Finite Group Databasebeta

← #28497989

Subgroups — #28497989

Order 15104.

382 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 99 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#21432788C211yesmaximal, characteristic
2#21425717C211yesmaximal, characteristic, complemented
3#21422048C212yesmaximal, complemented
4#21422031C212yesmaximal, complemented
5C59×#2115C211yesmaximal, characteristic, complemented
6C22×C8×D59C411yescharacteristic
7#16837909C411yescharacteristic
8C2×#14273667—21no—
9#16837471—21no—
10#16837471—21no—
11C8×C22⋊D59—21no—
12C8×C22⋊D59—21no—
13#16837430C2211yescharacteristic
14#16837369C411yescharacteristic
15#16837365C2211yescharacteristic
16#16837357C411yescharacteristic
17#16837184—21no—
18#16837184C2212yes—
19C2×#131×C59C2211yescharacteristic
20C2×M32×C59—21no—
21C59×#349—21no—
22C59×#349C2212yes—
23C22×C4×D59C2×C411yescharacteristic
24C2×#11745664C2×C411yescharacteristic
25C22×C59⋊C8C2×C411yescharacteristic
26C2×C8×D59—21no—
27C2×C8×D59—21no—
28C2×C8×D59—41no—
29C2×C8×D59—41no—
30#14273708C811yescharacteristic
31#14273708C811yescharacteristic
32#14273703—21no—
33#14273698—21no—
34#14273695C811yescharacteristic
35#14273695C811yescharacteristic
36C4×C22⋊D59—21no—
37C4×C22⋊D59—21no—
38#14273686C2×C411yescharacteristic
39#14273667—22no—
40#14273667—41no—
41C2×C59⋊C16—41no—
42Dic59⋊C8—21no—
43Dic59⋊C8—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46(C2×D59)⋊C8—21no—
47(C2×D59)⋊C8—21no—
48#14273637—21no—
49C8×Dic59—21no—
50C2×#41×C59C2×C411yescharacteristic
51#131×C59—41no—
52#131×C59—41no—
53C2×M16×C59D411yescharacteristic
54C2×M16×C59D411yescharacteristic
55C2×M16×C59C2311yescharacteristic
56C22×C8×C59C2×C411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59M32×C59—22no—
60M32×C59—41no—
61C2×C16×C59—41no—
62C23×D59C2×C811yescharacteristic
63#41×C59—41no—
64#41×C59—41no—
65C2×Q8×C59M1611yescharacteristic
66C2×D4×C59C2×C811yescharacteristic
67C2×D4×C59C2×C811yescharacteristic
68C2×D4×C59M1611yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C2×C22⋊D59—21no—
73C2×C22⋊D59—21no—
74C22×Dic59C2×C811yescharacteristic
75C2×C4×D59—21no—
76C2×C4×D59—21no—
77C2×C4×D59—41no—
78C2×C4×D59—41no—
79M16×C59—41no—
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59C22⋊C411yescharacteristic
89C2×C8×C59C22⋊C411yescharacteristic
90C2×C8×C59C2×D411yescharacteristic
91C2×C8×C59C2×D411yescharacteristic
92C22⋊Dic59—21no—
93C22⋊Dic59—21no—
94C22⋊Dic59C2×C811yescharacteristic
95C22⋊Dic59C2×C811yescharacteristic
96(C2×D59)⋊C4—21no—
97(C2×D59)⋊C4—21no—
98C4⋊Dic59—21no—
99Dic59⋊C4—21no—
100Dic59⋊C4—21no—
101C4×Dic59—21no—
102#11745664—41no—
103C2×C59⋊C8—21no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—41no—
106C8×D59—42no—
107C8×D59—41no—
108C8×D59—41no—
109C16×C59—42no—
110C59⋊C16—42no—
111C23×C59C22⋊C811yescharacteristic
112C23×C59C22⋊C811yescharacteristic
113C23×C59C22×C811yescharacteristic
114C22×D59—21no—
115C22×D59—42no—
116C22×D59—41no—
117Q8×C59—41no—
118D4×C59—41no—
119D4×C59—41no—
120D4×C59—41no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59C22⋊C811yescharacteristic
128C2×C4×C59C22⋊C811yescharacteristic
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59C2×M1611yescharacteristic
131C22⋊D59—42no—
132C22⋊D59—42no—
133C2×Dic59—21no—
134C2×Dic59—21no—
135C2×Dic59—21no—
136C2×Dic59—41no—
137C4×D59—42no—
138C4×D59—41no—
139C4×D59—41no—
140C8×C59—21no—
141C8×C59—21no—
142C8×C59—21no—
143C8×C59#9911yescharacteristic
144C8×C59#9911yescharacteristic
145C59⋊C8—41no—
146C59⋊C8—41no—
147#11483—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59C2×C22⋊C811yescharacteristic
155C2×D59—22no—
156C2×D59—41no—
157C2×D59—42no—
158C2×D59—41no—
159C2×D59—42no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59C2×#9911yescharacteristic
164C4×C59#41311yescharacteristic
165Dic59—41no—
166Dic59—41no—
167Dic59—41no—
168#2929—591no—
169#2115—591no—
170#2115D5911yescharacteristic, complemented
171#1328—592no—
