Finite Group Databasebeta

← #28485376

Subgroups — #28485376

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#21432814C211yesmaximal, characteristic, complemented
2#21424921C211yesmaximal, characteristic, complemented
3#21422764C211yesmaximal, characteristic, complemented
4#21422656C211yesmaximal, characteristic, complemented
5#21421955C211yesmaximal, characteristic, complemented
6#21421926C211yesmaximal, characteristic, complemented
7C59×#3205C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837474—22no—
10#16837474—22no—
11#16837455C2211yescharacteristic
12#16837426C2211yescharacteristic
13C2×#14273648—21no—
14C2×#14273648—21no—
15#16837369—21no—
16C2×#14273638C2211yescharacteristic
17#16837268C2211yescharacteristic
18#16837146—22no—
19#16837136—22no—
20C59×#532C2211yescharacteristic
21C2×#127×C59C2211yescharacteristic
22C59×#493C412yescomplemented
23C59×#486C412yescomplemented
24C59×#459—22no—
25C59×#447—22no—
26C59×#431C2211yescharacteristic
27#14273734—42no—
28#14273734—42no—
29C2×#11745664—22no—
30C2×#11745664—21no—
31C22×C59⋊C8—22no—
32C22×C59⋊C8D411yescharacteristic
33C22×C59⋊C8D411yescharacteristic
34C2×C4⋊Dic59—21no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)D412yes—
38#14273648—42no—
39#14273648—42no—
40C22⋊(C59⋊C8)—24no—
41C22⋊(C59⋊C8)—21no—
42#14273638—21no—
43#14273638—21no—
44C4×C59⋊C8—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C2×#41×C59—21no—
48C22×D4×C59—21no—
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C2×D8×C59—24no—
52C4⋊Q8×C59C2×C411yescharacteristic
53#127×C59—21no—
54#127×C59—21no—
55#127×C59—21no—
56#127×C59C2×C412yes—
57#127×C59C2×C411yescharacteristic
58#124×C59—21no—
59#121×C59—21no—
60#121×C59D412yes—
61C4×D4×C59—21no—
62C4×D4×C59D412yes—
63C2×C42×C59C2311yescharacteristic
64C4⋊C8×C59—21no—
65#102×C59—24no—
66C22⋊C8×C59—21no—
67C4×C8×C59C2×C412yes—
68C24×C59—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—22no—
76C2×D4×C59—21no—
77C2×D4×C59—22no—
78C2×D4×C59—21no—
79C2×D4×C59—22no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—82no—
88C2×D4×C59—82no—
89C2×D4×C59C22⋊C412yes—
90C22×C4×C59—22no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×Dic59—41no—
95D8×C59—84no—
96C2×C8×C59—22no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59C22⋊C412yes—
99C22⋊C4×C59—22no—
100C22⋊C4×C59—22no—
101C42×C59C22×C411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59#4111yescharacteristic
105C4⋊Dic59—41no—
106C4⋊Dic59—41no—
107#11745664—44no—
108#11745664—41no—
109#11745664—41no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—21no—
115C2×C59⋊C8—21no—
116C2×C59⋊C8—44no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8D812yescomplemented
119C23×C59—22no—
120C23×C59—42no—
121C23×C59—82no—
122C23×C59—82no—
123C23×C59—81no—
124C23×C59—81no—
125C23×C59#12111yescharacteristic
126Q8×C59—41no—
127Q8×C59—41no—
128D4×C59—42no—
129D4×C59—44no—
130D4×C59—42no—
131D4×C59—42no—
132D4×C59—41no—
133D4×C59—42no—
134D4×C59—41no—
135D4×C59—42no—
136D4×C59—81no—
137D4×C59—82no—
138D4×C59—82no—
139D4×C59—82no—
140D4×C59—81no—
141D4×C59—81no—
142C2×C4×C59—22no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59C2×C22⋊C411yescharacteristic
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59C4×D411yescharacteristic
150C2×C4×C59#12011yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59#12311yescharacteristic
153C2×C4×C59C2×D811yescharacteristic
154C2×C4×C59#13511yescharacteristic
155C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
157C2×Dic59—81no—
158C2×Dic59—81no—
159C2×Dic59—81no—
160C8×C59—42no—
161C59⋊C8—22no—
162C59⋊C8—42no—
163C59⋊C8—41no—
164C59⋊C8—81no—
165#9293—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—82no—
172C22×C59—82no—
173C22×C59—82no—
174C22×C59—162no—
175C22×C59—162no—
176C22×C59#38611yescharacteristic
177C22×C59#45911yescharacteristic
178C22×C59#46311yescharacteristic
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#3205D5911yescharacteristic
