×C59—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C59×#339—21no—
29C2×#11745664—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32C4⋊(C59⋊C8)—21no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)C2×C411yescharacteristic
35C4⋊(C59⋊C8)C2×C411yescharacteristic
36C4⋊(C59⋊C8)D412yes—
37C4⋊(C59⋊C8)D411yescharacteristic
38C4⋊(C59⋊C8)D411yescharacteristic
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—21no—
42#14273638—22no—
43C4×C59⋊C8—22no—
44C4×C59⋊C8—21no—
45C4×C59⋊C8—21no—
46C4×C59⋊C8C2×C411yescharacteristic
47C4×C59⋊C8C2×C411yescharacteristic
48C2×#41×C59D411yescharacteristic
49C2×M16×C59—21no—
50C4×Q8×C59C2×C411yescharacteristic
51C4×Q8×C59Q811yescharacteristic
52C4×D4×C59—21no—
53C4×D4×C59—21no—
54C4×D4×C59C2×C411yescharacteristic
55C4×D4×C59Q811yescharacteristic
56#117×C59—21no—
57#117×C59D411yescharacteristic
58C2×C4⋊C4×C59—21no—
59C2×C42×C59—21no—
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64#104×C59—42no—
65#95×C59—21no—
66#41×C59—21no—
67#41×C59—21no—
68#41×C59—21no—
69#41×C59—21no—
70C2×Q8×C59C4⋊C411yescharacteristic
71C2×D4×C59—21no—
72C2×D4×C59C4⋊C411yescharacteristic
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—41no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4111yescharacteristic
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—41no—
83C4⋊C4×C59—21no—
84C4⋊C4×C59—21no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59C4⋊C411yescharacteristic
88C4⋊C4×C59C4⋊C411yescharacteristic
89C22⋊C4×C59—21no—
90C22⋊C4×C59—21no—
91C22⋊C4×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—41no—
96C42×C59—41no—
97C42×C59C22×C411yescharacteristic
98C42×C59C2×D411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59C2×Q811yescharacteristic
101#11745664—41no—
102#11745664—41no—
103C2×C59⋊C8—21no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—42no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C23×C59—21no—
114C23×C59#12311yescharacteristic
115Q8×C59—21no—
116Q8×C59—21no—
117D4×C59—21no—
118D4×C59—21no—
119D4×C59—41no—
120D4×C59—41no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—81no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59C2×C4⋊C411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59C22⋊Q811yescharacteristic
140C2×C4×C59C22⋊Q811yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C8×C59—81no—
146C59⋊C8—22no—
147C59⋊C8—41no—
148C59⋊C8—41no—
149C59⋊C8—41no—
150C59⋊C8—41no—
151C59⋊C8—41no—
152C59⋊C8—42no—
153#8905—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59#38511yescharacteristic
160C22×C59#48211yescharacteristic
161C22×C59#48511yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—82no—
169C4×C59#42711yescharacteristic
170C4×C59#42811yescharacteristic
171C4×C59#44311yescharacteristic
172C4×C59#44411yescharacteristic
173C4×C59#44911yescharacteristic
174C4×C59#45211yescharacteristic
175#2925—591no—
176#1806D5911yescharacteristic
177#1571—591no—
178#1566—591no—
179#1476—591no—
180#1303—591no—
181#1279—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—81no—
185C2×C59#185411yescharacteristic
186C2×C59#186111yescharacteristic
187C2×C59#186611yescharacteristic
188C4×#41C2×D5911yescharacteristic
189#433—1182no—
190#432—1181no—
191#431—591no—
192#431—1181no—
193#423—591no—
194#423C2×D5911yescharacteristic
195C2×C4⋊C8—591no—
196C2×#104—21no—
197#408—1181no—
198#405—1181no—
199C2×C4×C8—591no—
200C4×C4⋊C4C2×D5911yescharacteristic
201#339—21no—
202#335—1181no—
203#335—1181no—
204#334—1182no—
205Q8⋊C8—591no—
206Q8⋊C8—591no—
207D4⋊C8—591no—
208D4⋊C8—591no—
209C59#890511yescharacteristic, complemented, Sylow
210C2×#41C22⋊D5911yescharacteristic
211C2×M16—21no—
212C2×M16—1181no—
213C22×C8—1181no—
214C22×C8—1181no—
215C4×Q8Dic11811yescharacteristic
216C4×Q8C4×D5911yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic11811yescharacteristic
220C4×D4C4×D5911yescharacteristic
221#117—21no—
222#117C22⋊D5911yescharacteristic
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×D5911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—591no—
230C4⋊C8—591no—
231C4⋊C8—592no—
232C4⋊C8—591no—
233C4⋊C8—591no—
234C4⋊C8—1181no—
235C4⋊C8—1181no—
236#104—42no—
237C22⋊C8—1182no—
238C22⋊C8—1182no—
239C22⋊C8—1181no—
240#97—1182no—
241C4×C8—591no—
242C4×C8—591no—
243C4×C8—1182no—
244C4×C8—1181no—
245C4×C8—1181no—
246#95—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic59⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic59⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4#1174568711yescharacteristic
259C22×C4#1174568711yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261M16—41no—
262M16—41no—
263M16—2361no—
264M16—2361no—
265C2×C8—41no—
266C2×C8—1181no—
267C2×C8—1181no—
268C2×C8—1181no—
269C2×C8—1181no—
270C2×C8—1181no—
271C2×C8—1182no—
272C2×C8—1181no—
273C2×C8—2362no—
274C2×C8—2361no—
275C2×C8—2361no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4Dic59⋊C411yescharacteristic
281C4⋊C4Dic59⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C2×Dic11811yescharacteristic
291C42C2×C4×D5911yescharacteristic
292C42D4×D5911yescharacteristic
293C42D4×D5911yescharacteristic
294C23—21no—
295C23#1427370811yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×Dic59⋊C411yescharacteristic
317C2×C4(C22⋊D59)⋊C411yescharacteristic
318C2×C4(C22⋊D59)⋊C411yescharacteristic
319C2×C4#1427369511yescharacteristic
320C2×C4#1427370411yescharacteristic
321C2×C4#1427370411yescharacteristic
322C2×C4#1427374711yescharacteristic
323C2×C4#1427374711yescharacteristic
324C2×C4#1427375311yescharacteristic
325C2×C4#1427375811yescharacteristic
326C8—81no—
327C8—1182no—
328C8—2361no—
329C8—2361no—
330C8—2361no—
331C8—2361no—
332C8—2361no—
333C8—2362no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1683727411yescharacteristic
340C22#1683772311yescharacteristic
341C22#1683774011yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4#1683743811yescharacteristic
350C4#1683744211yescharacteristic
351C4#1683751611yescharacteristic
352C4#1683752111yescharacteristic
353C4#1683754511yescharacteristic
354C4#1683756211yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2142429311yescharacteristic
359C2#2142433411yescharacteristic
360C2#2142436911yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1806C2
socleC22×C59#385
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#1806D59
hypocenterC59#8905
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21422872C2semidirect
3. #21422851C2semidirect
5. #21421999C2semidirect
6. #21421937C2semidirect
20. Q8⋊(C59⋊C8)C4semidirect
21. Q8⋊(C59⋊C8)C4semidirect
22. D4⋊(C59⋊C8)C4semidirect
23. D4⋊(C59⋊C8)C4semidirect
209. C59#8905semidirect

Maximal subgroups

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

subgroupindex
#214327622
#214228722
#214228512
#214224782
#214219992
#214219372
C59×#18062
#890559