Finite Group Databasebeta

← #28506133

Subgroups — #28506133

Order 15104.

478 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
1C2×#16837819C211yesmaximal, characteristic, complemented
2C2×#16837215C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4#21424160C211yesmaximal, characteristic, complemented
5#21422206C211yesmaximal, characteristic, complemented
6#21422189C211yesmaximal, characteristic, complemented
7C2×C59×#472C211yesmaximal, characteristic
8#16837819—22no—
9#16837819—22no—
10C2×#14273667—21no—
11C2×#14273667C2211yescharacteristic
12C22×C59⋊C16—21no—
13C22×C59⋊C16C2211yescharacteristic
14C2×#14273663C2211yescharacteristic
15C2×#14273661C2211yescharacteristic
16#16837403—21no—
17#16837215—22no—
18#16837203—21no—
19#16837203—21no—
20#16837170—21no—
21C2×AGL(1,ℤ/8)×C59—21no—
22C22×D8×C59C2211yescharacteristic
23C2×#131×C59C2211yescharacteristic
24C59×#472—22no—
25C59×#472C414yescomplemented
26C2×#108×C59C2211yescharacteristic
27C2×#100×C59—21no—
28C2×#11745664—21no—
29C2×C4⋊Dic59—21no—
30#14273667—24no—
31#14273667—21no—
32#14273667D412yes—
33C2×C59⋊C16—24no—
34C2×C59⋊C16—21no—
35C2×C59⋊C16—21no—
36C2×C59⋊C16—21no—
37C2×C59⋊C16D411yescharacteristic
38C2×C59⋊C16D411yescharacteristic
39#14273663—21no—
40#14273663—21no—
41#14273661—21no—
42#14273661—21no—
43#14273651—41no—
44#14273648—41no—
45C2×#41×C59D411yescharacteristic
46C22×D4×C59—21no—
47AGL(1,ℤ/8)×C59—24no—
48AGL(1,ℤ/8)×C59—42no—
49C2×SD16×C59—41no—
50C2×D8×C59—21no—
51C2×D8×C59—21no—
52C2×D8×C59—21no—
53C2×D8×C59—21no—
54C2×D8×C59—21no—
55C2×D8×C59—41no—
56C2×D8×C59C2×C412yes—
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#108×C59—21no—
67#108×C59C2×C412yes—
68#100×C59—24no—
69C24×C59—41no—
70#41×C59—21no—
71#41×C59—21no—
72#41×C59C22⋊C412yes—
73#41×C59SD1612yes—
74C2×Q8×C59D811yescharacteristic
75C2×D4×C59—21no—
76C2×D4×C59—22no—
77C2×D4×C59—41no—
78C2×D4×C59—41no—
79C2×D4×C59—41no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59D811yescharacteristic
83C22×C4×C59—21no—
84C22×C4×C59C2×D411yescharacteristic
85C22×Dic59—41no—
86SD16×C59—44no—
87D8×C59—22no—
88D8×C59—22no—
89D8×C59—44no—
90D8×C59—41no—
91D8×C59—41no—
92M16×C59—21no—
93M16×C59—22no—
94M16×C59—22no—
95M16×C59—22no—
96M16×C59—41no—
97M16×C59C22⋊C412yes—
98C2×C8×C59—21no—
99C2×C8×C59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—41no—
102C2×C8×C59C22×C411yescharacteristic
103C2×C8×C59C2×D411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59#4111yescharacteristic
107C2×C8×C59#4111yescharacteristic
108C4⋊Dic59—41no—
109C4⋊Dic59—41no—
110#11745664—41no—
111#11745664—41no—
112C2×C59⋊C8—41no—
113C59⋊C16—21no—
114C59⋊C16—22no—
115C59⋊C16—21no—
116C23×C59—21no—
117C23×C59—42no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59—161no—
121C23×C59—161no—
122C23×C59#13511yescharacteristic
123Q8×C59—21no—
124Q8×C59#10212yes—
125D4×C59—22no—
126D4×C59—22no—
127D4×C59—21no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—81no—
131D4×C59—81no—
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
142C2×Dic59—81no—
143C2×Dic59—161no—
144C8×C59—22no—
145C8×C59—42no—
146C8×C59C4×D411yescharacteristic
147C8×C59C4×D411yescharacteristic
148C8×C59#12011yescharacteristic
149C8×C59#12311yescharacteristic
150C59⋊C8—81no—
151#13073—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—82no—
158C22×C59—162no—
159C22×C59—162no—
160C22×C59—161no—
161C22×C59—161no—
162C22×C59#41611yescharacteristic
163C22×C59#46511yescharacteristic
164C22×C59#46611yescharacteristic
165C4×C59—22no—
166C4×C59#38611yescharacteristic
167C4×C59C2×#10211yescharacteristic
168C4×C59#46511yescharacteristic
169C4×C59#46611yescharacteristic
170Dic59—161no—
171C2×#504—591no—
172C2×#472D5911yescharacteristic
173C2×#361—591no—
174C2×#357—591no—
175#1816—591no—
176#1387—591no—
177#1381—591no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59—42no—
