Finite Group Databasebeta

← #28485374

Subgroups — #28485374

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435011C211yesmaximal, characteristic
2#21424920C211yesmaximal, characteristic, complemented
3#21422765C211yesmaximal, characteristic, complemented
4#21422656C211yesmaximal, characteristic, complemented
5#21421956C211yesmaximal, characteristic, complemented
6#21421927C211yesmaximal, characteristic, complemented
7C59×#2932C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9#16837778C412yescomplemented
10#16837750C412yescomplemented
11#16837597—22no—
12#16837535—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15C2×#14273649—21no—
16C2×#14273649—21no—
17#16837268C2211yescharacteristic
18#16837145—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×D236—21no—
28C22×D236—21no—
29C22×D236—21no—
30C2×#11745669—24no—
31#14273724—21no—
32#14273724—21no—
33#14273724—21no—
34#14273724C2×C412yes—
35C2×C4⋊Dic59—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38#14273649—42no—
39#14273649—42no—
40#14273647—24no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)—21no—
43C4×C59⋊C8—21no—
44C4×C59⋊C8C2×C412yes—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C2×#41×C59—21no—
48#131×C59—42no—
49#131×C59—42no—
50C2×M16×C59—22no—
51C2×M16×C59—21no—
52C22×C8×C59—22no—
53C22×C8×C59D411yescharacteristic
54C22×C8×C59D411yescharacteristic
55C4⋊Q8×C59C2×C411yescharacteristic
56#127×C59C2×C411yescharacteristic
57#124×C59—21no—
58#121×C59—21no—
59#121×C59D412yes—
60C4×D4×C59—21no—
61C4×D4×C59D412yes—
62C2×C42×C59C2311yescharacteristic
63C4⋊C8×C59—21no—
64C4⋊C8×C59D412yes—
65C22⋊C8×C59—24no—
66#97×C59—21no—
67#97×C59—21no—
68C23×D59—41no—
69#41×C59—41no—
70#41×C59—41no—
71#41×C59—41no—
72#41×C59—41no—
73C2×Q8×C59—21no—
74C2×D4×C59—22no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59C22⋊C412yes—
78C22×C4×C59—22no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×Dic59—41no—
83C2×D236—22no—
84C2×D236—22no—
85C2×D236—22no—
86C2×D236—41no—
87C2×D236—41no—
88C2×D236—41no—
89C2×D236—41no—
90C2×D236—41no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—82no—
94C2×D236—82no—
95M16×C59—44no—
96M16×C59—41no—
97M16×C59—41no—
98C2×C8×C59—21no—
99C2×C8×C59—21no—
100C2×C8×C59—21no—
101C2×C8×C59—22no—
102C2×C8×C59—21no—
103C2×C8×C59—21no—
104C2×C8×C59—44no—
105C2×C8×C59D812yescomplemented
106C4⋊C4×C59—21no—
107C4⋊C4×C59C22⋊C412yes—
108C22⋊C4×C59—22no—
109C22⋊C4×C59—22no—
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114#11745669—84no—
115C4⋊Dic59—41no—
116C4⋊Dic59—41no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8—41no—
119C23×C59—22no—
120C23×C59#12111yescharacteristic
121C22×D59—42no—
122C22×D59—82no—
123C22×D59—82no—
124C22×D59—81no—
125C22×D59—81no—
126Q8×C59—41no—
127Q8×C59—41no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—42no—
131D4×C59—41no—
132D4×C59—41no—
133D4×C59—81no—
134C2×C4×C59—22no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59C4×D411yescharacteristic
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×Dic59—81no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152D236—44no—
153D236—42no—
154D236—42no—
155D236—82no—
156D236—82no—
157D236—82no—
158D236—81no—
159D236—81no—
160C8×C59—22no—
161C8×C59—42no—
162C8×C59—41no—
163C59⋊C8—42no—
164C59⋊C8—81no—
165#9293—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59#38611yescharacteristic
172C22×C59#45911yescharacteristic
173C22×C59#46311yescharacteristic
174C2×D59—82no—
175C2×D59—82no—
176C2×D59—82no—
177C2×D59—162no—
178C2×D59—162no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—42no—
184C4×C59C4×D811yescharacteristic
185C4×C59#44211yescharacteristic
186C4×C59#44711yescharacteristic
187C4×C59#44911yescharacteristic
188Dic59—161no—
189#3205—591no—
190#2932D5911yescharacteristic, complemented
191#1961—591no—
192#1545—591no—
193#1517—591no—
194#1286—591no—
195#1275—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—82no—
199C2×C59#187911yescharacteristic
200C2×C59#190311yescharacteristic
201C2×C59#191011yescharacteristic
202D59—162no—
203C2×#131—21no—
