Finite Group Databasebeta

← #28485239

Subgroups — #28485239

Order 15104.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435034C211yesmaximal, characteristic
2#21433045C211yesmaximal, characteristic, complemented
3#21424935C211yesmaximal, characteristic
4C2×#16837145C211yesmaximal, characteristic
5#21421991C211yesmaximal, characteristic, complemented
6#21421946C211yesmaximal, characteristic, complemented
7C59×#1550C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9#16837776—21no—
10#16837750—21no—
11#16837602—22no—
12#16837552C414yes—
13#16837533—21no—
14#16837533—22no—
15Q8×C59⋊C8—21no—
16#16837455C2211yescharacteristic
17#16837429—21no—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×#14273643—21no—
20C2×#14273643—21no—
21#16837358—21no—
22C4×#11745664C2211yescharacteristic
23#16837145—21no—
24#16837145—21no—
25C59×#533C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C2×C4⋊C8×C59C2211yescharacteristic
28C59×#333—21no—
29C59×#332—21no—
30C22×D236—21no—
31C22×D236—21no—
32C22×D236C2×C412yes—
33C2×#11745671—24no—
34C2×#11745664D411yescharacteristic
35C2×#11745664D411yescharacteristic
36C22×C59⋊C8—21no—
37#14273724—21no—
38#14273724—21no—
39#14273724—21no—
40#14273724—21no—
41C4⋊(C59⋊C8)—22no—
42C4⋊(C59⋊C8)—22no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D411yescharacteristic
47#14273647—24no—
48#14273643—42no—
49#14273643—42no—
50C22⋊(C59⋊C8)C2×C412yes—
51#14273638—22no—
52#14273638—21no—
53C4×C59⋊C8—22no—
54C4×C59⋊C8—21no—
55C4×C59⋊C8—21no—
56C2×M16×C59—21no—
57C22×C8×C59—21no—
58C4⋊Q8×C59D411yescharacteristic
59C4⋊Q8×C59D411yescharacteristic
60#125×C59—21no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59C2×C412yes—
63C4×Q8×C59—21no—
64C4×Q8×C59D412yes—
65#117×C59—21no—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59—21no—
70C4⋊C8×C59—21no—
71C23×D59—22no—
72C2×Q8×C59—21no—
73C2×Q8×C59C22⋊C412yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59#4111yescharacteristic
77C2×D236—22no—
78C2×D236—22no—
79C2×D236—41no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—41no—
86C2×D236—41no—
87C2×D236—82no—
88C2×D236—81no—
89C2×D236—81no—
90M16×C59—41no—
91M16×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—81no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59C22⋊C412yes—
101C22⋊C4×C59—22no—
102C42×C59—22no—
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59C2×D411yescharacteristic
106C42×C59#4111yescharacteristic
107#11745671—84no—
108#11745664—21no—
109#11745664—21no—
110#11745664—21no—
111#11745664SD1612yescomplemented
112C2×C59⋊C8—21no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—21no—
115C2×C59⋊C8—21no—
116C2×C59⋊C8—21no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C23×C59C4×D411yescharacteristic
120C22×D59—42no—
121C22×D59—42no—
122C22×D59—42no—
123C22×D59—82no—
124C22×D59—82no—
125Q8×C59—42no—
126Q8×C59—41no—
127Q8×C59—41no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—41no—
133C2×C4×C59C2×C22⋊C411yescharacteristic
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59#12011yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59#12111yescharacteristic
138C2×C4×C59#12311yescharacteristic
139C2×C4×C59C2×SD1611yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C59#13711yescharacteristic
143D236—44no—
144D236—82no—
145D236—82no—
146D236—81no—
147D236—81no—
148D236—81no—
149D236—81no—
150D236—81no—
151D236—81no—
152C8×C59—81no—
153C8×C59—81no—
154C59⋊C8—22no—
155C59⋊C8—42no—
156C59⋊C8—41no—
157C59⋊C8—41no—
158C59⋊C8—41no—
159#9274—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59#38611yescharacteristic
163C22×C59C4×SD1611yescharacteristic
164C22×C59#44011yescharacteristic
165C2×D59—82no—
166C2×D59—82no—
167C2×D59—82no—
168C2×D59—162no—
169C2×D59—162no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—42no—
174C4×C59—42no—
175C4×C59#44811yescharacteristic
176C4×C59#44911yescharacteristic
177C4×C59#46011yescharacteristic
178C4×C59#46411yescharacteristic
179C4×C59#47111yescharacteristic
180C4×C59#47211yescharacteristic
181#3208—591no—
182#2962—591no—
183#1965—591no—
184#1550D5911yescharacteristic, complemented
185C2×#331—591no—
186#1301—591no—
187#1283—591no—
188C2×C59—21no—
