Finite Group Databasebeta

← #350×Dic59 #28503739

Subgroups — #350×Dic59 #28503739

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#131×Dic59C211yesmaximal, characteristic
2C2×#16837186C211yesmaximal, characteristic
3M32×Dic59C211yesmaximal, characteristic, complemented
4C2×C16×Dic59C211yesmaximal, characteristic, complemented
5#21422186C211yesmaximal, characteristic, complemented
6#21422184C211yesmaximal, characteristic, complemented
7C2×C59×#350C211yesmaximal, characteristic
8C2×#14273734C411yescharacteristic
9#41×Dic59C411yescharacteristic
10C2×#14273667C2211yescharacteristic
11C22×C59⋊C16C2211yescharacteristic
12#16837360—21no—
13#16837360C411yescharacteristic
14#16837357—21no—
15#16837357C2211yescharacteristic
16M16×Dic59—21no—
17M16×Dic59C411yescharacteristic
18C2×C8×Dic59—21no—
19C2×C8×Dic59C2211yescharacteristic
20#16837186C414yescomplemented
21#16837177—21no—
22C16×Dic59—21no—
23C16×Dic59—21no—
24C16×Dic59—21no—
25C2×#131×C59C2211yescharacteristic
26C2×M32×C59C2211yescharacteristic
27C22×C16×C59C2211yescharacteristic
28C59×#350C414yescomplemented
29#14273734—22no—
30#14273734C814yes—
31C2×#11745664—21no—
32C2×#11745664C2×C411yescharacteristic
33C22×C59⋊C8—21no—
34C22×C59⋊C8C2×C411yescharacteristic
35Q8×Dic59C812yes—
36D4×Dic59—22no—
37D4×Dic59C812yes—
38#14273686—21no—
39#14273686C2×C411yescharacteristic
40C2×C4×Dic59—21no—
41C2×C4×Dic59C2×C411yescharacteristic
42#14273667—21no—
43#14273667C2×C412yescomplemented
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16C2×C412yescomplemented
48#14273637—21no—
49#14273637—21no—
50#14273637—21no—
51C8×Dic59—21no—
52C8×Dic59—21no—
53C8×Dic59—21no—
54C8×Dic59D411yescharacteristic
55C8×Dic59D411yescharacteristic
56C8×Dic59D411yescharacteristic
57C8×Dic59D411yescharacteristic
58C2×#41×C59C2×C411yescharacteristic
59#131×C59—21no—
60#131×C59—21no—
61#131×C59C2×C412yes—
62#131×C59C2×C412yes—
63C2×M16×C59—21no—
64C2×M16×C59C2×C411yescharacteristic
65C22×C8×C59—21no—
66C22×C8×C59C2311yescharacteristic
67M32×C59—21no—
68M32×C59C2×C412yescomplemented
69C2×C16×C59—21no—
70C2×C16×C59—21no—
71C2×C16×C59—21no—
72C2×C16×C59C2×C412yescomplemented
73#41×C59—21no—
74#41×C59—21no—
75#41×C59C4212yes—
76#41×C59C2×C812yes—
77C2×Q8×C59C2×C811yescharacteristic
78C2×D4×C59—21no—
79C2×D4×C59C2×C811yescharacteristic
80C22×C4×C59—21no—
81C22×C4×C59C22×C411yescharacteristic
82C22×Dic59—22no—
83C22×Dic59M1612yes—
84M16×C59—21no—
85M16×C59—22no—
86M16×C59—22no—
87M16×C59C4212yes—
88C2×C8×C59—21no—
89C2×C8×C59—22no—
90C2×C8×C59—22no—
91C2×C8×C59C22×C411yescharacteristic
92C2×C8×C59C22×C411yescharacteristic
93C2×C8×C59C2×D411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59#4111yescharacteristic
96C2×C8×C59#4111yescharacteristic
97C22⋊Dic59—21no—
98C22⋊Dic59—22no—
99C4⋊Dic59—22no—
100C4⋊Dic59C2×C812yes—
101C4×Dic59—21no—
102C4×Dic59—22no—
103C4×Dic59C22⋊C411yescharacteristic
104C4×Dic59C22⋊C411yescharacteristic
105C4×Dic59C2×C812yes—
