Finite Group Databasebeta

← #28483005

Subgroups — #28483005

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432664C211yesmaximal, characteristic
2#21424116C211yesmaximal, characteristic
3#21422867C211yesmaximal, characteristic, complemented
4#21422852C211yesmaximal, characteristic, complemented
5#21421998C211yesmaximal, characteristic, complemented
6#21421936C211yesmaximal, characteristic, complemented
7C59×#1476C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837429—22no—
10#16837429—21no—
11#16837426—21no—
12#16837426C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)—21no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273654—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837161—21no—
19#16837153—22no—
20#16837153—21no—
21#16837153—21no—
22C4×#41×C59C2211yescharacteristic
23C59×#431C2211yescharacteristic
24C2×C4×C8×C59C2211yescharacteristic
25C59×Q8⋊C8C411yescharacteristic, complemented
26C59×Q8⋊C8C411yescharacteristic, complemented
27C59×D4⋊C8C411yescharacteristic, complemented
28C59×D4⋊C8C411yescharacteristic, complemented
29#14273734—42no—
30#14273734—42no—
31C2×#11745664—22no—
32C2×#11745664—21no—
33C2×#11745664—21no—
34C22×C59⋊C8—22no—
35C22×C59⋊C8—21no—
36C2×C4⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—22no—
39C4⋊(C59⋊C8)—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)D411yescharacteristic
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D412yes—
47#14273654—42no—
48C59⋊#95—21no—
49C2×#41×C59D411yescharacteristic
50C22×C8×C59—21no—
51C4×Q8×C59C2×C411yescharacteristic
52C4×Q8×C59Q811yescharacteristic
53C4×D4×C59—21no—
54C4×D4×C59—21no—
55C4×D4×C59C2×C411yescharacteristic
56C4×D4×C59Q811yescharacteristic
57#117×C59—21no—
58#117×C59D411yescharacteristic
59C2×C42×C59—21no—
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59C2×C411yescharacteristic
62C4⋊C8×C59C2×C411yescharacteristic
63C22⋊C8×C59—21no—
64C4×C8×C59—21no—
65C4×C8×C59—21no—
66C4×C8×C59C2×C411yescharacteristic
67C4×C8×C59C2×C411yescharacteristic
68#41×C59—21no—
69#41×C59—21no—
70#41×C59SD1612yes—
71#41×C59SD1612yes—
72C2×Q8×C59C4⋊C411yescharacteristic
73C2×D4×C59—21no—
74C2×D4×C59C4⋊C411yescharacteristic
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×Dic59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—42no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59C4⋊C411yescharacteristic
90C4⋊C4×C59C4⋊C411yescharacteristic
91C22⋊C4×C59—21no—
92C22⋊C4×C59—21no—
93C22⋊C4×C59—21no—
94C42×C59—21no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59C22×C411yescharacteristic
98C42×C59C2×D411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59C2×Q811yescharacteristic
101C4⋊Dic59—22no—
102C4⋊Dic59—41no—
103C4×Dic59—41no—
104C4×Dic59—41no—
105#11745664—44no—
106#11745664—41no—
107#11745664—41no—
108#11745664—41no—
109#11745664—41no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—44no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C23×C59—21no—
118C23×C59#12311yescharacteristic
119Q8×C59—21no—
120Q8×C59#10612yes—
121D4×C59—22no—
122D4×C59—22no—
123D4×C59—21no—
124D4×C59#10612yes—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—81no—
136C2×C4×C59C2×C4⋊C411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59C22⋊Q811yescharacteristic
141C2×C4×C59C22⋊Q811yescharacteristic
142C2×C4×C59C2×SD1611yescharacteristic
143C2×C4×C59C2×SD1611yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C59#13711yescharacteristic
146C2×Dic59—42no—
147C2×Dic59—81no—
148C2×Dic59—81no—
149C8×C59—22no—
150C8×C59—41no—
151C8×C59—41no—
152C8×C59—41no—
153C59⋊C8—41no—
154C59⋊C8—42no—
155C59⋊C8—41no—
156C59⋊C8—81no—
157#8904—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—22no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59#38511yescharacteristic
164C22×C59#48111yescharacteristic
165C22×C59#48311yescharacteristic
166C4×C59—22no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59C2×#10611yescharacteristic
173C4×C59#42811yescharacteristic
174C4×C59#44311yescharacteristic
175C4×C59#44411yescharacteristic
176C4×C59#44811yescharacteristic
177C4×C59#45011yescharacteristic
178Dic59—82no—
179#2906—591no—
180#1806—591no—
181#1570—591no—
182#1566—591no—
183#1476D5911yescharacteristic
184#1303—591no—
185#1279—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—42no—
189C2×C59#185311yescharacteristic
