Finite Group Databasebeta

← #28507047

Subgroups — #28507047

Order 15104.

432 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433751C211yesmaximal, characteristic
2#21425790C212yesmaximal
3#21425751C212yesmaximal
4#21423896C211yesmaximal, characteristic
5C59×#1813C211yesmaximal, characteristic
6C22×Dic236C411yescharacteristic, complemented
7#16837973C2212yes—
8#16837660—21no—
9#16837660—21no—
10#16837643—21no—
11#16837643—21no—
12#16837440C411yescharacteristic, complemented
13#16837412C412yescomplemented
14#16837386C2211yescharacteristic
15#16837376C2211yescharacteristic
16#16837357—21no—
17#16837200—24no—
18#16837193—24no—
19#16837173—21no—
20C2×#131×C59C2211yescharacteristic
21C2×#108×C59—21no—
22C59×#413C2212yes—
23C59×#338—21no—
24C22×Dic118C2×C412yes—
25C2×Dic236—21no—
26C2×Dic236—21no—
27C2×Dic236—41no—
28C2×Dic236—41no—
29C2×#11745664—21no—
30#14273725—22no—
31#14273714—22no—
32#14273708—22no—
33#14273708—22no—
34#14273706—22no—
35#14273706—22no—
36#14273686—21no—
37#14273686C2×C412yes—
38#14273661—21no—
39#14273659—21no—
40#14273652—22no—
41#14273652—22no—
42#14273646D412yes—
43#14273644—21no—
44#14273642—21no—
45#14273642D412yes—
46#14273637—41no—
47C8×Dic59—41no—
48C2×#41×C59C2311yescharacteristic
49#131×C59—41no—
50#131×C59—41no—
51C2×M16×C59—22no—
52C2×M16×C59C2×C411yescharacteristic
53C2×M16×C59D411yescharacteristic
54C2×M16×C59D411yescharacteristic
55C22×C8×C59C2×C411yescharacteristic
56C22×C8×C59D411yescharacteristic
57C22×C8×C59D411yescharacteristic
58#108×C59—41no—
59#108×C59—41no—
60#101×C59—22no—
61#100×C59—22no—
62#41×C59—41no—
63#41×C59—41no—
64C2×Q8×C59C2×D411yescharacteristic
65C2×D4×C59C2×D411yescharacteristic
66C2×D4×C59#4111yescharacteristic
67C2×D4×C59#4111yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—22no—
72C2×Dic118—22no—
73C2×Dic118—22no—
74C2×Dic118—42no—
75C2×Dic118—42no—
76M16×C59—41no—
77M16×C59—41no—
78M16×C59—41no—
79M16×C59—42no—
80M16×C59—42no—
81C2×C8×C59—21no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—42no—
88C2×C8×C59C22⋊C411yescharacteristic
89C2×C8×C59C22⋊C411yescharacteristic
90C2×C8×C59C22⋊C411yescharacteristic
91C2×C8×C59C22⋊C411yescharacteristic
92C22⋊Dic59—22no—
93C22⋊Dic59—22no—
94C22⋊Dic59—42no—
95C22⋊Dic59—42no—
96C4⋊Dic59—22no—
97C4⋊Dic59—41no—
98Dic59⋊C4—42no—
99Dic59⋊C4—42no—
100C4×Dic59—22no—
101C4×Dic59—41no—
102#11745664—22no—
103#11745664—41no—
104C2×C59⋊C8—41no—
105Dic236—42no—
106Dic236—41no—
107Dic236—41no—
108C23×C59C4×D411yescharacteristic
109C23×C59#12111yescharacteristic
110C23×C59#12411yescharacteristic
111Q8×C59—41no—
112D4×C59—41no—
113D4×C59—41no—
114D4×C59—41no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59C2×C22⋊C411yescharacteristic
122C2×C4×C59C4×D411yescharacteristic
123C2×C4×C59#12111yescharacteristic
124C2×C4×C59#12711yescharacteristic
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—82no—
129Dic118—42no—
130Dic118—42no—
131Dic118—82no—
132C8×C59—21no—
133C8×C59—21no—
134C8×C59—21no—
135C8×C59—82no—
136C8×C59#9911yescharacteristic
137C8×C59#9911yescharacteristic
138C59⋊C8—42no—
139#13233—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59#39011yescharacteristic
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59C2×#9911yescharacteristic
151C4×C59#41011yescharacteristic
152Dic59—82no—
153Dic59—82no—
154Dic59—82no—
155#3059—591no—
156#2135—592no—
157#2123—592no—
158#1813D5911yescharacteristic
159#1757—591no—
160C2×C59—21no—
161C2×C59—41no—
162C2×C59—41no—
163C2×C59—41no—
164C2×C59#196411yescharacteristic
165C22×Q16—591no—
