Finite Group Databasebeta

← #28484002

Subgroups — #28484002

Order 15104.

404 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 97 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#21434606C211yesmaximal, characteristic
2#21433197C211yesmaximal, characteristic
3#21422690C211yesmaximal, characteristic
4#21422517C211yesmaximal, characteristic
5#21421934C211yesmaximal, characteristic
6#21421932C211yesmaximal, characteristic
7C59×#1485C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837805C411yescharacteristic
10#16837805C411yescharacteristic
11#16837801—22no—
12#16837798C411yescharacteristic
13#16837798C411yescharacteristic
14#16837532—21no—
15#16837529—21no—
16C8×Dic118—21no—
17#16837526—21no—
18#16837460—22no—
19#16837455C2211yescharacteristic
20#16837438—22no—
21C4×#11745664C2211yescharacteristic
22C2×C4×C59⋊C8C2211yescharacteristic
23#16837143—22no—
24Q8⋊(C59⋊C8)—21no—
25Q8⋊(C59⋊C8)—21no—
26C59×#533C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C4×M16×C59C2211yescharacteristic
29C59×Q8⋊C8—21no—
30C59×Q8⋊C8—21no—
31C2×#11745664—21no—
32C22×C59⋊C8—21no—
33#14273727C2×C411yescharacteristic
34#14273727C2×C411yescharacteristic
35#14273719D412yes—
36#14273706—21no—
37#14273706—21no—
38C4×Dic118—21no—
39C4×Dic118—21no—
40#14273686D412yes—
41#14273660—22no—
42#14273660—22no—
43#14273658—22no—
44#14273658—22no—
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D411yescharacteristic
47Dic59⋊C8—22no—
48Dic59⋊C8—22no—
49C22⋊(C59⋊C8)—21no—
50#14273638—21no—
51#14273637—22no—
52C4×C59⋊C8—21no—
53C4×C59⋊C8—21no—
54C4×C59⋊C8—21no—
55C4×C59⋊C8C2×C411yescharacteristic
56C4×C59⋊C8C2×C411yescharacteristic
57C8×Dic59—22no—
58C2×M16×C59—21no—
59C4⋊Q8×C59C2×C411yescharacteristic
60C4⋊Q8×C59C2×C411yescharacteristic
61#125×C59—21no—
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C4×Q8×C59—21no—
65C4×Q8×C59—21no—
66#117×C59—21no—
67C2×C42×C59C2311yescharacteristic
68C4⋊C8×C59D411yescharacteristic
69C4⋊C8×C59D411yescharacteristic
70C22⋊C8×C59—21no—
71#97×C59—21no—
72C4×C8×C59—21no—
73C4×C8×C59—21no—
74C2×Q8×C59—21no—
75C2×Q8×C59—21no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C2×Dic118—21no—
80C2×Dic118—21no—
81M16×C59—22no—
82M16×C59—41no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—41no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C42×C59—41no—
94C42×C59C22×C411yescharacteristic
95C42×C59C2×D411yescharacteristic
96C42×C59C2×D411yescharacteristic
97C42×C59C2×D411yescharacteristic
98C22⋊Dic59—22no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59C22⋊C412yes—
102C4⋊Dic59C22⋊C412yes—
103Dic59⋊C4—22no—
104Dic59⋊C4—22no—
105C4×Dic59—22no—
106#11745664—22no—
107#11745664—41no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—21no—
114C2×C59⋊C8—41no—
115C23×C59#12311yescharacteristic
116Q8×C59—41no—
117Q8×C59—41no—
118Q8×C59—41no—
119Q8×C59—41no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59C2×C22⋊C411yescharacteristic
129C2×C4×C59C4×D411yescharacteristic
130C2×C4×C59C4×D411yescharacteristic
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—22no—
136Dic118—41no—
137Dic118—41no—
138Dic118—41no—
139Dic118—41no—
140C8×C59—42no—
141C8×C59—41no—
142C8×C59—41no—
143C59⋊C8—22no—
144C59⋊C8—22no—
145C59⋊C8—42no—
146C59⋊C8—41no—
147C59⋊C8—41no—
148#9068—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#38611yescharacteristic
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—81no—
157C4×C59—81no—
158C4×C59#44311yescharacteristic
159C4×C59#44311yescharacteristic
160C4×C59#44411yescharacteristic
161C4×C59#44411yescharacteristic
162C4×C59#45611yescharacteristic
163C4×C59#45611yescharacteristic
164Dic59—42no—
165Dic59—42no—
166#3160—591no—
167#2986—591no—
168#1526—591no—
