Finite Group Databasebeta

← #28506117

Subgroups — #28506117

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
1#21433816C211yesmaximal, characteristic
2#21425833C211yesmaximal, characteristic, complemented
3C2×#16837210C211yesmaximal, characteristic, complemented
4#21424150C211yesmaximal, characteristic, complemented
5#21422208C211yesmaximal, characteristic, complemented
6#21422192C211yesmaximal, characteristic, complemented
7C2×C59×#504C211yesmaximal, characteristic, complemented
8#16838200—21no—
9C2×#14273749C2211yescharacteristic
10#16837677C412yescomplemented
11#16837672C412yescomplemented
12#16837665—22no—
13C2×#14273663C2211yescharacteristic
14C2×#14273663C2211yescharacteristic
15#16837440C2211yescharacteristic
16#16837419—21no—
17#16837384—21no—
18#16837213—22no—
19#16837210—21no—
20#16837210—21no—
21#16837174—21no—
22C2×#131×C59C2211yescharacteristic
23C59×#504—22no—
24C59×#504—22no—
25C2×M32×C59—21no—
26C2×M32×C59C2211yescharacteristic
27C22×C16×C59—21no—
28C22×C16×C59C2211yescharacteristic
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—
43C2×#11745664—21no—
44#14273686—21no—
45#14273663—21no—
46#14273663—21no—
47#14273663—21no—
48#14273663C2×C412yes—
49#14273661—21no—
50#14273659—21no—
51#14273654—42no—
52#14273645—22no—
53#14273643—22no—
54C2×#41×C59D411yescharacteristic
55#131×C59—21no—
56#131×C59—21no—
57#131×C59C2×C412yes—
58#131×C59D412yes—
59C2×M16×C59—21no—
60C2×M16×C59D411yescharacteristic
61C22×C8×C59—21no—
62C22×C8×C59C2311yescharacteristic
63M32×C59—24no—
64M32×C59—21no—
65M32×C59D412yes—
66C2×C16×C59—24no—
67C2×C16×C59—21no—
68C2×C16×C59—21no—
69C2×C16×C59—21no—
70C2×C16×C59D411yescharacteristic
71C2×C16×C59D411yescharacteristic
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×C59C22⋊C412yes—
93C2×C8×C59—21no—
94C2×C8×C59—22no—
95C2×C8×C59—22no—
96C2×C8×C59C22×C411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C22⋊Dic59—41no—
103#11745672—42no—
104#11745671—42no—
105#11745670—42no—
106#11745669—42no—
107C4⋊Dic59—41no—
108C4×Dic59—41no—
109#11745664—22no—
110#11745664—41no—
111#11745664—41no—
112#11745664—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115Dic236—21no—
116Dic236—21no—
117D472—21no—
118D472—21no—
119#11745660—41no—
120#11745660—41no—
121C16×C59—21no—
122C16×C59—22no—
123C16×C59—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×C59C2×C22⋊C411yescharacteristic
137C2×C4×C59#12111yescharacteristic
138C2×C4×C59#12111yescharacteristic
139C2×C4×C59C2×D811yescharacteristic
140C2×C4×C59C2×SD1611yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C22⋊D59—81no—
143C22⋊D59—81no—
144C2×Dic59—41no—
145C2×Dic59—81no—
146D236—41no—
147D236—41no—
148C4×D59—81no—
149C4×D59—81no—
150Dic118—41no—
151Dic118—41no—
152C8×C59—22no—
153C8×C59C4×D411yescharacteristic
154C8×C59C4×D411yescharacteristic
155C8×C59#12011yescharacteristic
156C8×C59#12311yescharacteristic
157C59⋊C8—42no—
158C59⋊C8—81no—
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#38611yescharacteristic
170C4×C59C2×#10211yescharacteristic
171C4×C59#46511yescharacteristic
172C4×C59#46611yescharacteristic
173Dic59—81no—
174Dic59—161no—
175C2×#504D5911yescharacteristic, complemented
176#3067—591no—
177#2145—591no—
178C2×#359—591no—
179#1814—591no—
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—22no—
192#504—22no—
193C2×M32—21no—
194C2×M32C2×D5911yescharacteristic
195C22×C16—21no—
196C22×C16C2×D5911yescharacteristic
197#473—592no—
198#472—592no—
199#471—1182no—
