Finite Group Databasebeta

← #28505559

Subgroups — #28505559

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432997C211yesmaximal, characteristic
2#21425725C211yesmaximal, characteristic, complemented
3C2×#16837186C211yesmaximal, characteristic, complemented
4#21425711C211yesmaximal, characteristic, complemented
5#21422186C211yesmaximal, characteristic, complemented
6#21422181C211yesmaximal, characteristic, complemented
7C59×#3407C211yesmaximal, characteristic, complemented
8#16838200C411yescharacteristic
9C2×#14273733C2211yescharacteristic
10C2×#14273667C2211yescharacteristic
11C22×C59⋊C16C2211yescharacteristic
12#16837522—22no—
13#16837517—22no—
14#16837450C411yescharacteristic
15#16837419C411yescharacteristic
16#16837418C411yescharacteristic
17#16837357C2211yescharacteristic
18#16837187—22no—
19#16837186—22no—
20(C2×D59)⋊C16—22no—
21C2×#131×C59C2211yescharacteristic
22C59×#504—22no—
23C59×#504—22no—
24C2×M32×C59—22no—
25C2×M32×C59C2211yescharacteristic
26C2×M32×C59C2211yescharacteristic
27C2×#11745685C2×C411yescharacteristic
28#14273758—21no—
29#14273757C812yes—
30#14273753—21no—
31#14273752C812yes—
32#14273747—22no—
33#14273733—21no—
34#14273733—21no—
35#14273733—21no—
36#14273733—21no—
37C2×#11745664C2×C411yescharacteristic
38C2×#11745664C2×C411yescharacteristic
39C2×#11745659—21no—
40C2×C8×D59—21no—
41#14273686C2×C411yescharacteristic
42#14273667—21no—
43#14273667—21no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16—21no—
48#14273662—22no—
49#14273654—22no—
50#14273653—22no—
51#14273637—21no—
52C8×Dic59—21no—
53C2×#41×C59C2×C411yescharacteristic
54#131×C59—21no—
55#131×C59—21no—
56#131×C59D412yes—
57#131×C59D412yes—
58C2×M16×C59—21no—
59C2×M16×C59C2×C411yescharacteristic
60C22×C8×C59—21no—
61C22×C8×C59C2311yescharacteristic
62M32×C59—24no—
63M32×C59—21no—
64M32×C59—21no—
65M32×C59—21no—
66M32×C59D412yes—
67C2×C16×C59—24no—
68C2×C16×C59—21no—
69C2×C16×C59D412yes—
70#41×C59—21no—
71#41×C59—21no—
72#41×C59C22⋊C412yes—
73#41×C59C2×C812yes—
74C2×Q8×C59M1611yescharacteristic
75C2×D4×C59—21no—
76C2×D4×C59M1611yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59C22×C411yescharacteristic
79C2×C22⋊D59—21no—
80#11745685—21no—
81#11745685—21no—
82#11745685—21no—
83#11745685—21no—
84C2×D236C2×C811yescharacteristic
85C2×C4×D59—21no—
86C2×Dic118C2×C811yescharacteristic
87M16×C59—21no—
88M16×C59—22no—
89M16×C59—22no—
90M16×C59C22⋊C412yes—
91C2×C8×C59—21no—
92C2×C8×C59—22no—
93C2×C8×C59—22no—
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59#4111yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C22⋊Dic59—21no—
101#11745672—22no—
102#11745671—22no—
103#11745670—22no—
104#11745669—22no—
105C4⋊Dic59—21no—
106C4×Dic59—21no—
107#11745664—21no—
108#11745664—21no—
109#11745664—21no—
110#11745664C2×C812yes—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113#11745659—41no—
114#11745659—41no—
115C8×D59—41no—
116C8×D59—41no—
117C16×C59—22no—
118C16×C59—22no—
119C59⋊C16—41no—
120C59⋊C16—41no—
121C23×C59—21no—
122C23×C59#13111yescharacteristic
123C22×D59—21no—
124Q8×C59—21no—
125Q8×C59C22⋊C812yes—
126D4×C59—22no—
127D4×C59—22no—
128D4×C59—21no—
129D4×C59C22⋊C812yes—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59C2×C22⋊C411yescharacteristic
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59C4×D411yescharacteristic
136C2×C4×C59C22×C811yescharacteristic
137C2×C4×C59C2×M1611yescharacteristic
138C2×C4×C59#13111yescharacteristic
139C22⋊D59—41no—
140C22⋊D59—41no—
141C2×Dic59—21no—
142C2×Dic59—41no—
143D236—21no—
144D236—21no—
145C4×D59—41no—
146C4×D59—41no—
147Dic118—21no—
148Dic118—21no—
149C8×C59—22no—
150C8×C59#12011yescharacteristic
151C8×C59#12111yescharacteristic
152C8×C59#12111yescharacteristic
153C8×C59#12311yescharacteristic
154C59⋊C8—22no—
155C59⋊C8—41no—
156#12979—591nomaximal, Sylow
157C22×C59—22no—
158C22×C59—41no—
159C22×C59—81no—
160C22×C59#40811yescharacteristic
161C22×C59C8×D411yescharacteristic
162C22×C59#43511yescharacteristic
163C2×D59—42no—
164C2×D59—41no—
165C4×C59—22no—
166C4×C59#38611yescharacteristic
167C4×C59C2×C22⋊C811yescharacteristic
168C4×C59C8×D411yescharacteristic
169C4×C59#43511yescharacteristic
170Dic59—41no—
171Dic59—81no—
172#3407D5911yescharacteristic, complemented