169#1485—591no—
170#1485D5911yescharacteristic
171#1278—591no—
172#1278—591no—
173C2×C59—41no—
174C2×C59#187611yescharacteristic
175C2×C59#187611yescharacteristic
176C2×C59#188611yescharacteristic
177#533—591no—
178#533C2×D5911yescharacteristic
179#500—591no—
180#500—591no—
181#499—1182no—
182#498—591no—
183#498—591no—
184#446—1181no—
185#446—1181no—
186C8×Q8—1181no—
187C8×Q8—1181no—
188#433—1182no—
189#431—591no—
190#431C2×D5911yescharacteristic
191#427—1182no—
192C4×M16—591no—
193C4×M16C2×D5911yescharacteristic
194C2×C4×C8—591no—
195#330—1182no—
196Q8⋊C8—21no—
197Q8⋊C8—21no—
198Q8⋊C8—1181no—
199Q8⋊C8—1181no—
200C59#906811yescharacteristic, complemented, Sylow
201C2×M16—21no—
202C2×M16—1181no—
203C22×C8—1181no—
204C4⋊Q8—591no—
205C4⋊Q8—591no—
206C4⋊Q8C4×D5911yescharacteristic
207C4⋊Q8C4×D5911yescharacteristic
208#125—21no—
209#125—592no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—1181no—
213C22⋊Q8—1181no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—1181no—
217C4×Q8—1181no—
218#117—21no—
219#117—592no—
220C2×C42C22×D5911yescharacteristic
221#107—1182no—
222#107—1182no—
223#106—1182no—
224#106—1182no—
225C4⋊C8—591no—
226C4⋊C8—591no—
227C4⋊C8—1182no—
228C4⋊C8—1182no—
229C4⋊C8C22⋊D5911yescharacteristic
230C4⋊C8C22⋊D5911yescharacteristic
231C22⋊C8—21no—
232C22⋊C8—1181no—
233#97—21no—
234#97—1182no—
235#97—1181no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—591no—
239C4×C8—591no—
240C4×C8—1182no—
241C4×C8—1181no—
242C4×C8—1181no—
243C4×C8—1181no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—1181no—
247C2×Q8—1181no—
248C22×C4#1174568511yescharacteristic
249C22×C4D4×D5911yescharacteristic
250C22×C4#1174568711yescharacteristic
251M16—22no—
252M16—41no—
253M16—1182no—
254M16—2361no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—1182no—
259C2×C8—1182no—
260C2×C8—1181no—
261C2×C8—1181no—
262C2×C8—1181no—
263C2×C8—1181no—
264C2×C8—2361no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—592no—
272C4⋊C4—592no—
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C4⋊C4—1182no—
276C4⋊C4—1182no—
277C22⋊C4—41no—
278C22⋊C4—1182no—
279C42—41no—
280C42—1182no—
281C42C2×C4×D5911yescharacteristic
282C42D4×D5911yescharacteristic
283C42D4×D5911yescharacteristic
284C42C2×C22⋊D5911yescharacteristic
285C23#1427370911yescharacteristic
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—2361no—
291Q8—2361no—
292Q8—2361no—
293Q8—2361no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—1182no—
303C2×C4—1182no—
304C2×C4C22⋊C4×D5911yescharacteristic
305C2×C4(C22⋊D59)⋊C411yescharacteristic
306C2×C4C4×C22⋊D5911yescharacteristic
307C2×C4#1427369511yescharacteristic
308C2×C4#1427369811yescharacteristic
309C2×C4#1427370011yescharacteristic
310C8—42no—
311C8—41no—
312C8—41no—
313C8—1182no—
314C8—1182no—
315C8—2362no—
316C8—2361no—
317C8—2361no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22#1683727811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—81no—
326C4—81no—
327C4—2362no—
328C4—2362no—
329C4#1683751611yescharacteristic
330C4#1683751711yescharacteristic
331C4#1683752111yescharacteristic
332C4#1683752211yescharacteristic
333C4#1683758311yescharacteristic
334C4#1683758311yescharacteristic
335C2—41no—
336C2#2142442011yescharacteristic
337C2#2142442611yescharacteristic
338C2#2142448611yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1485C2
socleC22×C59#386
prime radicalC23×C59#123
cosocleC2×C42×C59C23
hypercenter#1485D59
hypocenterC59#9068
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
200. C59#9068semidirect

Maximal subgroups

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

subgroupindex
#214346062
#214331972
#214226902
#214225172
#214219342
#214219322
C59×#14852
#906859