200C2×#108—591no—
201C2×#108—591no—
202#427—591no—
203#421—1181no—
204#413—1181no—
205#360—1182no—
206#359—1181no—
207#359—1181no—
208#344—1181no—
209C59#1307111yescharacteristic, complemented, Sylow
210C2×#41—1181no—
211C2×#41D23611yescharacteristic
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#131C22⋊D5912yes—
228C2×M16—21no—
229C2×M16—1181no—
230C2×M16—1181no—
231C2×M16D23611yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5911yescharacteristic
234#117—1181no—
235M32—24no—
236M32—21no—
237M32C22⋊D5912yes—
238C2×C16—24no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22⋊D5911yescharacteristic
243C2×C16C22⋊D5911yescharacteristic
244#108—592no—
245#108—1181no—
246#108—1181no—
247#108—1181no—
248#107—1181no—
249#106—1181no—
250#104—2362no—
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#41D47212yes—
260#41(C2×D59)⋊C412yes—
261C2×Q8—1181no—
262C2×Q8#1174566011yescharacteristic
263C2×D4—21no—
264C2×D4—1181no—
265C2×D4—2361no—
266C2×D4#1174566011yescharacteristic
267C22×C4—21no—
268C22×C4—2361no—
269C22×C4C2×D23611yescharacteristic
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—1182no—
284M16—2361no—
285M16—2361no—
286M16—2361no—
287M16(C2×D59)⋊C412yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—2361no—
292C2×C8—2361no—
293C2×C8C2×C4×D5911yescharacteristic
294C2×C8#1174568511yescharacteristic
295C2×C8#1174568511yescharacteristic
296C2×C8C2×C22⋊D5911yescharacteristic
297C2×C8C2×C22⋊D5911yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C4⋊C4—2361no—
300C22⋊C4—2361no—
301C42—2361no—
302C16—21no—
303C16—22no—
304C16—21no—
305C23—21no—
306C23—2361no—
307C23#1427374911yescharacteristic
308Q8—21no—
309Q8—2361no—
310Q8—2361no—
311Q8#1427364912yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—4721no—
316D4—4721no—
317D4—2361no—
318D4—2361no—
319D4#1427364912yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—4721no—
324C2×C4—4721no—
325C2×C4—4721no—
326C2×C4—2361no—
327C2×C4C2×(C2×D59)⋊C411yescharacteristic
328C2×C4#1427370111yescharacteristic
329C2×C4#1427370111yescharacteristic
330C2×C4C2×D47211yescharacteristic
331C2×C4C2×#1174566011yescharacteristic
332C2×C4#1427374911yescharacteristic
333C8—22no—
334C8—4721no—
335C8—2362no—
336C8C4×C22⋊D5911yescharacteristic
337C8C4×C22⋊D5911yescharacteristic
338C8#1427369611yescharacteristic
339C8#1427371011yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—9441no—
344C22—4721no—
345C22#1683739911yescharacteristic
346C22#1683763111yescharacteristic
347C22#1683763811yescharacteristic
348C4—22no—
349C4—9441no—
350C4—4721no—
351C4#1683728111yescharacteristic
352C4C2×#1427364911yescharacteristic
353C4#1683763111yescharacteristic
354C4#1683763811yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—9441no—
360C2#2142452411yescharacteristic
subgroupG/H
centerC4#16837638
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#504C2
socleC2×C59#1893
prime radical#21433816C2
cosocleC22×C8×C59C23
hypercenterC2×#504D59
hypocenterC59#13071
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #21425833C2semidirect
3. C2×#16837210C2semidirect
4. #21424150C2semidirect
5. #21422208C2semidirect
6. #21422192C2semidirect
7. C2×C59×#504C2semidirect
10. #16837677C4semidirect
11. #16837672C4semidirect
175. C2×#504D59semidirect
209. C59#13071semidirect

Maximal subgroups

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

subgroupindex
#214338162
#214258332
C2×#168372102
#214241502
#214222082
#214221922
C2×C59×#5042
#1307159