173#2955—591no—
174#2117—591no—
175C2×#350—591no—
176#2113—591no—
177#1379—591no—
178#1378—591no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—42no—
183C2×C59#186911yescharacteristic
184D59—81no—
185#575—591no—
186C2×#131—591no—
187C2×#131C2×D5911yescharacteristic
188#504—22no—
189#504—22no—
190C2×M32—22no—
191C2×M32—591no—
192C2×M32C2×D5911yescharacteristic
193C2×M32C2×D5911yescharacteristic
194C22×C16—591no—
195#444—1182no—
196#443—1182no—
197#430—591no—
198#421—591no—
199#421—591no—
200#405—591no—
201#350—1182no—
202#350—1182no—
203C22⋊C16—1182no—
204C59#1297911yescharacteristic, complemented, Sylow
205C2×#41—591no—
206C2×#41C4×D5911yescharacteristic
207#137—592no—
208#137—1181no—
209AGL(1,ℤ/8)—592no—
210AGL(1,ℤ/8)—1181no—
211#135—1182no—
212#131—21no—
213#131—21no—
214#131—1181no—
215#131—1181no—
216#131—1181no—
217#131—1181no—
218#131D23612yes—
219#131C22⋊D5912yes—
220C2×M16—21no—
221C2×M16—591no—
222C2×M16—591no—
223C2×M16—1181no—
224C2×M16C4×D5911yescharacteristic
225C22×C8—21no—
226C22×C8—1181no—
227C22×C8C22×D5911yescharacteristic
228#117—591no—
229M32—24no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—1181no—
234M32—1181no—
235M32C22⋊D5912yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16—1181no—
239C2×C16—1181no—
240C2×C16—1181no—
241C2×C16—1181no—
242C2×C16C22⋊D5912yes—
243#108—1182no—
244#104—1182no—
245#104—1182no—
246#97—1181no—
247C4×C8—1181no—
248#41—21no—
249#41—21no—
250#41—1181no—
251#41—1181no—
252#41—1181no—
253#41—1181no—
254#41C8×D5912yes—
255#41(C2×D59)⋊C412yes—
256C2×Q8—591no—
257C2×Q8#1174565911yescharacteristic
258C2×D4—21no—
259C2×D4—591no—
260C2×D4—1181no—
261C2×D4#1174565911yescharacteristic
262C22×C4—21no—
263C22×C4—1181no—
264C22×C4C2×C4×D5911yescharacteristic
265Q16—1182no—
266SD16—1182no—
267SD16—1182no—
268D8—1182no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—592no—
273M16—1181no—
274M16—1181no—
275M16—1181no—
276M16—2361no—
277M16—2361no—
278M16(C2×D59)⋊C412yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—1181no—
283C2×C8—1181no—
284C2×C8—2361no—
285C2×C8—2361no—
286C2×C8C2×D23611yescharacteristic
287C2×C8#1174568511yescharacteristic
288C2×C8#1174568511yescharacteristic
289C2×C8C2×C22⋊D5911yescharacteristic
290C2×C8C2×C22⋊D5911yescharacteristic
291C2×C8C2×C22⋊D5911yescharacteristic
292C4⋊C4—1181no—
293C22⋊C4—1181no—
294C42—1181no—
295C16—22no—
296C16—22no—
297C16—2361no—
298C16—2361no—
299C23—21no—
300C23—1181no—
301C23#1427373311yescharacteristic
302Q8—21no—
303Q8—1181no—
304Q8—1181no—
305Q8(C2×D59)⋊C812yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—1181no—
310D4—1181no—
311D4—2361no—
312D4—2361no—
313D4(C2×D59)⋊C812yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—1181no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4C2×(C2×D59)⋊C411yescharacteristic
322C2×C4C4×C22⋊D5911yescharacteristic
323C2×C4C4×C22⋊D5911yescharacteristic
324C2×C4C2×C8×D5911yescharacteristic
325C2×C4C2×#1174565911yescharacteristic
326C2×C4#1427373311yescharacteristic
327C8—22no—
328C8—1182no—
329C8—2361no—
330C8#1427369611yescharacteristic
331C8#1427370111yescharacteristic
332C8#1427370111yescharacteristic
333C8#1427371011yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—2362no—
338C22—2361no—
339C22#1683736811yescharacteristic
340C22C8×C22⋊D5911yescharacteristic
341C22#1683747311yescharacteristic
342C4—22no—
343C4—4721no—
344C4—2361no—
345C4#1683728111yescharacteristic
346C4C2×(C2×D59)⋊C811yescharacteristic
347C4C8×C22⋊D5911yescharacteristic
348C4#1683747311yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—4721no—
354C2#2142438811yescharacteristic
subgroupG/H
centerC8#14273696
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#3407C2
socleC2×C59#1869
prime radical#16838200C4
cosocleC22×C8×C59C23
hypercenter#3407D59
hypocenterC59#12979
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. #21425725C2semidirect
3. C2×#16837186C2semidirect
4. #21425711C2semidirect
5. #21422186C2semidirect
6. #21422181C2semidirect
7. C59×#3407C2semidirect
172. #3407D59semidirect
204. C59#12979semidirect

Maximal subgroups

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

subgroupindex
#214329972
#214257252
C2×#168371862
#214257112
#214221862
#214221812
C59×#34072
#1297959