Finite Group Databasebeta

← #28505809

Subgroups — #28505809

Order 15104.

462 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#21437230C211yesmaximal, characteristic
2C2×#16837208C211yesmaximal, characteristic, complemented
3#21425803C211yesmaximal, characteristic, complemented
4#21424994C211yesmaximal, characteristic, complemented
5#21422210C211yesmaximal, characteristic, complemented
6#21422185C211yesmaximal, characteristic, complemented
7C59×#2957C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9#16837843—21no—
10#16837842C412yescomplemented
11#16837838—21no—
12#16837837C412yescomplemented
13#16837832—22no—
14C2×#14273667C2211yescharacteristic
15C22×C59⋊C16C2211yescharacteristic
16#16837440C2211yescharacteristic
17#16837418—21no—
18#16837375—21no—
19#16837208—22no—
20#16837206—21no—
21#16837203—21no—
22C2×#135×C59C2211yescharacteristic
23C59×#568—21no—
24C59×#444—22no—
25C59×#443—22no—
26C2×#108×C59C2211yescharacteristic
27C59×#421—21no—
28C59×#405C2211yescharacteristic
29C2×#11745685—21no—
30#14273749—21no—
31#14273749—21no—
32#14273749—21no—
33#14273749—21no—
34C2×Dic236C2×C411yescharacteristic
35C2×#11745660—21no—
36C2×D472C2×C411yescharacteristic
37#14273686—21no—
38#14273673—22no—
39#14273671—22no—
40#14273670—22no—
41#14273668—22no—
42#14273667—21no—
43#14273667C2×C412yes—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16—21no—
48#14273661—21no—
49#14273659—21no—
50#14273653—42no—
51#14273641—42no—
52C2×#41×C59—21no—
53#135×C59—21no—
54#135×C59—21no—
55#135×C59C2×C412yes—
56#135×C59D412yes—
57C2×Q16×C59D411yescharacteristic
58C2×SD16×C59—21no—
59C2×D8×C59D411yescharacteristic
60#131×C59—24no—
61#131×C59—42no—
62C2×M16×C59—21no—
63C2×M16×C59—42no—
64C2×M16×C59D411yescharacteristic
65C22×C8×C59C2311yescharacteristic
66#117×C59D411yescharacteristic
67#108×C59—21no—
68#108×C59D412yes—
69#104×C59—24no—
70#97×C59—21no—
71C4×C8×C59—21no—
72#41×C59—22no—
73#41×C59—22no—
74#41×C59—41no—
75#41×C59—41no—
76C2×Q8×C59—21no—
77C2×D4×C59—21no—
78C2×D4×C59—41no—
79C22×C4×C59—41no—
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—
89Q16×C59—21no—
90Q16×C59C22⋊C412yes—
91SD16×C59—22no—
92SD16×C59—22no—
93D8×C59—21no—
94D8×C59C22⋊C412yes—
95M16×C59—21no—
96M16×C59—22no—
97M16×C59—44no—
98M16×C59—41no—
99M16×C59D812yes—
100C2×C8×C59—21no—
101C2×C8×C59—44no—
102C2×C8×C59—41no—
103C2×C8×C59C22×C411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59C2×D411yescharacteristic
107C2×C8×C59#4111yescharacteristic
108C2×C8×C59#4111yescharacteristic
109C4⋊C4×C59—21no—
110C22⋊C4×C59—21no—
111C42×C59—21no—
112C22⋊Dic59—41no—
113C4⋊Dic59—41no—
114C4×Dic59—41no—
115Dic236—21no—
116Dic236—21no—
117D472—21no—
118D472—21no—
119#11745660—41no—
120#11745660—41no—
121C59⋊C16—22no—
122C59⋊C16—41no—
123C23×C59—41no—
124C23×C59AGL(1,ℤ/8)11yescharacteristic
125C22×D59—41no—
126Q8×C59—22no—
127Q8×C59—41no—
128D4×C59—22no—
129D4×C59—42no—
130D4×C59—42no—
131D4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—42no—
135C2×C4×C59—41no—
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59#12011yescharacteristic
138C2×C4×C59#12111yescharacteristic
139C2×C4×C59C2×D811yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
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×C59—42no—
154C8×C59C2×C22⋊C411yescharacteristic
155C8×C59C4×D411yescharacteristic
156C8×C59#12111yescharacteristic
157C8×C59#12311yescharacteristic
158#13021—591nomaximal, Sylow
159C22×C59—41no—
160C22×C59—42no—
161C22×C59—161no—
162C22×C59#44211yescharacteristic
163C22×C59#44911yescharacteristic
164C22×C59#45911yescharacteristic
165C2×D59—81no—
166C2×D59—161no—
167C4×C59—42no—
168C4×C59—81no—
169C4×C59#38611yescharacteristic
170C4×C59C4×D811yescharacteristic
171C4×C59#44711yescharacteristic
172C4×C59#46311yescharacteristic
173Dic59—81no—
174Dic59—161no—
175#3414—591no—
176#2957D5911yescharacteristic, complemented
177C2×#359—591no—
178#2138—591no—
179#1976—591no—
180#1389—591no—
181#1379—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—82no—
186C2×C59#187911yescharacteristic
187D59—161no—