172#1320—592no—
173C2×C59—21no—
174C2×C59—41no—
175C2×C59—41no—
176C2×C59—41no—
177C2×C59#145911yescharacteristic
178D59—42no—
179C2×#131C2×D5911yescharacteristic
180C23×C8—591no—
181#525—591no—
182C2×M32—21no—
183C2×M32—1181no—
184#435—1181no—
185#435—1181no—
186C8×D4—1181no—
187C8×D4—1181no—
188#424—591no—
189#408—591no—
190#407—591no—
191#405—591no—
192#349—21no—
193#349—592no—
194#349—1181no—
195#349C2×D5912yes—
196C59#1148311yescharacteristic, complemented, Sylow
197C2×#41C4×D5911yescharacteristic
198C23×C4—591no—
199#131—41no—
200#131—41no—
201C2×M16—591no—
202C2×M16D23611yescharacteristic
203C2×M16C22⋊D5911yescharacteristic
204C2×M16C22×D5911yescharacteristic
205C22×C8—591no—
206C22×C8—1181no—
207C22×C8—1181no—
208C22×C8—2361no—
209C22×C8—2361no—
210C22×C8C4×D5911yescharacteristic
211C22×C8D23611yescharacteristic
212C22×C8C22⋊D5911yescharacteristic
213#123—591no—
214#123—591no—
215C22⋊Q8—1181no—
216#121—1181no—
217#120—591no—
218#120—591no—
219C4×D4—1181no—
220C4×D4—1181no—
221#117—591no—
222M32—22no—
223M32—41no—
224M32—1182no—
225M32—2361no—
226C2×C16—41no—
227C2×C16—2361no—
228C4⋊C8—1181no—
229C4⋊C8—1181no—
230C22⋊C8—1181no—
231C22⋊C8—1181no—
232C22⋊C8—1181no—
233C22⋊C8—1181no—
234#97—1181no—
235C4×C8—1181no—
236C24—591no—
237#41—41no—
238#41—41no—
239C2×Q8#1174565911yescharacteristic
240C2×D4—1181no—
241C2×D4—1181no—
242C2×D4C8×D5911yescharacteristic
243C2×D4C8×D5911yescharacteristic
244C2×D4#1174565911yescharacteristic
245C22×C4—591no—
246C22×C4—1181no—
247C22×C4—1181no—
248C22×C4—2361no—
249C22×C4—2361no—
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×C4×D5911yescharacteristic
253M16—41no—
254M16—41no—
255M16—41no—
256M16—2361no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—1181no—
264C2×C8—1181no—
265C2×C8—2362no—
266C2×C8—2361no—
267C2×C8—2361no—
268C2×C8—2361no—
269C2×C8(C2×D59)⋊C411yescharacteristic
270C2×C8(C2×D59)⋊C411yescharacteristic
271C2×C8C2×D23611yescharacteristic
272C2×C8C2×C22⋊D5911yescharacteristic
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C4⋊C4—1181no—
276C22⋊C4—591no—
277C22⋊C4—591no—
278C22⋊C4—1181no—
279C22⋊C4—1181no—
280C22⋊C4—1181no—
281C22⋊C4—1181no—
282C42—1181no—
283C16—42no—
284C16—2362no—
285C23—1181no—
286C23—2361no—
287C23—2362no—
288C23(C2×D59)⋊C811yescharacteristic
289C23(C2×D59)⋊C811yescharacteristic
290C23C2×C8×D5911yescharacteristic
291Q8—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—2362no—
296D4—2362no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—1181no—
304C2×C4—1181no—
305C2×C4—1181no—
306C2×C4—2362no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4(C2×D59)⋊C811yescharacteristic
311C2×C4(C2×D59)⋊C811yescharacteristic
312C2×C4C2×(C2×D59)⋊C411yescharacteristic
313C2×C4C2×#1174565911yescharacteristic
314C8—21no—
315C8—21no—
316C8—21no—
317C8—2361no—
318C8—2361no—
319C8#1427364111yescharacteristic
320C8#1427364111yescharacteristic
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—1182no—
328C22—2361no—
329C22—2362no—
330C22—2362no—
331C22—2361no—
332C22C2×(C2×D59)⋊C811yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—2361no—
337C4—2361no—
338C4—2361no—
339C4C2×#1427364111yescharacteristic
340C4#1683738411yescharacteristic
341C2—21no—
342C2—41no—
343C2—41no—
344C2—41no—
345C2—2362no—
346C2#2142241211yescharacteristic
subgroupG/H
centerC8#14273641
commutatorC23×C59C22×C8
FrattiniC2×M16C22×D59
FittingC59×#2115C2
socleC2×C59#1459
prime radical#16837909C4
cosocleC2×M16×C59C23
hypercenter#2115D59
hypocenterC59#11483
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ #2115

Complements

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

NH ≅ G/Ntype
2. #21425717C2semidirect
3. #21422048C2semidirect
4. #21422031C2semidirect
5. C59×#2115C2semidirect
170. #2115D59semidirect
196. C59#11483semidirect

Maximal subgroups

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

subgroupindex
#214327882
#214257172
#214220482
#214220312
C59×#21152
#1148359