Finite Group Databasebeta

← #28505811

Subgroups — #28505811

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#21433013C211yesmaximal, characteristic, complemented
2C2×#16837209C211yesmaximal, characteristic, complemented
3#21425803C211yesmaximal, characteristic, complemented
4#21424995C211yesmaximal, characteristic, complemented
5#21422211C211yesmaximal, characteristic, complemented
6#21422186C211yesmaximal, characteristic, complemented
7C59×#3414C211yesmaximal, characteristic
8#16838169—21no—
9C22×C59⋊C16C2211yescharacteristic
10#16837524—22no—
11#16837519—22no—
12C2×#14273663C2211yescharacteristic
13#16837440C2211yescharacteristic
14#16837419—21no—
15#16837419—21no—
16#16837376—21no—
17#16837357C2211yescharacteristic
18#16837209—22no—
19#16837206—21no—
20#16837203—21no—
21C2×#135×C59C2211yescharacteristic
22C2×#135×C59C2211yescharacteristic
23C59×#510—21no—
24C59×#510C412yescomplemented
25C59×#509—21no—
26C59×#509C412yescomplemented
27C59×#508—22no—
28C2×M32×C59C2211yescharacteristic
29#14273734—24no—
30#14273734—42no—
31C2×#11745664—21no—
32C2×#11745664—42no—
33C2×#11745664D411yescharacteristic
34#14273686—21no—
35#14273686D411yescharacteristic
36C2×C59⋊C16—21no—
37C2×C59⋊C16—21no—
38C2×C59⋊C16—21no—
39#14273663—21no—
40#14273663D412yes—
41#14273661—21no—
42#14273659—21no—
43#14273654—24no—
44#14273654—42no—
45#14273642—42no—
46#14273637—21no—
47C8×Dic59—21no—
48C2×#41×C59—21no—
49C2×#41×C59—21no—
50#135×C59—21no—
51#135×C59—21no—
52#135×C59—21no—
53#135×C59—21no—
54#135×C59—21no—
55#135×C59—21no—
56#135×C59C2×C412yes—
57#135×C59D412yes—
58C2×Q16×C59C2×C411yescharacteristic
59C2×Q16×C59D411yescharacteristic
60C2×SD16×C59—21no—
61C2×SD16×C59—21no—
62C2×D8×C59C2×C411yescharacteristic
63C2×D8×C59D411yescharacteristic
64C22×C8×C59C2311yescharacteristic
65Q32×C59—22no—
66SD32×C59—22no—
67SD32×C59—22no—
68D16×C59—22no—
69M32×C59—21no—
70M32×C59C2×C412yes—
71C2×C16×C59—21no—
72#41×C59—22no—
73#41×C59—22no—
74#41×C59—41no—
75#41×C59—41no—
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79#41×C59—41no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89Q16×C59—21no—
90Q16×C59—21no—
91Q16×C59—21no—
92Q16×C59C22⋊C412yes—
93SD16×C59—22no—
94SD16×C59—22no—
95SD16×C59—41no—
96SD16×C59—41no—
97D8×C59—21no—
98D8×C59—21no—
99D8×C59—21no—
100D8×C59C22⋊C412yes—
101C2×C8×C59C22×C411yescharacteristic
102C2×C8×C59C2×D411yescharacteristic
103C2×C8×C59C2×D411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59#4111yescharacteristic
106C2×C8×C59#4111yescharacteristic
107C22⋊Dic59—21no—
108C22⋊Dic59—41no—
109C4⋊Dic59—21no—
110C4⋊Dic59—41no—
111C4×Dic59—21no—
112C4×Dic59—41no—
113#11745664—21no—
114#11745664—22no—
115#11745664—44no—
116#11745664—41no—
117#11745664D812yes—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—44no—
120C2×C59⋊C8—41no—
121C16×C59—22no—
122C59⋊C16—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59AGL(1,ℤ/8)11yescharacteristic
126Q8×C59—22no—
127Q8×C59—41no—
128Q8×C59—41no—
129Q8×C59—41no—
130D4×C59—22no—
131D4×C59—42no—
132D4×C59—42no—
133D4×C59—41no—
134D4×C59—41no—
135D4×C59—41no—
136D4×C59—81no—
137D4×C59—81no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59C2×D811yescharacteristic
148C2×C4×C59#13511yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×Dic59—41no—
151C2×Dic59—81no—
152C8×C59C2×C22⋊C411yescharacteristic
153C8×C59C4×D411yescharacteristic
154C8×C59#12111yescharacteristic
155C8×C59#12311yescharacteristic
156C59⋊C8—22no—
157C59⋊C8—42no—
158#13021—591nomaximal, Sylow
159C22×C59—41no—
160C22×C59—42no—
161C22×C59—81no—
162C22×C59—161no—
163C22×C59—161no—
164C22×C59#44211yescharacteristic
165C22×C59#44911yescharacteristic
166C22×C59#45911yescharacteristic
167C4×C59—42no—
168C4×C59—81no—
169C4×C59#38611yescharacteristic