181C2×C59—162no—
182C2×C59#189311yescharacteristic
183C2×C59#222111yescharacteristic
184C2×C59#222311yescharacteristic
185C2×AGL(1,ℤ/8)—21no—
186C22×D8C2×D5911yescharacteristic
187C2×#131C2×D5911yescharacteristic
188#504—1182no—
189#504—1182no—
190C2×M32—591no—
191C2×M32—1181no—
192C22×C16—591no—
193C22×C16—1181no—
194#472—22no—
195#472Dic5914yescomplemented
196C2×#108—591no—
197C2×#108C2×D5911yescharacteristic
198C2×#107—591no—
199#417—1181no—
200C2×#100—21no—
201#361—1182no—
202#357—1181no—
203#357—1181no—
204#343—1181no—
205C59#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D5911yescharacteristic
207C22×D4—21no—
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—41no—
217C2×D8C2×Dic5912yes—
218#131—21no—
219#131—21no—
220#131C2×Dic5912yes—
221#131C22⋊D5912yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1181no—
225C2×M16C22⋊D5911yescharacteristic
226C22×C8—21no—
227C22×C8C22×D5911yescharacteristic
228C2×C4⋊C4—1181no—
229M32—592no—
230M32—1184no—
231M32—1181no—
232C2×C16—591no—
233C2×C16—591no—
234C2×C16—1184no—
235C2×C16—1181no—
236C2×C16—1181no—
237C2×C16—1181no—
238#108—21no—
239#108—1181no—
240#108—1181no—
241#108C2×Dic5912yes—
242#107—1181no—
243#107—1181no—
244#103—2361no—
245#102—2361no—
246#100—24no—
247C24—41no—
248#41—21no—
249#41—21no—
250#41#1174567012yes—
251#41C22⋊Dic5912yes—
252C2×Q8#1174566911yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4#1174566911yescharacteristic
261C22×C4—21no—
262C22×C4—2361no—
263C22×C4C2×C22⋊D5911yescharacteristic
264SD16—44no—
265D8—22no—
266D8—22no—
267D8—44no—
268D8—41no—
269D8—41no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—2361no—
276M16—2361no—
277M16C22⋊Dic5912yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—2361no—
283C2×C8D4×D5911yescharacteristic
284C2×C8D4×D5911yescharacteristic
285C2×C8#1174568711yescharacteristic
286C2×C8#1174568711yescharacteristic
287C2×C8C22×Dic5911yescharacteristic
288C2×C8C2×C22⋊D5911yescharacteristic
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C16—1181no—
292C16—1181no—
293C16—1182no—
294C23—21no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—161no—
299C23—161no—
300C23#1427374711yescharacteristic
301Q8—21no—
302Q8#1427364812yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4#1427364812yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—9441no—
315C2×C4—4721no—
316C2×C4C2×C22⋊Dic5911yescharacteristic
317C2×C4#1427369911yescharacteristic
318C2×C4#1427369911yescharacteristic
319C2×C4C2×#1174566911yescharacteristic
320C2×C4C2×#1174567011yescharacteristic
321C2×C4#1427374711yescharacteristic
322C8—22no—
323C8—42no—
324C8—4721no—
325C8D4×Dic5911yescharacteristic
326C8D4×Dic5911yescharacteristic
327C8#1427369511yescharacteristic
328C8#1427370811yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1683739811yescharacteristic
340C22#1683762911yescharacteristic
341C22#1683763611yescharacteristic
342C4—22no—
343C4—9441no—
344C4#1683728211yescharacteristic
345C4C2×#1427364811yescharacteristic
346C4#1683762911yescharacteristic
347C4#1683763611yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#2142452511yescharacteristic
353C2#2142621411yescharacteristic
354C2#2142622411yescharacteristic
subgroupG/H
centerC22#16837636
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC2×C59×#472C2
socleC22×C59#466
cosocleC22×C8×C59C23
hypercenterC2×#472D59
hypocenterC59#13073
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C2×#16837819C2semidirect
2. C2×#16837215C2semidirect
3. C2×#16837203C2semidirect
4. #21424160C2semidirect
5. #21422206C2semidirect
6. #21422189C2semidirect
25. C59×#472C4semidirect
195. #472Dic59semidirect
205. C59#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#168378192
C2×#168372152
C2×#168372032
#214241602
#214222062
#214221892
C2×C59×#4722
#1307359