Finite Group Databasebeta

← #28506112

Subgroups — #28506112

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437199C211yesmaximal, characteristic, complemented
2C2×#16837212C211yesmaximal, characteristic, complemented
3#21425824C211yesmaximal, characteristic, complemented
4#21424155C211yesmaximal, characteristic, complemented
5#21422207C211yesmaximal, characteristic, complemented
6#21422196C211yesmaximal, characteristic, complemented
7C59×#3067C211yesmaximal, characteristic
8#16837819—22no—
9#16837819—22no—
10C2×#14273667—22no—
11C2×#14273667C2211yescharacteristic
12C2×#14273667C2211yescharacteristic
13C2×#14273663C2211yescharacteristic
14C2×#14273659C2211yescharacteristic
15#16837403—21no—
16#16837212—22no—
17#16837209—22no—
18#16837170—21no—
19C59×#575—21no—
20C2×#135×C59C2211yescharacteristic
21C2×#131×C59C2211yescharacteristic
22C59×#473C412yescomplemented
23C59×#472C412yescomplemented
24C59×#471—22no—
25C2×#108×C59C2211yescharacteristic
26C59×#413—21no—
27C2×#11745664—21no—
28C2×C4⋊Dic59—21no—
29#14273667—24no—
30#14273667—21no—
31#14273667—21no—
32#14273667—21no—
33#14273667D412yes—
34C2×C59⋊C16—24no—
35C2×C59⋊C16—21no—
36C2×C59⋊C16D412yes—
37#14273663—21no—
38#14273663—21no—
39#14273659—21no—
40#14273659—21no—
41#14273651—41no—
42#14273648—41no—
43C2×#41×C59—21no—
44C2×#41×C59D411yescharacteristic
45#137×C59—22no—
46#137×C59—41no—
47AGL(1,ℤ/8)×C59—22no—
48AGL(1,ℤ/8)×C59—41no—
49#135×C59—21no—
50#135×C59—21no—
51#135×C59—21no—
52#135×C59—21no—
53#135×C59—42no—
54C2×Q16×C59C2×C411yescharacteristic
55C2×SD16×C59—21no—
56C2×D8×C59C2×C411yescharacteristic
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#101×C59—22no—
69#100×C59—22no—
70#41×C59—21no—
71#41×C59—21no—
72#41×C59—41no—
73#41×C59—41no—
74#41×C59—41no—
75#41×C59—41no—
76#41×C59C22⋊C412yes—
77#41×C59SD1612yes—
78C2×Q8×C59—21no—
79C2×Q8×C59D811yescharacteristic
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—41no—
83C2×D4×C59D811yescharacteristic
84C22×C4×C59—21no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C22×Dic59—41no—
88Q16×C59—21no—
89Q16×C59—21no—
90Q16×C59—42no—
91SD16×C59—42no—
92SD16×C59—42no—
93SD16×C59—41no—
94SD16×C59—41no—
95D8×C59—21no—
96D8×C59—21no—
97D8×C59—42no—
98M16×C59—21no—
99M16×C59—22no—
100M16×C59—22no—
101M16×C59—22no—
102M16×C59—41no—
103M16×C59C22⋊C412yes—
104C2×C8×C59—21no—
105C2×C8×C59—22no—
106C2×C8×C59—22no—
107C2×C8×C59—41no—
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59#4111yescharacteristic
113C2×C8×C59#4111yescharacteristic
114C4⋊Dic59—41no—
115C4⋊Dic59—41no—
116#11745664—41no—
117#11745664—41no—
118C2×C59⋊C8—41no—
119C59⋊C16—22no—
120C59⋊C16—22no—
121C23×C59—21no—
122C23×C59—41no—
123C23×C59#13511yescharacteristic
124Q8×C59—21no—
125Q8×C59—41no—
126Q8×C59—41no—
127Q8×C59#10212yes—
128D4×C59—22no—
129D4×C59—22no—
130D4×C59—21no—
131D4×C59—41no—
132D4×C59—41no—
133D4×C59—81no—
134D4×C59—81no—
135D4×C59#10212yes—
136C2×C4×C59—21no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—41no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59C2×C22⋊C411yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×Dic59—81no—
149C2×Dic59—161no—
150C8×C59—22no—
151C8×C59—42no—
152C8×C59C4×D411yescharacteristic
153C8×C59C4×D411yescharacteristic
154C8×C59#12011yescharacteristic
155C8×C59#12311yescharacteristic
156C59⋊C8—81no—
157#13070—591nomaximal, Sylow
158C22×C59—22no—
159C22×C59—41no—
160C22×C59—81no—
161C22×C59—81no—
162C22×C59—161no—
163C22×C59#41611yescharacteristic
164C22×C59#46511yescharacteristic
165C22×C59#46611yescharacteristic
166C4×C59—22no—
167C4×C59—81no—
168C4×C59#38611yescharacteristic
169C4×C59C2×#10211yescharacteristic
170C4×C59#46511yescharacteristic
171C4×C59#46611yescharacteristic
172Dic59—161no—
173#3407—591no—
174#3067D5911yescharacteristic
175C2×#360—591no—
176#2143—591no—