190#2932—591no—
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—162no—
200C2×C59#187911yescharacteristic
201C2×C59#190311yescharacteristic
202C2×C59#191011yescharacteristic
203C2×#131—1181no—
204#532C2×D5911yescharacteristic
205C2×#127C2×D5911yescharacteristic
206#493Dic5912yescomplemented
207#486Dic5912yescomplemented
208#459—22no—
209#447—22no—
210#435—1182no—
211#435—1182no—
212#431—591no—
213#431C2×D5911yescharacteristic
214#423—591no—
215C2×#102—1181no—
216C2×#102—1181no—
217#408—1181no—
218C2×#97—591no—
219#382—591no—
220#331—1182no—
221#327—1182no—
222C59#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C2×D8—24no—
228#131—2362no—
229#131—2362no—
230C2×M16—1182no—
231C2×M16—1181no—
232C22×C8—591no—
233C22×C8—591no—
234C22×C8—1182no—
235C4⋊Q8C2×Dic5911yescharacteristic
236#127—21no—
237#127—21no—
238#127—21no—
239#127C2×Dic5912yes—
240#127C2×Dic5911yescharacteristic
241#124—21no—
242#121—21no—
243#121C22⋊D5912yes—
244C4×D4—21no—
245C4×D4C22⋊D5912yes—
246C2×C4⋊C4—1181no—
247C2×C42C22×D5911yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—592no—
250C4⋊C8—1181no—
251C4⋊C8—1181no—
252#102—24no—
253#102—2362no—
254#102—2362no—
255C22⋊C8—21no—
256C22⋊C8—1184no—
257C22⋊C8—1181no—
258#97—1181no—
259#97—1181no—
260C4×C8—1181no—
261C4×C8C2×Dic5912yes—
262#95—1181no—
263#95—1181no—
264C24—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C22⋊Dic5912yes—
286C22×C4—22no—
287C22×C4—2361no—
288C22×C4D4×D5911yescharacteristic
289C22×C4#1174568711yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291D8—84no—
292M16—2364no—
293M16—2361no—
294M16—2361no—
295C2×C8—22no—
296C2×C8—592no—
297C2×C8—1181no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1182no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—2364no—
304C2×C8—2361no—
305C4⋊C4—21no—
306C4⋊C4—2361no—
307C4⋊C4—2361no—
308C4⋊C4C22⋊Dic5912yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42D4×D5911yescharacteristic
312C42#1174568711yescharacteristic
313C42C22×Dic5911yescharacteristic
314C42C2×C22⋊D5911yescharacteristic
315C23—22no—
316C23—42no—
317C23—82no—
318C23—82no—
319C23—81no—
320C23—81no—
321C23#1427369911yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—42no—
330D4—41no—
331D4—42no—
332D4—81no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—81no—
337D4—81no—
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×C22⋊Dic5911yescharacteristic
347C2×C4D4×Dic5911yescharacteristic
348C2×C4D4×Dic5911yescharacteristic
349C2×C4#1427369511yescharacteristic
350C2×C4#1427369911yescharacteristic
351C2×C4#1427370811yescharacteristic
352C2×C4D8×D5911yescharacteristic
353C2×C4#1427374611yescharacteristic
354C2×C4#1427375111yescharacteristic
355C2×C4#1427375111yescharacteristic
356C8—42no—
357C8—1182no—
358C8—4721no—
359C8—2362no—
360C8—2361no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—162no—
370C22—162no—
371C22#1683728211yescharacteristic
372C22#1683759511yescharacteristic
373C22#1683761511yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—9441no—
380C4D8×Dic5911yescharacteristic
381C4#1683751411yescharacteristic
382C4#1683753411yescharacteristic
383C4#1683754411yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—162no—
388C2#2142444611yescharacteristic
389C2#2142457811yescharacteristic
390C2#2142462411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3205C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3205D59
hypocenterC59#9293
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21432814C2semidirect
2. #21424921C2semidirect
3. #21422764C2semidirect
4. #21422656C2semidirect
5. #21421955C2semidirect
6. #21421926C2semidirect
22. C59×#493C4semidirect
23. C59×#486C4semidirect
118. C2×C59⋊C8D8semidirect
206. #493Dic59semidirect
207. #486Dic59semidirect
222. C59#9293semidirect

Maximal subgroups

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

subgroupindex
#214328142
#214249212
#214227642
#214226562
#214219552
#214219262
C59×#32052
#929359