Finite Group Databasebeta

← #28506115

Subgroups — #28506115

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837819C211yesmaximal, characteristic, complemented
2#21433816C211yesmaximal, characteristic
3#21425831C211yesmaximal, characteristic, complemented
4C2×#16837208C211yesmaximal, characteristic, complemented
5#21422207C211yesmaximal, characteristic, complemented
6#21422191C211yesmaximal, characteristic, complemented
7C59×#1814C211yesmaximal, characteristic, complemented
8#16838200—21no—
9C2×#14273749C2211yescharacteristic
10#16837819—22no—
11#16837819—22no—
12C2×#14273667—21no—
13C2×#14273667C2211yescharacteristic
14C22×C59⋊C16—21no—
15C22×C59⋊C16C2211yescharacteristic
16#16837677C412yescomplemented
17#16837672C412yescomplemented
18#16837665—22no—
19C2×#14273663C2211yescharacteristic
20#16837384—21no—
21#16837211—22no—
22#16837208—21no—
23#16837208—21no—
24C2×#131×C59C2211yescharacteristic
25C2×#108×C59C2211yescharacteristic
26C59×#427C2211yescharacteristic
27C59×#421—21no—
28C59×#344—21no—
29C2×#11745685—21no—
30#14273758—41no—
31#14273757—22no—
32#14273753—41no—
33#14273752—22no—
34#14273749—21no—
35#14273749—21no—
36#14273749—21no—
37#14273749—21no—
38#14273747—42no—
39C2×Dic236C2×C411yescharacteristic
40C2×#11745660—21no—
41C2×D472C2×C411yescharacteristic
42C2×#11745664—21no—
43#14273667—24no—
44#14273667—21no—
45#14273667D412yes—
46C2×C59⋊C16—24no—
47C2×C59⋊C16—21no—
48C2×C59⋊C16—21no—
49C2×C59⋊C16—21no—
50C2×C59⋊C16D411yescharacteristic
51C2×C59⋊C16D411yescharacteristic
52#14273663—21no—
53#14273663C2×C412yes—
54#14273645—22no—
55#14273643—22no—
56C2×#41×C59D411yescharacteristic
57#131×C59—21no—
58#131×C59—21no—
59#131×C59C2×C412yes—
60#131×C59D412yes—
61C2×M16×C59—21no—
62C2×M16×C59—21no—
63C2×M16×C59D411yescharacteristic
64C22×C8×C59—21no—
65C22×C8×C59C2311yescharacteristic
66#117×C59—21no—
67#108×C59—21no—
68#108×C59—21no—
69#107×C59—21no—
70#106×C59—21no—
71#104×C59—42no—
72#41×C59—21no—
73#41×C59—21no—
74#41×C59C22⋊C412yes—
75#41×C59D812yes—
76C2×Q8×C59SD1611yescharacteristic
77C2×D4×C59—21no—
78C2×D4×C59SD1611yescharacteristic
79C22×C4×C59—21no—
80C22×C4×C59C2×D411yescharacteristic
81C2×C22⋊D59—41no—
82#11745685—41no—
83#11745685—41no—
84#11745685—41no—
85#11745685—41no—
86C2×D236—21no—
87C2×C4×D59—41no—
88C2×Dic118—21no—
89M16×C59—21no—
90M16×C59—22no—
91M16×C59—22no—
92M16×C59—41no—
93M16×C59—41no—
94M16×C59C22⋊C412yes—
95C2×C8×C59—21no—
96C2×C8×C59—22no—
97C2×C8×C59—22no—
98C2×C8×C59—41no—
99C2×C8×C59C22×C411yescharacteristic
100C2×C8×C59C2×D411yescharacteristic
101C2×C8×C59C2×D411yescharacteristic
102C2×C8×C59C2×D411yescharacteristic
103C2×C8×C59#4111yescharacteristic
104C2×C8×C59#4111yescharacteristic
105C4⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C42×C59—41no—
108#11745672—42no—
109#11745671—42no—
110#11745670—42no—
111#11745669—42no—
112#11745664—22no—
113#11745664—41no—
114C2×C59⋊C8—41no—
115Dic236—21no—
116Dic236—21no—
117D472—21no—
118D472—21no—
119#11745660—41no—
120#11745660—41no—
121C59⋊C16—21no—
122C59⋊C16—22no—
123C59⋊C16—21no—
124C23×C59—21no—
125C23×C59#13511yescharacteristic
126C22×D59—41no—
127Q8×C59—21no—
128Q8×C59#10212yes—
129D4×C59—22no—
130D4×C59—22no—
131D4×C59—21no—
132D4×C59#10212yes—
133C2×C4×C59—21no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—81no—
137C2×C4×C59C2×C22⋊C411yescharacteristic
138C2×C4×C59#12111yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59C2×D811yescharacteristic
141C2×C4×C59C2×SD1611yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C22⋊D59—81no—
144C22⋊D59—81no—
145C2×Dic59—41no—
146D236—41no—
147D236—41no—
148C4×D59—81no—
149C4×D59—81no—
150Dic118—41no—
151Dic118—41no—
152C8×C59—22no—
153C8×C59—81no—
154C8×C59C4×D411yescharacteristic
155C8×C59C4×D411yescharacteristic
156C8×C59#12011yescharacteristic
157C8×C59#12311yescharacteristic
158C59⋊C8—42no—
159#13071—591nomaximal, Sylow
160C22×C59—22no—
161C22×C59—41no—
162C22×C59—42no—
163C22×C59#41611yescharacteristic
164C22×C59#46511yescharacteristic
165C22×C59#46611yescharacteristic
166C2×D59—81no—
167C2×D59—161no—
168C4×C59—22no—
169C4×C59—161no—
170C4×C59#38611yescharacteristic
171C4×C59C2×#10211yescharacteristic
172C4×C59#46511yescharacteristic
173C4×C59#46611yescharacteristic
174Dic59—81no—
175C2×#504—591no—
176#3067—591no—
177#2145—591no—
178C2×#359—591no—
179#1814D5911yescharacteristic, complemented
