Finite Group Databasebeta

← #28505608

Subgroups — #28505608

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837818C211yesmaximal, characteristic, complemented
2#21432997C211yesmaximal, characteristic
3#21425725C211yesmaximal, characteristic, complemented
4#21425717C211yesmaximal, characteristic, complemented
5#21422184C211yesmaximal, characteristic, complemented
6#21422181C211yesmaximal, characteristic, complemented
7C2×C59×#350C211yesmaximal, characteristic, complemented
8#16838200C411yescharacteristic
9C2×#14273733C2211yescharacteristic
10#16837818—22no—
11#16837818—22no—
12C2×#14273667C2211yescharacteristic
13C2×#14273667C2211yescharacteristic
14C2×#14273666—21no—
15C2×C16×D59—21no—
16#16837522—22no—
17#16837517—22no—
18#16837450C411yescharacteristic
19#16837419C411yescharacteristic
20#16837418C411yescharacteristic
21#16837357C2211yescharacteristic
22#16837187—22no—
23#16837184—22no—
24C2×#131×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C22×C16×C59C2211yescharacteristic
27C59×#350—21no—
28C59×#350—21no—
29C2×#11745685C2×C411yescharacteristic
30#14273758C812yes—
31#14273757—21no—
32#14273753C812yes—
33#14273752—21no—
34#14273747—22no—
35#14273733—21no—
36#14273733—21no—
37#14273733D412yes—
38#14273733D412yes—
39C2×#11745664C2×C411yescharacteristic
40C2×#11745664C2×C411yescharacteristic
41C2×#11745659—21no—
42C2×C8×D59—21no—
43#14273686C2×C411yescharacteristic
44#14273667—21no—
45#14273667—21no—
46#14273667—21no—
47#14273667D412yes—
48#14273666—24no—
49C2×C59⋊C16—21no—
50C2×C59⋊C16—21no—
51C16×D59—24no—
52#14273662—22no—
53#14273654—22no—
54#14273653—22no—
55#14273637—21no—
56C8×Dic59—21no—
57C2×#41×C59C2×C411yescharacteristic
58#131×C59—21no—
59#131×C59—21no—
60#131×C59—21no—
61#131×C59—21no—
62C2×M16×C59—21no—
63C2×M16×C59C2×C411yescharacteristic
64C22×C8×C59—21no—
65C22×C8×C59C2311yescharacteristic
66M32×C59—21no—
67M32×C59—21no—
68C2×C16×C59—21no—
69C2×C16×C59—21no—
70C2×C16×C59—21no—
71C2×C16×C59D411yescharacteristic
72C2×C16×C59D411yescharacteristic
73#41×C59—21no—
74#41×C59—21no—
75#41×C59—21no—
76#41×C59—21no—
77C2×Q8×C59C2×C811yescharacteristic
78C2×D4×C59—21no—
79C2×D4×C59C2×C811yescharacteristic
80C22×C4×C59—21no—
81C22×C4×C59C22×C411yescharacteristic
82C2×C22⋊D59—21no—
83#11745685—21no—
84#11745685—21no—
85#11745685C22⋊C412yes—
86#11745685C2×C812yes—
87C2×D236M1611yescharacteristic
88C2×C4×D59—21no—
89C2×Dic118M1611yescharacteristic
90M16×C59—21no—
91M16×C59—21no—
92M16×C59—41no—
93M16×C59—41no—
94C2×C8×C59—21no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59C2×D411yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C2×C8×C59#4111yescharacteristic
103C22⋊Dic59—21no—
104#11745672—22no—
105#11745671—22no—
106#11745670—22no—
107#11745669—22no—
108C4⋊Dic59—21no—
109C4×Dic59—21no—
110#11745664—21no—
111#11745664—21no—
112#11745664C22⋊C412yes—
113#11745664C2×C812yes—
114C2×C59⋊C8—21no—
115C2×C59⋊C8—21no—
116#11745659—22no—
117#11745659—22no—
118C8×D59—22no—
119C8×D59—22no—
120C16×C59—21no—
121C16×C59—21no—
122C16×C59—41no—
123C59⋊C16—22no—
124C59⋊C16—41no—
125C23×C59—21no—
126C23×C59#13111yescharacteristic
127C22×D59—21no—
128Q8×C59—21no—
129Q8×C59—21no—
130D4×C59—21no—
131D4×C59—21no—
132D4×C59—41no—
133D4×C59—41no—
134C2×C4×C59—21no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59C2×C22⋊C411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C22×C811yescharacteristic
141C2×C4×C59C2×M1611yescharacteristic
142C2×C4×C59#13111yescharacteristic
143C22⋊D59—22no—
144C22⋊D59—22no—
145C2×Dic59—21no—
146C2×Dic59—41no—
147D236—21no—
148D236C22⋊C812yes—
149C4×D59—22no—
150C4×D59—22no—
151Dic118—21no—
152Dic118C22⋊C812yes—
153C8×C59—41no—
154C8×C59#12011yescharacteristic
155C8×C59#12111yescharacteristic
156C8×C59#12111yescharacteristic
157C8×C59#12311yescharacteristic
158C59⋊C8—22no—
159C59⋊C8—22no—
160#12987—591nomaximal, Sylow
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59#40811yescharacteristic
165C22×C59C8×D411yescharacteristic
166C22×C59#43511yescharacteristic
167C2×D59—22no—
168C2×D59—42no—
169C4×C59—41no—
170C4×C59#38611yescharacteristic
171C4×C59C2×C22⋊C811yescharacteristic
172C4×C59C8×D411yescharacteristic
173C4×C59#43511yescharacteristic
174Dic59—22no—
175Dic59—81no—