177#1815—591no—
178#1388—591no—
179#1383—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—42no—
184C2×C59—161no—
185C2×C59#189311yescharacteristic
186#575—21no—
187C2×#135C2×D5911yescharacteristic
188C2×#131C2×D5911yescharacteristic
189#504—1182no—
190#504—1182no—
191C2×M32—591no—
192C2×M32—591no—
193C2×M32—1182no—
194#473Dic5912yescomplemented
195#472Dic5912yescomplemented
196#471—22no—
197C2×#108—591no—
198C2×#108C2×D5911yescharacteristic
199C2×#106—591no—
200#417—1181no—
201#413—21no—
202#360—1182no—
203#359—1182no—
204#343—1181no—
205C59#1307011yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C2×#41C22⋊D5911yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic5911yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic5911yescharacteristic
220#131—21no—
221#131—21no—
222#131C2×Dic5912yes—
223#131C22⋊D5912yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—1181no—
227C2×M16C22⋊D5911yescharacteristic
228C22×C8—21no—
229C22×C8C22×D5911yescharacteristic
230C2×C4⋊C4—1181no—
231M32—592no—
232M32—1184no—
233M32—1181no—
234M32—1181no—
235M32—1181no—
236C2×C16—592no—
237C2×C16—1184no—
238C2×C16—1181no—
239#108—21no—
240#108—1181no—
241#108—1181no—
242#108C2×Dic5912yes—
243#106—1181no—
244#106—1181no—
245#103—2361no—
246#102—2361no—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41#1174567012yes—
256#41C22⋊Dic5912yes—
257C2×Q8—21no—
258C2×Q8#1174566911yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1174566911yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—2361no—
266C22×C4C2×C22⋊D5911yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—42no—
270SD16—42no—
271SD16—42no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—41no—
282M16—2361no—
283M16—2361no—
284M16C22⋊Dic5912yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—2361no—
290C2×C8D4×D5911yescharacteristic
291C2×C8D4×D5911yescharacteristic
292C2×C8#1174568711yescharacteristic
293C2×C8#1174568711yescharacteristic
294C2×C8C22×Dic5911yescharacteristic
295C2×C8C2×C22⋊D5911yescharacteristic
296C4⋊C4—2361no—
297C4⋊C4—2361no—
298C16—1182no—
299C16—1182no—
300C23—21no—
301C23—41no—
302C23#1427374711yescharacteristic
303Q8—21no—
304Q8—41no—
305Q8—41no—
306Q8#1427364812yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4#1427364812yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—9441no—
322C2×C4—4721no—
323C2×C4C2×C22⋊Dic5911yescharacteristic
324C2×C4#1427369911yescharacteristic
325C2×C4#1427369911yescharacteristic
326C2×C4C2×#1174566911yescharacteristic
327C2×C4C2×#1174567011yescharacteristic
328C2×C4#1427374711yescharacteristic
329C8—22no—
330C8—42no—
331C8—4721no—
332C8D4×Dic5911yescharacteristic
333C8D4×Dic5911yescharacteristic
334C8#1427369511yescharacteristic
335C8#1427370811yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22#1683739811yescharacteristic
342C22#1683762911yescharacteristic
343C22#1683763611yescharacteristic
344C4—22no—
345C4—81no—
346C4—9441no—
347C4#1683728211yescharacteristic
348C4C2×#1427364811yescharacteristic
349C4#1683762911yescharacteristic
350C4#1683763611yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2142452511yescharacteristic
subgroupG/H
centerC2#21424525
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3067C2
socleC2×C59#1893
cosocleC22×C8×C59C23
hypercenter#3067D59
hypocenterC59#13070
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #21437199C2semidirect
2. C2×#16837212C2semidirect
3. #21425824C2semidirect
4. #21424155C2semidirect
5. #21422207C2semidirect
6. #21422196C2semidirect
22. C59×#473C4semidirect
23. C59×#472C4semidirect
194. #473Dic59semidirect
195. #472Dic59semidirect
205. C59#13070semidirect

Maximal subgroups

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

subgroupindex
#214371992
C2×#168372122
#214258242
#214241552
#214222072
#214221962
C59×#30672
#1307059