Finite Group Databasebeta

← #28504972

Subgroups — #28504972

Order 15104.

334 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 79 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#21432743C211yesmaximal, characteristic
2#21424201C211yesmaximal, characteristic
3#21424197C211yesmaximal, characteristic
4#21422225C211yesmaximal, characteristic
5#21422218C211yesmaximal, characteristic
6#21422216C211yesmaximal, characteristic
7C59×#1395C211yesmaximal, characteristic
8#41×Dic59C2211yescharacteristic
9#16837460—21no—
10#16837460—21no—
11#16837456—21no—
12#16837456—21no—
13#16837430C411yescharacteristic, complemented
14#16837428C411yescharacteristic, complemented
15#16837386C2211yescharacteristic
16#16837376C2211yescharacteristic
17#16837369C2211yescharacteristic
18M16×Dic59C411yescharacteristic, complemented
19C2×#14273637C411yescharacteristic, complemented
20#16837163—21no—
21#16837161—21no—
22#16837159—21no—
23#16837158—21no—
24C59×#413C2211yescharacteristic
25C59×#410C2211yescharacteristic
26C59×#408C2211yescharacteristic
27C2×#11745664—21no—
28C2×#11745664C2×C411yescharacteristic
29C22×C59⋊C8C2×C411yescharacteristic
30Q8×Dic59—21no—
31D4×Dic59—21no—
32D4×Dic59—21no—
33D4×Dic59—21no—
34#14273686—21no—
35#14273686C2×C411yescharacteristic
36#14273686D411yescharacteristic
37#14273686D411yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C4×Dic59D411yescharacteristic
40C2×C4×Dic59D411yescharacteristic
41Dic59⋊C8—21no—
42Dic59⋊C8—21no—
43Dic59⋊C8—21no—
44Dic59⋊C8—21no—
45#14273646—21no—
46#14273644—21no—
47#14273642—21no—
48#14273642—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51#14273637—21no—
52#14273637—21no—
53#14273637—21no—
54C8×Dic59—21no—
55C2×#41×C59C2311yescharacteristic
56C2×M16×C59—21no—
57C2×M16×C59C2×C411yescharacteristic
58C22×C8×C59C2×C411yescharacteristic
59#117×C59—21no—
60#101×C59—21no—
61#100×C59—21no—
62#99×C59—21no—
63#99×C59—21no—
64C22⋊C8×C59—21no—
65C22⋊C8×C59—21no—
66#41×C59—41no—
67#41×C59—41no—
68C2×Q8×C59#4111yescharacteristic
69C2×D4×C59#4111yescharacteristic
70C2×D4×C59#4111yescharacteristic
71C2×D4×C59#4111yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78M16×C59—41no—
79M16×C59—41no—
80M16×C59—41no—
81C2×C8×C59—21no—
82C2×C8×C59—21no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C4⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C42×C59—41no—
88C22⋊Dic59—21no—
89C22⋊Dic59—21no—
90C22⋊Dic59—21no—
91C22⋊Dic59—41no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—41no—
96C4×Dic59—21no—
97C4×Dic59—21no—
98C4×Dic59—21no—
99C4×Dic59—41no—
100C4×Dic59C22⋊C411yescharacteristic
101C4×Dic59C22⋊C411yescharacteristic
102C4×Dic59C22⋊C411yescharacteristic
103C4×Dic59C22⋊C411yescharacteristic
104#11745664—41no—
105#11745664—41no—
106#11745664—41no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C23×C59#11711yescharacteristic
112C23×C59#12311yescharacteristic
113C23×C59#12311yescharacteristic
114Q8×C59—41no—
115D4×C59—41no—
116D4×C59—41no—
117D4×C59—41no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—81no—
125C2×C4×C59C2×C22⋊C411yescharacteristic
126C2×C4×C59#11711yescharacteristic
127C2×C4×C59#12311yescharacteristic
128C2×C4×C59#12311yescharacteristic
129C2×Dic59—21no—
130C2×Dic59—21no—
131C2×Dic59—21no—
132C2×Dic59—21no—
133C2×Dic59—81no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136C2×Dic59—81no—
137C8×C59—41no—
138C8×C59—41no—
139C8×C59—81no—
140C59⋊C8—41no—
141C59⋊C8—41no—
142C59⋊C8—81no—
143#12865—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59#38111yescharacteristic
151C4×C59—21no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—161no—
155C4×C59#41011yescharacteristic
156C4×C59#41011yescharacteristic
157Dic59—41no—