176C2×#504—591no—
177#2955—591no—
178#2117—591no—
179C2×#350D5911yescharacteristic, complemented
180#2115—591no—
181#1379—591no—
182#1378—591no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—81no—
187C2×C59#186911yescharacteristic
188D59—42no—
189#575—591no—
190C2×#131—591no—
191C2×#131C2×D5911yescharacteristic
192#504—1182no—
193#504—1182no—
194C2×M32—591no—
195C2×M32—591no—
196C2×M32—1181no—
197C2×M32C2×D5911yescharacteristic
198C22×C16—1181no—
199C22×C16C2×D5911yescharacteristic
200#444—1182no—
201#443—1182no—
202#430—591no—
203#421—591no—
204#421—591no—
205#405—591no—
206#350—21no—
207#350—21no—
208#350—1182no—
209#349—1182no—
210C59#1298711yescharacteristic, complemented, Sylow
211C2×#41—591no—
212C2×#41C4×D5911yescharacteristic
213#137—592no—
214#137—1181no—
215AGL(1,ℤ/8)—592no—
216AGL(1,ℤ/8)—1181no—
217#135—1182no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—592no—
223#131—592no—
224#131—1181no—
225#131—1181no—
226C2×M16—21no—
227C2×M16—591no—
228C2×M16—591no—
229C2×M16—1181no—
230C2×M16C4×D5911yescharacteristic
231C22×C8—21no—
232C22×C8—1181no—
233C22×C8C22×D5911yescharacteristic
234#117—591no—
235M32—21no—
236M32—21no—
237M32—592no—
238M32—1184no—
239M32—1181no—
240M32—1181no—
241M32—1181no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1184no—
246C2×C16—1181no—
247C2×C16—1181no—
248C2×C16C22⋊D5911yescharacteristic
249C2×C16C22⋊D5911yescharacteristic
250#108—1182no—
251#104—1182no—
252#104—1182no—
253#97—1181no—
254C4×C8—1181no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—592no—
260#41—592no—
261#41—1181no—
262#41—1181no—
263C2×Q8—591no—
264C2×Q8C8×D5911yescharacteristic
265C2×D4—21no—
266C2×D4—591no—
267C2×D4—1181no—
268C2×D4C8×D5911yescharacteristic
269C22×C4—21no—
270C22×C4—1181no—
271C22×C4C2×C4×D5911yescharacteristic
272Q16—1182no—
273SD16—1182no—
274SD16—1182no—
275D8—1182no—
276M16—21no—
277M16—21no—
278M16—41no—
279M16—41no—
280M16—592no—
281M16—592no—
282M16—1181no—
283M16—1181no—
284M16—1182no—
285M16—1182no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1181no—
290C2×C8—1181no—
291C2×C8—1182no—
292C2×C8—1182no—
293C2×C8#1174568511yescharacteristic
294C2×C8D4×D5911yescharacteristic
295C2×C8D4×D5911yescharacteristic
296C2×C8D4×D5911yescharacteristic
297C2×C8#1174568711yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C4⋊C4—1181no—
300C22⋊C4—1181no—
301C42—1181no—
302C16—21no—
303C16—21no—
304C16—41no—
305C16—1182no—
306C16—2361no—
307C23—21no—
308C23—1181no—
309C23#1427373511yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—592no—
313Q8—1181no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—41no—
318D4—592no—
319D4—1182no—
320D4—1182no—
321D4—1181no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—1181no—
326C2×C4—1182no—
327C2×C4—1182no—
328C2×C4—2361no—
329C2×C4C22⋊C4×D5911yescharacteristic
330C2×C4(C22⋊D59)⋊C411yescharacteristic
331C2×C4C4×C22⋊D5911yescharacteristic
332C2×C4C2×C8×D5911yescharacteristic
333C2×C4M16×D5911yescharacteristic
334C2×C4#1427373311yescharacteristic
335C8—41no—
336C8—1182no—
337C8—1182no—
338C8#1427369511yescharacteristic
339C8#1427369811yescharacteristic
340C8#1427370011yescharacteristic
341C8#1427370911yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—1182no—
346C22—2362no—
347C22#1683736711yescharacteristic
348C22(C22⋊D59)⋊C811yescharacteristic
349C22#1683747111yescharacteristic
350C4—41no—
351C4—1182no—
352C4—4721no—
353C4#1683727811yescharacteristic
354C4C22⋊C8×D5911yescharacteristic
355C4C8×C22⋊D5911yescharacteristic
356C4#1683747611yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—2362no—
362C2#2142438511yescharacteristic
subgroupG/H
centerC8#14273698
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC2×C59×#350C2
socleC2×C59#1869
prime radical#16838200C4
cosocleC22×C8×C59C23
hypercenterC2×#350D59
hypocenterC59#12987
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. C2×#16837818C2semidirect
3. #21425725C2semidirect
4. #21425717C2semidirect
5. #21422184C2semidirect
6. #21422181C2semidirect
7. C2×C59×#350C2semidirect
179. C2×#350D59semidirect
210. C59#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168378182
#214329972
#214257252
#214257172
#214221842
#214221812
C2×C59×#3502
#1298759