189C2×C59—21no—
190C2×C59#188011yescharacteristic
191C2×C59#190711yescharacteristic
192C2×C59#191411yescharacteristic
193D59—162no—
194#533C2×D5911yescharacteristic
195C2×#127—591no—
196#492—1181no—
197#486—1181no—
198#460—1182no—
199#450—594no—
200#446—1181no—
201#446—1182no—
202C8×Q8—1181no—
203#431—591no—
204#424—1181no—
205#423C2×D5911yescharacteristic
206C2×C4⋊C8—591no—
207C2×C4⋊C8C2×D5911yescharacteristic
208C2×#100—1181no—
209C2×#100—1181no—
210#405—1181no—
211C4×M16—591no—
212#333—21no—
213#332—21no—
214#331—1181no—
215#331—1181no—
216C59#927411yescharacteristic, complemented, Sylow
217C22×D4—592no—
218C22×D4—1181no—
219C22×D4—1181no—
220C2×SD16—1184no—
221C2×M16—21no—
222C2×M16—591no—
223C2×M16—591no—
224C22×C8—21no—
225C22×C8—1181no—
226C4⋊Q8C22⋊D5911yescharacteristic
227C4⋊Q8C22⋊D5911yescharacteristic
228#127—1181no—
229#127—1181no—
230#127—1181no—
231#127—1181no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8C4×D5912yes—
235C4×Q8—21no—
236C4×Q8D23612yes—
237#117—21no—
238C2×C42C22×D5911yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—591no—
244C4⋊C8—591no—
245C4⋊C8—1182no—
246C4⋊C8—1182no—
247C4⋊C8—1181no—
248C4⋊C8—1181no—
249#102—1184no—
250#100—2362no—
251#100—2362no—
252C22⋊C8—592no—
253#97—1182no—
254#97—1181no—
255C4×C8—1182no—
256C4×C8—1181no—
257C4×C8—1181no—
258C24—1182no—
259C2×Q8—21no—
260C2×Q8(C2×D59)⋊C412yes—
261C2×D4—1182no—
262C2×D4—1182no—
263C2×D4—4721no—
264C2×D4—4721no—
265C2×D4—4722no—
266C2×D4—2361no—
267C2×D4—2361no—
268C2×D4—2361no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2361no—
272C2×D4—2361no—
273C2×D4—2361no—
274C22×C4C2×C4×D5911yescharacteristic
275C22×C4D4×D5911yescharacteristic
276C22×C4#1174568711yescharacteristic
277SD16—4724no—
278M16—41no—
279M16—41no—
280M16—592no—
281M16—1181no—
282M16—1181no—
283M16—1181no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—81no—
287C2×C8—1181no—
288C2×C8—1182no—
289C2×C8—1181no—
290C2×C8—1181no—
291C2×C8—1181no—
292C2×C8—2361no—
293C2×C8—2361no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4(C2×D59)⋊C412yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D23611yescharacteristic
303C42D4×D5911yescharacteristic
304C42#1174568711yescharacteristic
305C42C2×C22⋊D5911yescharacteristic
306C23—4722no—
307C23—4722no—
308C23—2362no—
309C23—2362no—
310C23—2362no—
311C23(C22⋊D59)⋊C411yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—4722no—
316D4—4722no—
317D4—4721no—
318D4—4721no—
319D4—4721no—
320D4—4721no—
321D4—4721no—
322D4—4721no—
323D4—2364no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×(C2×D59)⋊C411yescharacteristic
330C2×C4(C22⋊D59)⋊C411yescharacteristic
331C2×C4#1427369711yescharacteristic
332C2×C4#1427369911yescharacteristic
333C2×C4#1427369911yescharacteristic
334C2×C4#1427371111yescharacteristic
335C2×C4SD16×D5911yescharacteristic
336C2×C4#1427375011yescharacteristic
337C2×C4#1427375411yescharacteristic
338C2×C4#1427375911yescharacteristic
339C8—81no—
340C8—81no—
341C8—1182no—
342C8—2362no—
343C8—2361no—
344C8—2361no—
345C8—2361no—
346C22—21no—
347C22—21no—
348C22—9442no—
349C22—9442no—
350C22—4722no—
351C22—4722no—
352C22—4722no—
353C22#1683727911yescharacteristic
354C22#1683748711yescharacteristic
355C22#1683750511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4#1683754211yescharacteristic
362C4#1683755011yescharacteristic
363C4#1683760011yescharacteristic
364C4#1683762611yescharacteristic
365C4#1683766711yescharacteristic
366C4#1683767011yescharacteristic
367C2—21no—
368C2—21no—
369C2—9442no—
370C2#2142444911yescharacteristic
371C2#2142460211yescharacteristic
372C2#2142464111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1550C2
socleC22×C59#386
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenter#1550D59
hypocenterC59#9274
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
2. #21433045C2semidirect
5. #21421991C2semidirect
6. #21421946C2semidirect
7. C59×#1550C2semidirect
111. #11745664SD16semidirect
184. #1550D59semidirect
216. C59#9274semidirect

Maximal subgroups

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

subgroupindex
#214350342
#214330452
#214249352
C2×#168371452
#214219912
#214219462
C59×#15502
#927459