106#11745664—21no—
107#11745664—24no—
108#11745664C2×C812yes—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—24no—
111C2×C59⋊C8C22⋊C411yescharacteristic
112C2×C59⋊C8C22⋊C411yescharacteristic
113C2×C59⋊C8C2×C812yes—
114C2×C59⋊C8M1612yes—
115C16×C59—22no—
116C16×C59—22no—
117C59⋊C16—22no—
118C59⋊C16—22no—
119C23×C59—21no—
120C23×C59C2×M1611yescharacteristic
121Q8×C59—21no—
122Q8×C59C4×C812yes—
123D4×C59—22no—
124D4×C59—22no—
125D4×C59—21no—
126D4×C59C4×C812yes—
127C2×C4×C59—21no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59C2×C4211yescharacteristic
131C2×C4×C59C2×C22⋊C411yescharacteristic
132C2×C4×C59#11711yescharacteristic
133C2×C4×C59C22×C811yescharacteristic
134C2×C4×C59C22×C811yescharacteristic
135C2×C4×C59C2×M1611yescharacteristic
136C2×Dic59—24no—
137C2×Dic59—22no—
138C2×Dic59—44no—
139C2×Dic59C22⋊C812yes—
140C2×Dic59C22⋊C812yes—
141C8×C59—22no—
142C8×C59C4×D411yescharacteristic
143C8×C59C4×D411yescharacteristic
144C8×C59C4×D411yescharacteristic
145C8×C59C4×D411yescharacteristic
146C59⋊C8—22no—
147C59⋊C8C22⋊C812yes—
148C59⋊C8C22⋊C812yes—
149C4×#350—591nomaximal, Sylow
150C22×C59—22no—
151C22×C59—22no—
152C22×C59—42no—
153C22×C59C4×M1611yescharacteristic
154C22×C59C2×C22⋊C811yescharacteristic
155C22×C59#43111yescharacteristic
156C4×C59—22no—
157C4×C59C4×C22⋊C411yescharacteristic
158C4×C59C2×C4×C811yescharacteristic
159C4×C59C2×C22⋊C811yescharacteristic
160C4×C59#43111yescharacteristic
161Dic59—22no—
162Dic59—42no—
163Dic59#35014yesdirect product factor
164C4×#131—591no—
165C2×#350—591no—
166C2×#350D5911yescharacteristic
167C4×M32—591no—
168C2×C4×C16—591no—
169#1379—591no—
170#1379—591no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—42no—
174C2×C59C4×C22⋊C811yescharacteristic
175C2×C59C2×#35012yes—
176C2×#131—591no—
177C2×#131C2×D5911yescharacteristic
178C4×#41—591no—
179C2×M32—591no—
180C2×M32C2×D5911yescharacteristic
181C22×C16—591no—
182C22×C16C2×D5911yescharacteristic
183#405—591no—
184#405—591no—
185#405—1181no—
186#405—1181no—
187C4×M16—591no—
188C4×M16—1181no—
189C2×C4×C8—591no—
190C2×C4×C8—1181no—
191#350—594no—
192#350Dic5914yesdirect product factor
193#346—1181no—
194C4×C16—1181no—
195C4×C16—1181no—
196C4×C16—1181no—
197C59C4×#35011yescharacteristic, complemented, Sylow
198C2×#41C4×D5911yescharacteristic
199#131—21no—
200#131—21no—
201#131—594no—
202#131—1182no—
203#131C4×D5912yes—
204#131C2×Dic5912yes—
205C2×M16—21no—
206C2×M16—591no—
207C2×M16—1181no—
208C2×M16C4×D5911yescharacteristic
209C22×C8—21no—
210C22×C8—591no—
211C22×C8—1181no—
212C22×C8C22×D5911yescharacteristic
213C4×Q8—592no—
214C4×D4—592no—
215C4×D4—1182no—
216#117—591no—
217#117—1181no—
218C2×C42—591no—
219C2×C42—1181no—
220M32—21no—
221M32—592no—
222M32—1181no—
223M32C2×Dic5912yescomplemented
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—592no—
228C2×C16—1181no—
229C2×C16—1181no—
230C2×C16—1181no—
231C2×C16C2×Dic5912yescomplemented
232#97—1181no—
233#97—1181no—
234#97—1181no—