180#1388—591no—
181#1382—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—42no—
186C2×C59#189311yescharacteristic
187D59—161no—
188#575—1181no—
189C2×#135—591no—
190C2×#131C2×D5911yescharacteristic
191#504—1182no—
192#504—1182no—
193C2×M32—591no—
194C2×M32—1181no—
195C22×C16—591no—
196C22×C16—1181no—
197#473—592no—
198#472—592no—
199#471—1182no—
200C2×#108—591no—
201C2×#108C2×D5911yescharacteristic
202#427C2×D5911yescharacteristic
203#421—21no—
204#413—1181no—
205#360—1182no—
206#359—1181no—
207#359—1181no—
208#344—21no—
209C59#1307111yescharacteristic, complemented, Sylow
210C2×#41—1181no—
211C2×#41C22⋊D5911yescharacteristic
212#137—1182no—
213#137—2361no—
214AGL(1,ℤ/8)—1182no—
215AGL(1,ℤ/8)—2361no—
216#135—1181no—
217#135—1181no—
218#135—1181no—
219#135—1181no—
220#135—2362no—
221C2×Q16—591no—
222C2×SD16—1181no—
223C2×D8—591no—
224#131—21no—
225#131—21no—
226#131C4×D5912yes—
227#131D23612yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1181no—
231C2×M16C22⋊D5911yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5911yescharacteristic
234#117—21no—
235M32—592no—
236M32—1184no—
237M32—1181no—
238C2×C16—591no—
239C2×C16—591no—
240C2×C16—1184no—
241C2×C16—1181no—
242C2×C16—1181no—
243C2×C16—1181no—
244#108—21no—
245#108—21no—
246#108—592no—
247#108—1181no—
248#107—21no—
249#106—21no—
250#104—42no—
251#101—1182no—
252#100—1182no—
253#41—21no—
254#41—21no—
255#41—2361no—
256#41—2361no—
257#41—2361no—
258#41—2361no—
259#41(C2×D59)⋊C412yes—
260#41#1174566912yes—
261C2×Q8—1181no—
262C2×Q8#1174567111yescharacteristic
263C2×D4—21no—
264C2×D4—1181no—
265C2×D4—2361no—
266C2×D4#1174567111yescharacteristic
267C22×C4—21no—
268C22×C4—2361no—
269C22×C4C2×C22⋊D5911yescharacteristic
270Q16—1181no—
271Q16—1181no—
272Q16—2362no—
273SD16—2362no—
274SD16—2362no—
275SD16—2361no—
276SD16—2361no—
277D8—1181no—
278D8—1181no—
279D8—2362no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—41no—
284M16—41no—
285M16—1182no—
286M16—2361no—
287M16(C2×D59)⋊C412yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—2361no—
293C2×C8C2×C4×D5911yescharacteristic
294C2×C8C2×D23611yescharacteristic
295C2×C8D4×D5911yescharacteristic
296C2×C8D4×D5911yescharacteristic
297C2×C8#1174568711yescharacteristic
298C2×C8#1174568711yescharacteristic
299C4⋊C4—41no—
300C22⋊C4—41no—
301C42—41no—
302C16—1181no—
303C16—1181no—
304C16—1182no—
305C23—21no—
306C23—2361no—
307C23#1427374711yescharacteristic
308Q8—21no—
309Q8—2361no—
310Q8—2361no—
311Q8#1427364712yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—4721no—
316D4—4721no—
317D4—2361no—
318D4—2361no—
319D4#1427364712yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—81no—
324C2×C4—4721no—
325C2×C4—4721no—
326C2×C4—2361no—
327C2×C4C2×(C2×D59)⋊C411yescharacteristic
328C2×C4#1427369911yescharacteristic
329C2×C4#1427369911yescharacteristic
330C2×C4C2×#1174566911yescharacteristic
331C2×C4C2×#1174567111yescharacteristic
332C2×C4#1427374711yescharacteristic
333C8—22no—
334C8—81no—
335C8—2362no—
336C8(C22⋊D59)⋊C411yescharacteristic
337C8(C22⋊D59)⋊C411yescharacteristic
338C8#1427369711yescharacteristic
339C8#1427371111yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—9441no—
344C22—4721no—
345C22#1683739611yescharacteristic
346C22#1683763311yescharacteristic
347C22#1683763611yescharacteristic
348C4—22no—
349C4—161no—
350C4—4721no—
351C4#1683727911yescharacteristic
352C4C2×#1427364711yescharacteristic
353C4#1683763311yescharacteristic
354C4#1683763611yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—9441no—
360C2#2142452211yescharacteristic
subgroupG/H
centerC4#16837636
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1814C2
socleC2×C59#1893
prime radical#21433816C2
cosocleC22×C8×C59C23
hypercenter#1814D59
hypocenterC59#13071
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #1814

Complements

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

NH ≅ G/Ntype
1. C2×#16837819C2semidirect
3. #21425831C2semidirect
4. C2×#16837208C2semidirect
5. #21422207C2semidirect
6. #21422191C2semidirect
7. C59×#1814C2semidirect
16. #16837677C4semidirect
17. #16837672C4semidirect
179. #1814D59semidirect
209. C59#13071semidirect

Maximal subgroups

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

subgroupindex
C2×#168378192
#214338162
#214258312
C2×#168372082
#214222072
#214221912
C59×#18142
#1307159