158Dic59—41no—
159Dic59—41no—
160Dic59—41no—
161Dic59—161no—
162#2923—591no—
163#1827—591no—
164#1827—591no—
165#1395—591no—
166#1395D5911yescharacteristic
167#1392—591no—
168#1391—591no—
169C2×C59—21no—
170C2×C59—41no—
171C2×C59—41no—
172C2×C59—41no—
173C2×C59#182011yescharacteristic
174C4×#41—591no—
175#433—1181no—
176#433—1181no—
177#432—1181no—
178#432—1181no—
179#424—591no—
180#424—591no—
181#413—591no—
182#413C2×D5911yescharacteristic
183#410—591no—
184#410C2×D5911yescharacteristic
185#408—591no—
186#408C2×D5911yescharacteristic
187C4×M16—591no—
188C2×#97—591no—
189#339—1181no—
190#339—1181no—
191H(3,ℤ/4)—1181no—
192H(3,ℤ/4)—1181no—
193C59#1286511yescharacteristic, complemented, Sylow
194C2×#41C22×D5911yescharacteristic
195C2×M16—21no—
196C2×M16—591no—
197C2×M16—1181no—
198C2×M16C4×D5911yescharacteristic
199C22×C8—591no—
200C22×C8C4×D5911yescharacteristic
201C4×Q8—1181no—
202C4×D4—1181no—
203C4×D4—1181no—
204C4×D4—1181no—
205#117—21no—
206#117—591no—
207#117—591no—
208#117—591no—
209#117—1181no—
210C2×C42—591no—
211C2×C42—591no—
212C2×C42—591no—
213C4⋊C8—1181no—
214C4⋊C8—1181no—
215C4⋊C8—1181no—
216C4⋊C8—1181no—
217#101—21no—
218#101—1181no—
219#100—21no—
220#100—1181no—
221#99—21no—
222#99—21no—
223#99—1181no—
224#99—1181no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—1181no—
228C22⋊C8—1181no—
229#97—1181no—
230#97—1181no—
231#97—1181no—
232C4×C8—1181no—
233#41—41no—
234#41—41no—
235C2×Q8#1174568511yescharacteristic
236C2×D4#1174568511yescharacteristic
237C2×D4#1174568711yescharacteristic
238C2×D4#1174568911yescharacteristic
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4C2×C4×D5911yescharacteristic
243C22×C4D4×D5911yescharacteristic
244C22×C4D4×D5911yescharacteristic
245M16—41no—
246M16—41no—
247M16—41no—
248M16—2361no—
249M16—2361no—
250M16—2361no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—1181no—
256C2×C8—1181no—
257C2×C8—2361no—
258C2×C8—2361no—
259C4⋊C4—41no—
260C4⋊C4—1181no—
261C4⋊C4—1181no—
262C4⋊C4—1181no—
263C4⋊C4—2361no—
264C22⋊C4—41no—
265C22⋊C4—1181no—
266C22⋊C4—1181no—
267C22⋊C4—1181no—
268C22⋊C4—2361no—
269C42—41no—
270C42—591no—
271C42—591no—
272C42—591no—
273C42—591no—
274C42—1181no—
275C42—1181no—
276C42—1181no—
277C42—2361no—
278C23#1427368511yescharacteristic
279C23#1427370911yescharacteristic
280C23#1427371211yescharacteristic
281Q8—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—1181no—
293C2×C4—1181no—
294C2×C4—1181no—
295C2×C4—1181no—
296C2×C4—4721no—
297C2×C4—4721no—
298C2×C4—4721no—
299C2×C4—4721no—
300C2×C4C22⋊C4×D5911yescharacteristic
301C2×C4#1427368411yescharacteristic
302C2×C4#1427370911yescharacteristic
303C2×C4#1427371211yescharacteristic
304C8—41no—
305C8—41no—
306C8—81no—
307C8—4721no—
308C8—2361no—
309C8—2361no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22#1683726411yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—161no—
321C4—9441no—
322C4—2361no—
323C4—2361no—
324C4—2361no—
325C4—2361no—
326C4#1683737411yescharacteristic
327C4#1683737411yescharacteristic
328C2—21no—
329C2—41no—
330C2—41no—
331C2—41no—
332C2#2142417211yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#1395C2
socleC2×C59#1820
prime radical, cosocleC2×#41×C59C23
hypercenter#1395D59
hypocenterC59#12865
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1395

Complements

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

NH ≅ G/Ntype
13. #16837430C4semidirect
14. #16837428C4semidirect
18. M16×Dic59C4semidirect
19. C2×#14273637C4semidirect
193. C59#12865semidirect

Maximal subgroups

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

subgroupindex
#214327432
#214242012
#214241972
#214222252
#214222182
#214222162
C59×#13952
#1286559