190C2×C59#186111yescharacteristic
191C2×C59#186611yescharacteristic
192C2×#131—1181no—
193C4×#41C2×D5911yescharacteristic
194#431C2×D5911yescharacteristic
195#424—1182no—
196#424—1181no—
197#423—591no—
198#423—591no—
199#423—1181no—
200C2×C4⋊C8—591no—
201C2×C4⋊C8—1181no—
202C2×#104—1181no—
203C2×C4×C8C2×D5911yescharacteristic
204C4×C4⋊C4—591no—
205#339—1181no—
206#334—1182no—
207#334—1181no—
208#334—1181no—
209Q8⋊C8Dic5911yescharacteristic, complemented
210Q8⋊C8Dic5911yescharacteristic, complemented
211D4⋊C8Dic5911yescharacteristic, complemented
212D4⋊C8Dic5911yescharacteristic, complemented
213C59#890411yescharacteristic, complemented, Sylow
214C2×#41D23611yescharacteristic
215#131—2362no—
216#131—2362no—
217C2×M16—1182no—
218C2×M16—1181no—
219C2×M16—1181no—
220C22×C8—21no—
221C22×C8—1182no—
222C22×C8—1181no—
223C4×Q8Dic11811yescharacteristic
224C4×Q8C2×Dic5911yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic11811yescharacteristic
228C4×D4C2×Dic5911yescharacteristic
229#117—21no—
230#117D23611yescharacteristic
231C2×C4⋊C4—1181no—
232C2×C42—21no—
233C2×C42—1181no—
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—591no—
236C4⋊C8—591no—
237C4⋊C8—592no—
238C4⋊C8—1182no—
239C4⋊C8—1182no—
240C4⋊C8—1181no—
241C4⋊C8—1181no—
242C4⋊C8—1181no—
243C4⋊C8—1181no—
244C4⋊C8C2×Dic5911yescharacteristic
245C4⋊C8C2×Dic5911yescharacteristic
246#104—2362no—
247C22⋊C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C2×Dic5911yescharacteristic
251C4×C8C2×Dic5911yescharacteristic
252#95—1181no—
253#41—21no—
254#41—21no—
255#41#1174566012yes—
256#41#1174566012yes—
257C2×Q8C4⋊Dic5911yescharacteristic
258C2×D4—21no—
259C2×D4C4⋊Dic5911yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—2361no—
264C22×C4C2×D23611yescharacteristic
265C22×C4#1174568711yescharacteristic
266C22×C4#1174568711yescharacteristic
267M16—2361no—
268M16—2361no—
269M16—2364no—
270M16—2361no—
271M16—2361no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—42no—
277C2×C8—1181no—
278C2×C8—1181no—
279C2×C8—1182no—
280C2×C8—2361no—
281C2×C8—2364no—
282C2×C8—2361no—
283C2×C8—2361no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—1182no—
287C4⋊C4—2361no—
288C4⋊C4C4⋊Dic5911yescharacteristic
289C4⋊C4C4⋊Dic5911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—2361no—
297C42—2361no—
298C42C2×Dic11811yescharacteristic
299C42D4×D5911yescharacteristic
300C42D4×D5911yescharacteristic
301C42C22×Dic5911yescharacteristic
302C23—21no—
303C23#1427371111yescharacteristic
304Q8—21no—
305Q8#1427365912yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1427365912yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4—2362no—
324C2×C4C2×C4⋊Dic5911yescharacteristic
325C2×C4D4×Dic5911yescharacteristic
326C2×C4D4×Dic5911yescharacteristic
327C2×C4#1427369711yescharacteristic
328C2×C4#1427370411yescharacteristic
329C2×C4#1427370411yescharacteristic
330C2×C4C2×#1174566011yescharacteristic
331C2×C4C2×#1174566011yescharacteristic
332C2×C4#1427375511yescharacteristic
333C2×C4#1427376011yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—4721no—
339C8—2361no—
340C8—2362no—
341C8—2361no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1683727711yescharacteristic
348C22#1683772111yescharacteristic
349C22#1683773311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—4722no—
357C4C2×#1427365911yescharacteristic
358C4#1683744411yescharacteristic
359C4#1683751911yescharacteristic
360C4#1683752411yescharacteristic
361C4#1683754311yescharacteristic
362C4#1683755511yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2142429111yescharacteristic
367C2#2142433811yescharacteristic
368C2#2142437311yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1476C2
socleC22×C59#385
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#1476D59
hypocenterC59#8904
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
3. #21422867C2semidirect
4. #21422852C2semidirect
5. #21421998C2semidirect
6. #21421936C2semidirect
25. C59×Q8⋊C8C4semidirect
26. C59×Q8⋊C8C4semidirect
27. C59×D4⋊C8C4semidirect
28. C59×D4⋊C8C4semidirect
209. Q8⋊C8Dic59semidirect
210. Q8⋊C8Dic59semidirect
211. D4⋊C8Dic59semidirect
212. D4⋊C8Dic59semidirect
213. C59#8904semidirect

Maximal subgroups

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

subgroupindex
#214326642
#214241162
#214228672
#214228522
#214219982
#214219362
C59×#14762
#890459