Finite Group Databasebeta

← #28506909

Subgroups — #28506909

Order 15104.

514 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×#16837785C211yesmaximal, characteristic
2C2×#16837207C211yesmaximal, characteristic, complemented
3C2×#16837205C211yesmaximal, characteristic, complemented
4C2×#16837203C211yesmaximal, characteristic
5C2×C16×Dic59C211yesmaximal, characteristic, complemented
6#21424259C211yesmaximal, characteristic, complemented
7C2×C59×#357C211yesmaximal, characteristic
8C22×Dic236C2211yescharacteristic
9C22×#11745670—21no—
10C2×#14273714—21no—
11C22×C59⋊C16C2211yescharacteristic
12#16837785—22no—
13#16837785C414yescomplemented
14C2×#14273661C2211yescharacteristic
15C2×#14273658—21no—
16C2×C8×Dic59C2211yescharacteristic
17#16837267—21no—
18#16837207—22no—
19#16837205—22no—
20#16837203—22no—
21C16×Dic59—22no—
22C22×D8×C59C2211yescharacteristic
23C22×C16×C59C2211yescharacteristic
24C2×#107×C59C2211yescharacteristic
25C59×#357—22no—
26C22×Dic118—21no—
27C2×Dic236—21no—
28C2×Dic236—21no—
29C2×Dic236—21no—
30C2×Dic236—21no—
31C2×Dic236—21no—
32C2×Dic236C2×C412yes—
33C2×#11745670—24no—
34C2×#11745670—42no—
35C2×#11745670—42no—
36C22×C59⋊C8D411yescharacteristic
37#14273714—24no—
38#14273714—42no—
39C2×C4⋊Dic59—21no—
40C2×C22⋊Dic59—42no—
41C2×C4×Dic59D411yescharacteristic
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16D412yes—
45#14273661—21no—
46#14273661—21no—
47#14273658—42no—
48C8×Dic59C2×C412yes—
49C8×Dic59D412yes—
50C59⋊#95—42no—
51C22×D4×C59—21no—
52C2×D8×C59—21no—
53C2×D8×C59—21no—
54C2×D8×C59—21no—
55C2×D8×C59—21no—
56C2×D8×C59—21no—
57C2×D8×C59C2×C412yes—
58C22×C8×C59C2311yescharacteristic
59C2×C4⋊C4×C59—21no—
60C2×C16×C59—21no—
61C2×C16×C59—21no—
62C2×C16×C59D412yes—
63#107×C59—21no—
64#107×C59—21no—
65C24×C59—41no—
66C2×D4×C59—22no—
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—41no—
75C22×Dic59—41no—
76C22×Dic59D812yes—
77C2×Dic118—22no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×Dic118—41no—
83D8×C59—22no—
84D8×C59—22no—
85D8×C59—41no—
86D8×C59—41no—
87C2×C8×C59C22×C411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C2×C8×C59C2×D411yescharacteristic
90C2×C8×C59C2×D411yescharacteristic
91C2×C8×C59#4111yescharacteristic
92C2×C8×C59#4111yescharacteristic
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C22⋊Dic59—48no—
96#11745670—48no—
97C4⋊Dic59—41no—
98C4⋊Dic59—41no—
99C4×Dic59C22⋊C412yes—
100C4×Dic59SD1612yes—
101C2×C59⋊C8C22⋊C412yes—
102C2×C59⋊C8D812yes—
103C2×C59⋊C8SD1612yes—
104Dic236—22no—
105Dic236—22no—
106Dic236—41no—
107Dic236—41no—
108C16×C59—22no—
109C59⋊C16—22no—
110C23×C59—42no—
111C23×C59—81no—
112C23×C59—81no—
113C23×C59—82no—
114C23×C59—82no—
115C23×C59C2×D811yescharacteristic
116D4×C59—42no—
117D4×C59—42no—
118D4×C59—81no—
119D4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59—82no—
122C2×C4×C59C2×C22⋊C411yescharacteristic
123C2×C4×C59#12411yescharacteristic
124C2×C4×C59#12711yescharacteristic
125C2×C4×C59C2×D811yescharacteristic
126C2×C4×C59C2×SD1611yescharacteristic
127C2×C4×C59C2×SD1611yescharacteristic
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59—42no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C2×Dic59—82no—
135C2×Dic59#10214yes—
136Dic118—42no—
137Dic118—42no—
138Dic118—81no—
139Dic118—81no—
140C8×C59C4×D411yescharacteristic
141C8×C59C4×D411yescharacteristic
142C8×C59#12111yescharacteristic
143C8×C59#12111yescharacteristic
144C59⋊C8#10214yes—
145#13204—591nomaximal, Sylow
146C22×C59—84no—
147C22×C59—82no—
148C22×C59—162no—
149C22×C59—161no—
150C22×C59—161no—
151C22×C59C2×#10211yescharacteristic
152C22×C59#48611yescharacteristic
153C22×C59#49311yescharacteristic
154C22×C59#50812yes—
155C22×C59#50812yes—
156C4×C59—161no—
157C4×C59#39011yescharacteristic
158C4×C59C2×#10211yescharacteristic
159C4×C59#48611yescharacteristic
160C4×C59#49211yescharacteristic
161Dic59—24no—
162Dic59—82no—
163Dic59—161no—
164C2×#495—591no—
165C2×#358—591no—
166C2×#358—591no—
167C2×#357—591no—
168C2×#357D5911yescharacteristic