204#532C2×D5911yescharacteristic
205C2×#127—591no—
206#493—592no—
207#486—592no—
208#459—1182no—
209#447—1182no—
210#435—22no—
211#435—22no—
212#431—591no—
213#431—591no—
214#423C2×D5911yescharacteristic
215C2×#102—1181no—
216C2×#102—1181no—
217#408—21no—
218C2×#97C2×D5911yescharacteristic
219#382—591no—
220#331—1182no—
221#327—1182no—
222C59#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—1181no—
225C22×D4—1181no—
226C22×D4—1181no—
227C2×D8—1184no—
228#131—42no—
229#131—42no—
230C2×M16—22no—
231C2×M16—21no—
232C22×C8—22no—
233C22×C8C22⋊D5911yescharacteristic
234C22×C8C22⋊D5911yescharacteristic
235C4⋊Q8C4×D5911yescharacteristic
236#127—592no—
237#127—1181no—
238#127—1181no—
239#127—1181no—
240#127C4×D5911yescharacteristic
241#124—21no—
242#121—21no—
243#121D23612yes—
244C4×D4—21no—
245C4×D4C22⋊D5912yes—
246C2×C4⋊C4—1181no—
247C2×C42C22×D5911yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—1181no—
250C4⋊C8—1181no—
251C4⋊C8C22⋊D5912yes—
252#102—1184no—
253#102—2362no—
254#102—2362no—
255C22⋊C8—24no—
256C22⋊C8—1181no—
257C22⋊C8—1181no—
258#97—21no—
259#97—21no—
260C4×C8—592no—
261C4×C8—1181no—
262#95—1181no—
263#95—1181no—
264C24—2361no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—1182no—
274C2×D4—1182no—
275C2×D4—1182no—
276C2×D4—4722no—
277C2×D4—4722no—
278C2×D4—2361no—
279C2×D4—2361no—
280C2×D4—2361no—
281C2×D4—2361no—
282C2×D4—2361no—
283C2×D4—2361no—
284C2×D4—2361no—
285C2×D4(C2×D59)⋊C412yes—
286C22×C4—22no—
287C22×C4—2361no—
288C22×C4C2×D23611yescharacteristic
289C22×C4#1174568511yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291D8—4724no—
292M16—44no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—1182no—
303C2×C8—2361no—
304C2×C8#1174566912yescomplemented
305C4⋊C4—21no—
306C4⋊C4—2361no—
307C4⋊C4—2361no—
308C4⋊C4(C2×D59)⋊C412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C2×C4×D5911yescharacteristic
312C42#1174568511yescharacteristic
313C42C2×C22⋊D5911yescharacteristic
314C42C2×C22⋊D5911yescharacteristic
315C23—22no—
316C23—4722no—
317C23—4722no—
318C23—4721no—
319C23—4721no—
320C23—2362no—
321C23#1427370111yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—4722no—
331D4—4722no—
332D4—4722no—
333D4—4721no—
334D4—4721no—
335D4—2364no—
336D4—2362no—
337D4—2362no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—4721no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4C2×(C2×D59)⋊C411yescharacteristic
347C2×C4C4×C22⋊D5911yescharacteristic
348C2×C4C4×C22⋊D5911yescharacteristic
349C2×C4#1427369611yescharacteristic
350C2×C4#1427370111yescharacteristic
351C2×C4#1427371011yescharacteristic
352C2×C4C2×#1174566911yescharacteristic
353C2×C4#1427374711yescharacteristic
354C2×C4#1427375211yescharacteristic
355C2×C4#1427375311yescharacteristic
356C8—22no—
357C8—42no—
358C8—41no—
359C8—4721no—
360C8—2362no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—9442no—
367C22—9442no—
368C22—4722no—
369C22—4722no—
370C22—4722no—
371C22#1683728111yescharacteristic
372C22#1683759711yescharacteristic
373C22#1683761711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—9441no—
380C4C4×#1174566911yescharacteristic
381C4#1683751211yescharacteristic
382C4#1683753611yescharacteristic
383C4#1683754711yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—9442no—
388C2#2142444411yescharacteristic
389C2#2142457611yescharacteristic
390C2#2142462311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2932C2
socleC22×C59#386
prime radical#21435011C2
cosocleC2×C42×C59C23
hypercenter#2932D59
hypocenterC59#9293
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. #21424920C2semidirect
3. #21422765C2semidirect
4. #21422656C2semidirect
5. #21421956C2semidirect
6. #21421927C2semidirect
7. C59×#2932C2semidirect
9. #16837778C4semidirect
10. #16837750C4semidirect
105. C2×C8×C59D8semidirect
190. #2932D59semidirect
222. C59#9293semidirect
304. C2×C8#11745669semidirect

Maximal subgroups

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

subgroupindex
#214350112
#214249202
#214227652
#214226562
#214219562
#214219272
C59×#29322
#929359