166C2×#131C2×D5911yescharacteristic
167#536—592no—
168#470—1181no—
169#470—1181no—
170#467—1181no—
171#467—1181no—
172C2×#108—21no—
173#427—591no—
174#419—592no—
175#413—591no—
176#413C2×D5912yes—
177#410—591no—
178#405—1181no—
179#356—1184no—
180#352—1184no—
181#344—1181no—
182#338—21no—
183C59#1323311yescharacteristic, complemented, Sylow
184C2×#41C22×D5911yescharacteristic
185C22×Q8—592no—
186C2×Q16—1181no—
187C2×Q16—1181no—
188C2×Q16—2361no—
189C2×Q16—2361no—
190#131—41no—
191#131—41no—
192C2×M16—22no—
193C2×M16—1181no—
194C2×M16C4×D5911yescharacteristic
195C2×M16D23611yescharacteristic
196C2×M16C22⋊D5911yescharacteristic
197C22×C8C4×D5911yescharacteristic
198C22×C8D23611yescharacteristic
199C22×C8C22⋊D5911yescharacteristic
200C4⋊Q8—1182no—
201#124—1182no—
202#123—1182no—
203#123—1182no—
204C22⋊Q8—1182no—
205C22⋊Q8—1182no—
206#117—592no—
207#117—1181no—
208#108—41no—
209#108—41no—
210#107—1181no—
211#106—1181no—
212#103—1182no—
213#103—1182no—
214#101—22no—
215#101—592no—
216#100—22no—
217#100—1181no—
218#99—592no—
219#99—1181no—
220#97—2361no—
221C4×C8—2361no—
222#41—41no—
223#41—41no—
224C2×Q8—1182no—
225C2×Q8—1182no—
226C2×Q8—2362no—
227C2×Q8—2362no—
228C2×Q8D4×D5911yescharacteristic
229C2×D4D4×D5911yescharacteristic
230C2×D4#1174568911yescharacteristic
231C2×D4#1174568911yescharacteristic
232C22×C4—1182no—
233C22×C4C2×C4×D5911yescharacteristic
234C22×C4C2×D23611yescharacteristic
235C22×C4C2×C22⋊D5911yescharacteristic
236Q16—2362no—
237Q16—2361no—
238Q16—2361no—
239M16—41no—
240M16—41no—
241M16—41no—
242M16—42no—
243M16—42no—
244M16—1182no—
245M16—2361no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—42no—
253C2×C8—2361no—
254C2×C8(C2×D59)⋊C411yescharacteristic
255C2×C8(C2×D59)⋊C411yescharacteristic
256C2×C8(C2×D59)⋊C411yescharacteristic
257C2×C8(C2×D59)⋊C411yescharacteristic
258C4⋊C4—1182no—
259C4⋊C4—2361no—
260C4⋊C4—2362no—
261C4⋊C4—2362no—
262C22⋊C4—1182no—
263C22⋊C4—1182no—
264C22⋊C4—2362no—
265C22⋊C4—2362no—
266C42—1182no—
267C42—2361no—
268C23#1427368911yescharacteristic
269C23#1427370211yescharacteristic
270C23#1427371611yescharacteristic
271Q8—41no—
272Q8—4722no—
273Q8—2362no—
274Q8—2362no—
275D4—41no—
276D4—41no—
277D4—41no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—4722no—
285C2×C4—2362no—
286C2×C4—2362no—
287C2×C4—2362no—
288C2×C4C2×(C2×D59)⋊C411yescharacteristic
289C2×C4#1427368911yescharacteristic
290C2×C4#1427370211yescharacteristic
291C2×C4#1427372311yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—82no—
296C8—2362no—
297C8#1427364111yescharacteristic
298C8#1427364111yescharacteristic
299C22—21no—
300C22—21no—
301C22—21no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22#1683730311yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—4722no—
310C4—4722no—
311C4—4722no—
312C4C2×#1427364111yescharacteristic
313C4#1683737511yescharacteristic
314C2—21no—
315C2—41no—
316C2—41no—
317C2—41no—
318C2#2142493111yescharacteristic
subgroupG/H
centerC2#21424931
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#1813C2
socleC2×C59#1964
prime radical, cosocleC2×#41×C59C23
hypercenter#1813D59
hypocenterC59#13233
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #1813

Complements

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

NH ≅ G/Ntype
6. C22×Dic236C4semidirect
12. #16837440C4semidirect
13. #16837412C4semidirect
183. C59#13233semidirect

Maximal subgroups

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

subgroupindex
#214337512
#214257902
#214257512
#214238962
C59×#18132
#1323359