169C2×C4×C16—591no—
170#1846—591no—
171C2×C59—162no—
172C2×C59#195711yescharacteristic
173C2×C59#213812yes—
174C2×C59#224212yes—
175C2×C59#224911yescharacteristic
176C2×C59#224911yescharacteristic
177C22×Q16—591no—
178C22×SD16—1181no—
179C22×D8C2×D5911yescharacteristic
180C2×#124—1181no—
181C22×C16—591no—
182C22×C16C2×D5911yescharacteristic
183#495—594no—
184#495—1182no—
185C2×#107—591no—
186C2×#107C2×D5911yescharacteristic
187C2×#106—1181no—
188C2×C4×C8—591no—
189#382—1181no—
190#358—1182no—
191#358—1182no—
192#357—22no—
193#357—1182no—
194C4×C16—1182no—
195C59#1320411yescharacteristic, complemented, Sylow
196C22×Q8—1181no—
197C22×D4—21no—
198C2×Q16—592no—
199C2×Q16—1181no—
200C2×Q16—1181no—
201C2×Q16—1181no—
202C2×Q16—1181no—
203C2×Q16—1181no—
204C2×SD16—1184no—
205C2×SD16—2362no—
206C2×SD16—2362no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8C4×D5912yes—
213C22×C8—591no—
214C22×C8C22×D5911yescharacteristic
215#124—1184no—
216#124—2362no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—1181no—
219C2×C22⋊C4—2362no—
220C2×C42—591no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—592no—
224C2×C16—1181no—
225C2×C16—1181no—
226C2×C16C22⋊D5912yes—
227#107—21no—
228#107—21no—
229#107—1181no—
230#107—1181no—
231#106—2362no—
232C4×C8—592no—
233C4×C8—592no—
234#95—2362no—
235C24—41no—
236C2×Q8—1182no—
237C2×Q8—2361no—
238C2×Q8—2361no—
239C2×Q8—2361no—
240C2×Q8—2361no—
241C2×Q8—2361no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—592no—
250C22×C4—2361no—
251C22×C4—2361no—
252C22×C4D4×D5911yescharacteristic
253Q16—1182no—
254Q16—1182no—
255Q16—2361no—
256Q16—2361no—
257SD16—2368no—
258D8—22no—
259D8—22no—
260D8—41no—
261D8—41no—
262C2×C8—592no—
263C2×C8—592no—
264C2×C8—592no—
265C2×C8C2×C4×D5911yescharacteristic
266C2×C8#1174568511yescharacteristic
267C2×C8D4×D5911yescharacteristic
268C2×C8D4×D5911yescharacteristic
269C2×C8#1174568711yescharacteristic
270C2×C8C2×C22⋊D5911yescharacteristic
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—2361no—
274C4⋊C4—2361no—
275C22⋊C4—2368no—
276C42—592no—
277C42—592no—
278C16—22no—
279C16—1182no—
280C23—42no—
281C23—81no—
282C23—81no—
283C23—82no—
284C23—82no—
285C23D8×D5911yescharacteristic
286Q8—4721no—
287Q8—4721no—
288Q8—2362no—
289Q8—2362no—
290D4—42no—
291D4—42no—
292D4—81no—
293D4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—594no—
297C2×C4—1182no—
298C2×C4—1182no—
299C2×C4—4721no—
300C2×C4—4722no—
301C2×C4—4721no—
302C2×C4—4721no—
303C2×C4—2362no—
304C2×C4C22⋊C4×D5911yescharacteristic
305C2×C4#1427371311yescharacteristic
306C2×C4#1427372311yescharacteristic
307C2×C4D8×D5911yescharacteristic
308C2×C4SD16×D5911yescharacteristic
309C2×C4SD16×D5911yescharacteristic
310C8—594no—
311C8(C22⋊D59)⋊C411yescharacteristic
312C8C4×C22⋊D5911yescharacteristic
313C8#1427369911yescharacteristic
314C8#1427370011yescharacteristic
315C22—82no—
316C22—84no—
317C22—162no—
318C22—161no—
319C22—161no—
320C22#102×D5911yescharacteristic
321C22#1683774711yescharacteristic
322C22#1683777711yescharacteristic
323C22#1683783112yes—
324C22#1683783312yes—
325C4—161no—
326C4—1184no—
327C4—9441no—
328C4—4722no—
329C4#1683730211yescharacteristic
330C4#102×D5911yescharacteristic
331C4#1683774711yescharacteristic
332C4#1683777311yescharacteristic
333C2—162no—
334C2#2142490011yescharacteristic
335C2#2142580012yes—
336C2#2142632612yes—
337C2#2142635411yescharacteristic
338C2#2142635611yescharacteristic
subgroupG/H
centerC23D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#357C2
socleC23×C59C2×D8
prime radicalC22×D8×C59C22
cosocleC22×C8×C59C23
hypercenterC2×#357D59
hypocenterC59#13204
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16837207C2semidirect
3. C2×#16837205C2semidirect
5. C2×C16×Dic59C2semidirect
6. #21424259C2semidirect
13. #16837785C4semidirect
195. C59#13204semidirect

Maximal subgroups

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

subgroupindex
C2×#168377852
C2×#168372072
C2×#168372052
C2×#168372032
C2×C16×Dic592
#214242592
C2×C59×#3572
#1320459