Finite Group Databasebeta

← #28506943

Subgroups — #28506943

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837780C211yesmaximal, characteristic
2C2×#16837219C211yesmaximal, characteristic
3C2×#16837218C211yesmaximal, characteristic
4C2×#16837217C211yesmaximal, characteristic
5M32×Dic59C211yesmaximal, characteristic
6#21424266C211yesmaximal, characteristic
7C2×C59×#362C211yesmaximal, characteristic
8C22×Dic236C2211yescharacteristic
9C22×#11745672—21no—
10C2×#14273725—21no—
11C2×#14273667C2211yescharacteristic
12#16837780—21no—
13#16837780—21no—
14#16837780C412yes—
15#16837780C412yes—
16C2×#14273663C2211yescharacteristic
17C2×#14273662—21no—
18#16837425—21no—
19C2×C8×Dic59C2211yescharacteristic
20#16837219—22no—
21#16837218—22no—
22#16837217—22no—
23#16837177—21no—
24C16×Dic59—21no—
25C22×Q16×C59C2211yescharacteristic
26C2×M32×C59C2211yescharacteristic
27C2×#108×C59C2211yescharacteristic
28C59×#362—22no—
29C22×Dic118—21no—
30C2×Dic236—21no—
31C2×Dic236—21no—
32C2×Dic236—21no—
33C2×Dic236—21no—
34C2×Dic236—21no—
35C2×Dic236C2×C412yes—
36C2×#11745672—22no—
37C2×#11745672—22no—
38C2×#11745672—42no—
39C2×#11745672—42no—
40C2×#11745664—21no—
41C22×C59⋊C8D411yescharacteristic
42#14273725—22no—
43#14273725—22no—
44#14273725—42no—
45C2×Dic59⋊C4—42no—
46C2×C4×Dic59D411yescharacteristic
47#14273667—21no—
48#14273667D412yes—
49C2×C59⋊C16—21no—
50#14273663—21no—
51#14273663—21no—
52#14273662—42no—
53Dic59⋊C8—42no—
54C8×Dic59C2×C411yescharacteristic
55C8×Dic59C2×C411yescharacteristic
56C8×Dic59D411yescharacteristic
57C8×Dic59D411yescharacteristic
58C22×Q8×C59—21no—
59C2×Q16×C59—21no—
60C2×Q16×C59—21no—
61C2×Q16×C59—21no—
62C2×Q16×C59—21no—
63C2×Q16×C59—21no—
64C2×Q16×C59C2×C412yes—
65C2×M16×C59—21no—
66C22×C8×C59C2311yescharacteristic
67M32×C59—21no—
68M32×C59D412yes—
69C2×C16×C59—21no—
70#108×C59—21no—
71#108×C59—21no—
72C2×Q8×C59—22no—
73C2×Q8×C59—41no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×Dic59—41no—
81C22×Dic59SD1612yes—
82C2×Dic118—22no—
83C2×Dic118—41no—
84C2×Dic118—41no—
85C2×Dic118—41no—
86C2×Dic118—41no—
87C2×Dic118—41no—
88Q16×C59—22no—
89Q16×C59—22no—
90Q16×C59—41no—
91Q16×C59—41no—
92M16×C59—41no—
93M16×C59—41no—
94C2×C8×C59—41no—
95C2×C8×C59C22×C411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101#11745672—44no—
102#11745672—44no—
103Dic59⋊C4—44no—
104Dic59⋊C4—44no—
105C4×Dic59C22⋊C411yescharacteristic
106C4×Dic59C22⋊C411yescharacteristic
107C4×Dic59D812yes—
108#11745664—41no—
109#11745664—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8C22⋊C411yescharacteristic
112C2×C59⋊C8C22⋊C411yescharacteristic
113C2×C59⋊C8D812yes—
114C2×C59⋊C8SD1612yes—
115Dic236—22no—
116Dic236—22no—
117Dic236—41no—
118Dic236—41no—
119C16×C59—22no—
120C59⋊C16—22no—
121C23×C59C2×SD1611yescharacteristic
122Q8×C59—42no—
123Q8×C59—42no—
124Q8×C59—81no—
125Q8×C59—81no—
126C2×C4×C59—42no—
127C2×C4×C59—81no—
128C2×C4×C59—81no—
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59#12411yescharacteristic
131C2×C4×C59#12711yescharacteristic
132C2×C4×C59C2×D811yescharacteristic
133C2×C4×C59C2×D811yescharacteristic
134C2×C4×C59C2×SD1611yescharacteristic
135C2×Dic59—24no—
136C2×Dic59—42no—
137C2×Dic59—81no—
138C2×Dic59—81no—
139C2×Dic59#10212yes—
140C2×Dic59#10212yes—
141Dic118—42no—
142Dic118—42no—
143Dic118—81no—
144Dic118—81no—
145C8×C59—81no—
146C8×C59C4×D411yescharacteristic
147C8×C59C4×D411yescharacteristic
148C8×C59#12111yescharacteristic
149C8×C59#12111yescharacteristic
150C59⋊C8—81no—
151C59⋊C8#10212yes—
152C59⋊C8#10212yes—
153#13212—591nomaximal, Sylow
154C22×C59—22no—
155C22×C59C2×#10211yescharacteristic
156C22×C59#48611yescharacteristic
157C22×C59#49211yescharacteristic
158C4×C59—82no—
159C4×C59#39011yescharacteristic
160C4×C59C2×#10211yescharacteristic
161C4×C59#48611yescharacteristic
162C4×C59#49311yescharacteristic
163Dic59—22no—