188C2×#135—591no—
189C2×#135C2×D5911yescharacteristic
190#568—21no—
191#510—592no—
192#510—1181no—
193#509—592no—
194#509—1181no—
195#508—1182no—
196C2×M32—591no—
197C22×C16—591no—
198#444—22no—
199#443—22no—
200C2×#108C2×D5911yescharacteristic
201#427—591no—
202#421—21no—
203#421—1181no—
204#410—1181no—
205#405C2×D5911yescharacteristic
206#359—1182no—
207#358—1181no—
208#357—1181no—
209C59#1302111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41—1181no—
212#135—21no—
213#135—21no—
214#135—1181no—
215#135—1181no—
216#135—1181no—
217#135—1181no—
218#135C4×D5912yes—
219#135C22⋊D5912yes—
220C2×Q16—591no—
221C2×Q16D23611yescharacteristic
222C2×SD16—21no—
223C2×SD16—1181no—
224C2×D8—591no—
225C2×D8D23611yescharacteristic
226#131—24no—
227#131—42no—
228C2×M16—21no—
229C2×M16—42no—
230C2×M16C22⋊D5911yescharacteristic
231C22×C8C22×D5911yescharacteristic
232#117—1181no—
233#117C22⋊D5911yescharacteristic
234Q32—1182no—
235SD32—1182no—
236SD32—1182no—
237D16—1182no—
238M32—592no—
239M32—1181no—
240C2×C16—1181no—
241C2×C16—1181no—
242C2×C16—1181no—
243C2×C16—1181no—
244#108—21no—
245#108C22⋊D5912yes—
246#107—1181no—
247#106—1181no—
248#104—24no—
249#104—2362no—
250#99—2362no—
251#97—21no—
252C4×C8—21no—
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—2361no—
258#41—2361no—
259#41—2361no—
260#41—2361no—
261C2×Q8—21no—
262C2×Q8—1181no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—1181no—
266C2×D4—2361no—
267C22×C4—41no—
268C22×C4—2361no—
269C22×C4C2×C22⋊D5911yescharacteristic
270Q16—21no—
271Q16—1181no—
272Q16—1181no—
273Q16(C2×D59)⋊C412yes—
274SD16—22no—
275SD16—22no—
276SD16—2361no—
277SD16—2361no—
278D8—21no—
279D8—1181no—
280D8—1181no—
281D8(C2×D59)⋊C412yes—
282M16—21no—
283M16—22no—
284M16—44no—
285M16—41no—
286M16#1174566912yes—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8—41no—
290C2×C8C2×C4×D5911yescharacteristic
291C2×C8C2×D23611yescharacteristic
292C2×C8#1174568511yescharacteristic
293C2×C8#1174568511yescharacteristic
294C2×C8C2×C22⋊D5911yescharacteristic
295C2×C8C2×C22⋊D5911yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—2361no—
298C22⋊C4—21no—
299C22⋊C4—2361no—
300C42—21no—
301C42—2361no—
302C16—1182no—
303C16—2361no—
304C23—41no—
305C23—2361no—
306C23#1427375211yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—2361no—
310Q8—2361no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—4721no—
316D4—4721no—
317D4—2361no—
318D4—2361no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—4721no—
324C2×C4—4721no—
325C2×C4—4721no—
326C2×C4—2361no—
327C2×C4C4×C22⋊D5911yescharacteristic
328C2×C4#1427369611yescharacteristic
329C2×C4#1427370111yescharacteristic
330C2×C4C2×#1174566911yescharacteristic
331C2×C4#1427374711yescharacteristic
332C2×C4#1427375311yescharacteristic
333C8—22no—
334C8—42no—
335C8C2×(C2×D59)⋊C411yescharacteristic
336C8C4×C22⋊D5911yescharacteristic
337C8#1427370111yescharacteristic
338C8#1427371011yescharacteristic
339C22—41no—
340C22—42no—
341C22—161no—
342C22—9441no—
343C22—4721no—
344C22#1683751211yescharacteristic
345C22#1683754711yescharacteristic
346C22#1683759711yescharacteristic
347C4—42no—
348C4—81no—
349C4—9441no—
350C4—4721no—
351C4#1683728111yescharacteristic
352C4C4×#1174566911yescharacteristic
353C4#1683753611yescharacteristic
354C4#1683761711yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—9441no—
360C2#2142444411yescharacteristic
subgroupG/H
centerC4#16837536
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2957C2
socleC2×C59#1879
prime radical#21437230C2
cosocleC22×C8×C59C23
hypercenter#2957D59
hypocenterC59#13021
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. C2×#16837208C2semidirect
3. #21425803C2semidirect
4. #21424994C2semidirect
5. #21422210C2semidirect
6. #21422185C2semidirect
7. C59×#2957C2semidirect
10. #16837842C4semidirect
12. #16837837C4semidirect
176. #2957D59semidirect
209. C59#13021semidirect

Maximal subgroups

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

subgroupindex
#214372302
C2×#168372082
#214258032
#214249942
#214222102
#214221852
C59×#29572
#1302159