170C4×C59C4×D811yescharacteristic
171C4×C59#44711yescharacteristic
172C4×C59#46311yescharacteristic
173Dic59—81no—
174Dic59—161no—
175#3414D5911yescharacteristic
176#2957—591no—
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—161no—
187C2×C59#187911yescharacteristic
188C2×#135C2×D5911yescharacteristic
189C2×#135C2×D5911yescharacteristic
190#568—1181no—
191#510—21no—
192#510Dic5912yescomplemented
193#509—21no—
194#509Dic5912yescomplemented
195#508—22no—
196C2×M32C2×D5911yescharacteristic
197C22×C16—591no—
198#444—1182no—
199#443—1182no—
200C2×#108—591no—
201#427—591no—
202#421—1181no—
203#421—1181no—
204#410—1181no—
205#405—591no—
206#359—1182no—
207#358—1181no—
208#357—1181no—
209C59#1302111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135C2×Dic5912yes—
219#135C22⋊D5912yes—
220C2×Q16C2×Dic5911yescharacteristic
221C2×Q16C22⋊D5911yescharacteristic
222C2×SD16—21no—
223C2×SD16—21no—
224C2×D8C2×Dic5911yescharacteristic
225C2×D8C22⋊D5911yescharacteristic
226#131—1184no—
227#131—2362no—
228C2×M16—591no—
229C2×M16—1181no—
230C2×M16—2362no—
231C22×C8C22×D5911yescharacteristic
232#117—591no—
233#117—1181no—
234Q32—22no—
235SD32—22no—
236SD32—22no—
237D16—22no—
238M32—21no—
239M32C2×Dic5912yes—
240C2×C16—21no—
241C2×C16—1181no—
242C2×C16—1181no—
243C2×C16—1181no—
244#108—592no—
245#108—1181no—
246#107—1181no—
247#106—1181no—
248#104—1184no—
249#104—2362no—
250#99—2362no—
251#97—1181no—
252C4×C8—1181no—
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4D4×D5911yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—21no—
273Q16C22⋊Dic5912yes—
274SD16—22no—
275SD16—22no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8C22⋊Dic5912yes—
282M16—592no—
283M16—1181no—
284M16—1182no—
285M16—2364no—
286M16—2361no—
287C2×C8—1181no—
288C2×C8—2364no—
289C2×C8—2361no—
290C2×C8D4×D5911yescharacteristic
291C2×C8#1174568711yescharacteristic
292C2×C8#1174568711yescharacteristic
293C2×C8C22×Dic5911yescharacteristic
294C2×C8C2×C22⋊D5911yescharacteristic
295C2×C8C2×C22⋊D5911yescharacteristic
296C4⋊C4—1181no—
297C4⋊C4—2361no—
298C22⋊C4—1181no—
299C22⋊C4—2361no—
300C42—1181no—
301C42—2361no—
302C16—22no—
303C16—2361no—
304C23—41no—
305C23—41no—
306C23#1427375111yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—4721no—
326C2×C4—2361no—
327C2×C4D4×Dic5911yescharacteristic
328C2×C4#1427369511yescharacteristic
329C2×C4#1427369911yescharacteristic
330C2×C4D8×D5911yescharacteristic
331C2×C4#1427374611yescharacteristic
332C2×C4#1427375111yescharacteristic
333C8—1182no—
334C8—2362no—
335C8C2×C22⋊Dic5911yescharacteristic
336C8D4×Dic5911yescharacteristic
337C8#1427369911yescharacteristic
338C8#1427370811yescharacteristic
339C22—41no—
340C22—42no—
341C22—81no—
342C22—161no—
343C22—161no—
344C22#1683751411yescharacteristic
345C22#1683754411yescharacteristic
346C22#1683759511yescharacteristic
347C4—42no—
348C4—81no—
349C4—9441no—
350C4—4721no—
351C4#1683728211yescharacteristic
352C4D8×Dic5911yescharacteristic
353C4#1683753411yescharacteristic
354C4#1683761511yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—161no—
360C2#2142444611yescharacteristic
subgroupG/H
centerC4#16837534
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3414C2
socleC2×C59#1879
cosocleC22×C8×C59C23
hypercenter#3414D59
hypocenterC59#13021
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #21433013C2semidirect
2. C2×#16837209C2semidirect
3. #21425803C2semidirect
4. #21424995C2semidirect
5. #21422211C2semidirect
6. #21422186C2semidirect
24. C59×#510C4semidirect
26. C59×#509C4semidirect
192. #510Dic59semidirect
194. #509Dic59semidirect
209. C59#13021semidirect

Maximal subgroups

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

subgroupindex
#214330132
C2×#168372092
#214258032
#214249952
#214222112
#214221862
C59×#34142
#1302159