235C4×C8—591no—
236C4×C8—591no—
237C4×C8—591no—
238C4×C8—591no—
239C4×C8—1181no—
240C4×C8—1181no—
241C4×C8—1181no—
242#41—21no—
243#41—21no—
244#41C8×D5912yes—
245#41C4×Dic5912yes—
246C2×Q8C8×D5911yescharacteristic
247C2×D4—21no—
248C2×D4C8×D5911yescharacteristic
249C22×C4—21no—
250C22×C4—592no—
251C22×C4—1182no—
252C22×C4C2×C4×D5911yescharacteristic
253M16—21no—
254M16—22no—
255M16—22no—
256M16—592no—
257M16—1184no—
258M16—1181no—
259M16C4×Dic5912yes—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—591no—
264C2×C8—591no—
265C2×C8—592no—
266C2×C8—592no—
267C2×C8—1184no—
268C2×C8—1181no—
269C2×C8C2×C4×D5911yescharacteristic
270C2×C8D4×D5911yescharacteristic
271C2×C8D4×D5911yescharacteristic
272C2×C8#1174568711yescharacteristic
273C2×C8#1174568711yescharacteristic
274C2×C8C22×Dic5911yescharacteristic
275C4⋊C4—592no—
276C4⋊C4—1182no—
277C22⋊C4—1182no—
278C22⋊C4—1181no—
279C42—591no—
280C42—591no—
281C42—592no—
282C42—1182no—
283C42—1181no—
284C16—22no—
285C16—22no—
286C16—1182no—
287C16—1182no—
288C23—21no—
289C23M16×D5911yescharacteristic
290Q8—21no—
291Q8C8×Dic5912yes—
292D4—22no—
293D4—22no—
294D4—21no—
295D4C8×Dic5912yes—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—592no—
300C2×C4—592no—
301C2×C4—1184no—
302C2×C4—1182no—
303C2×C4—2364no—
304C2×C4C2×C4×Dic5911yescharacteristic
305C2×C4C22⋊C4×D5911yescharacteristic
306C2×C4#1427368311yescharacteristic
307C2×C4C2×C8×D5911yescharacteristic
308C2×C4C2×C8×D5911yescharacteristic
309C2×C4M16×D5911yescharacteristic
310C8—22no—
311C8—592no—
312C8—592no—
313C8—1182no—
314C8(C22⋊D59)⋊C411yescharacteristic
315C8(C22⋊D59)⋊C411yescharacteristic
316C8D4×Dic5911yescharacteristic
317C8D4×Dic5911yescharacteristic
318C22—22no—
319C22—22no—
320C22—42no—
321C22M16×Dic5911yescharacteristic
322C22C22⋊C8×D5911yescharacteristic
323C22#1683745211yescharacteristic
324C4—22no—
325C4—594no—
326C4—1182no—
327C4—2362no—
328C4C22⋊C4×Dic5911yescharacteristic
329C4C2×C8×Dic5911yescharacteristic
330C4C22⋊C8×D5911yescharacteristic
331C4#1683745211yescharacteristic
332C2—21no—
333C2—21no—
334C2—42no—
335C2C22⋊C8×Dic5911yescharacteristic
336C2#350×D5912yes—
subgroupG/H
centerC2×C8D4×D59
commutatorC4×C59C2×C4×C8
FrattiniC22×C8C22×D59
FittingC2×C59×#350C2
socleC22×C59C2×C22⋊C8
prime radicalC2×#41×C59C2×C4
cosocleC22×C8×C59C23
hypercenterC2×#350D59
hypocenterC59C4×#350
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
3. M32×Dic59C2semidirect
4. C2×C16×Dic59C2semidirect
5. #21422186C2semidirect
6. #21422184C2semidirect
20. #16837186C4semidirect
28. C59×#350C4semidirect
43. #14273667C2×C4semidirect
47. C2×C59⋊C16C2×C4semidirect
68. M32×C59C2×C4semidirect
72. C2×C16×C59C2×C4semidirect
163. Dic59#350direct (N × H)
192. #350Dic59direct (N × H)
197. C59C4×#350semidirect
223. M32C2×Dic59semidirect
231. C2×C16C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#131×Dic592
C2×#168371862
M32×Dic592
C2×C16×Dic592
#214221862
#214221842
C2×C59×#3502
C4×#35059