164Dic59—82no—
165Dic59#36214yescomplemented
166C2×#494—591no—
167C2×#362—591no—
168C2×#362—591no—
169C2×#362—591no—
170C2×#362D5911yescharacteristic
171C4×M32—591no—
172#1848—591no—
173C2×C59—21no—
174C2×C59—21no—
175C2×C59#195711yescharacteristic
176C2×C59C2×#36212yes—
177C22×Q16—591no—
178C22×Q16—1181no—
179C22×Q16C2×D5911yescharacteristic
180C2×C4⋊Q8—1181no—
181C2×M32—591no—
182C2×M32C2×D5911yescharacteristic
183#494—592no—
184#494—592no—
185#494—1181no—
186#494—1181no—
187C2×#108—591no—
188C2×#108—1181no—
189C2×#108C2×D5911yescharacteristic
190#423—1181no—
191C2×C4×C8—591no—
192#362—22no—
193#362—1182no—
194#362—1182no—
195#362—1182no—
196#346—1181no—
197C4×C16—1181no—
198C59#1321211yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C22×Q8—1181no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—592no—
207C2×Q16—1182no—
208C2×Q16—1182no—
209C2×Q16—1181no—
210C2×Q16—1181no—
211C2×Q16—1181no—
212C2×Q16—1181no—
213C2×Q16—1181no—
214C2×Q16—2362no—
215C2×Q16—2362no—
216C2×Q16C4×D5912yes—
217C2×M16—21no—
218C2×M16—1181no—
219C22×C8—591no—
220C22×C8C22×D5911yescharacteristic
221C4⋊Q8—1182no—
222C4⋊Q8—1182no—
223C4⋊Q8—2362no—
224C2×C4⋊C4—2362no—
225C2×C42—591no—
226M32—21no—
227M32—592no—
228M32—1181no—
229M32C22⋊D5912yes—
230C2×C16—21no—
231C2×C16—1181no—
232#108—21no—
233#108—21no—
234#108—1181no—
235#108—1181no—
236#108—2362no—
237C4⋊C8—2362no—
238C4×C8—591no—
239C4×C8—591no—
240C4×C8—591no—
241C4×C8—591no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—1182no—
249C2×Q8—2361no—
250C2×Q8—2361no—
251C2×Q8—2361no—
252C2×Q8—2361no—
253C2×Q8—2361no—
254C22×C4—41no—
255C22×C4—592no—
256C22×C4—2361no—
257C22×C4D4×D5911yescharacteristic
258Q16—22no—
259Q16—22no—
260Q16—41no—
261Q16—41no—
262Q16—1182no—
263Q16—1182no—
264Q16—2364no—
265Q16—2364no—
266Q16—2361no—
267Q16—2361no—
268M16—41no—
269M16—41no—
270M16—2361no—
271M16—2361no—
272C2×C8—41no—
273C2×C8—591no—
274C2×C8—591no—
275C2×C8—592no—
276C2×C8—592no—
277C2×C8—2361no—
278C2×C8C2×C4×D5911yescharacteristic
279C2×C8#1174568511yescharacteristic
280C2×C8D4×D5911yescharacteristic
281C2×C8D4×D5911yescharacteristic
282C2×C8#1174568711yescharacteristic
283C2×C8C2×C22⋊D5911yescharacteristic
284C4⋊C4—2364no—
285C4⋊C4—2364no—
286C42—591no—
287C42—591no—
288C42—592no—
289C16—22no—
290C16—1182no—
291C23SD16×D5911yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—81no—
295Q8—81no—
296Q8—4721no—
297Q8—4721no—
298Q8—2362no—
299Q8—2362no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—592no—
304C2×C4—592no—
305C2×C4—1184no—
306C2×C4—4721no—
307C2×C4—4721no—
308C2×C4—2362no—
309C2×C4C22⋊C4×D5911yescharacteristic
310C2×C4#1427371311yescharacteristic
311C2×C4#1427372311yescharacteristic
312C2×C4D8×D5911yescharacteristic
313C2×C4D8×D5911yescharacteristic
314C2×C4SD16×D5911yescharacteristic
315C8—81no—
316C8—592no—
317C8—592no—
318C8—4721no—
319C8(C22⋊D59)⋊C411yescharacteristic
320C8C4×C22⋊D5911yescharacteristic
321C8#1427369911yescharacteristic
322C8#1427370011yescharacteristic
323C22—22no—
324C22#102×D5911yescharacteristic
325C22#1683774711yescharacteristic
326C22#1683777311yescharacteristic
327C4—82no—
328C4—594no—
329C4—1182no—
330C4—4722no—
331C4#1683730211yescharacteristic
332C4#102×D5911yescharacteristic
333C4#1683774711yescharacteristic
334C4#1683777711yescharacteristic
335C2—21no—
336C2—21no—
337C2#2142490011yescharacteristic
338C2#362×D5912yes—
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#362C2
socleC22×C59C2×#102
prime radicalC23×C59C2×SD16
cosocleC22×C8×C59C23
hypercenterC2×#362D59
hypocenterC59#13212
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
165. Dic59#362semidirect
198. C59#13212semidirect

Maximal subgroups

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

subgroupindex
C2×#168377802
C2×#168372192
C2×#168372182
C2×#168372172
M32×Dic592
#214242662
C2×C59×#3622
#1321259