Finite Group Databasebeta

← #28506131

Subgroups — #28506131

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×#16837672C211yesmaximal, characteristic
2C2×#16837216C211yesmaximal, characteristic, complemented
3C2×#16837204C211yesmaximal, characteristic, complemented
4#21424160C211yesmaximal, characteristic, complemented
5#21422205C211yesmaximal, characteristic, complemented
6#21422190C211yesmaximal, characteristic, complemented
7C2×C59×#504C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C22×D472C2211yescharacteristic
10#16837672—22no—
11#16837672C414yescomplemented
12C2×#14273663C2211yescharacteristic
13C2×#14273663C2211yescharacteristic
14C2×#14273661C2211yescharacteristic
15#16837403—21no—
16C2×#14273643—21no—
17#16837216—22no—
18#16837204—21no—
19#16837204—21no—
20#16837170—21no—
21C2×#131×C59C2211yescharacteristic
22C59×#504—22no—
23C59×#504—22no—
24C2×M32×C59—21no—
25C2×M32×C59C2211yescharacteristic
26C22×C16×C59—21no—
27C22×C16×C59C2211yescharacteristic
28C22×D236—21no—
29#14273752—24no—
30#14273752—42no—
31C2×#11745671—41no—
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472C2×C412yes—
38C2×#11745669—41no—
39C2×#11745664—21no—
40C2×#11745664—21no—
41C2×C4⋊Dic59—21no—
42#14273663—21no—
43#14273663—21no—
44#14273663—21no—
45#14273663C2×C412yes—
46#14273661—21no—
47#14273661—21no—
48#14273651—41no—
49#14273648—41no—
50#14273643—24no—
51C2×#41×C59D411yescharacteristic
52#131×C59—21no—
53#131×C59—21no—
54#131×C59C2×C412yes—
55#131×C59D412yes—
56C2×M16×C59—21no—
57C2×M16×C59D411yescharacteristic
58C22×C8×C59—21no—
59C22×C8×C59C2311yescharacteristic
60M32×C59—24no—
61M32×C59—21no—
62M32×C59D412yes—
63C2×C16×C59—24no—
64C2×C16×C59—21no—
65C2×C16×C59—21no—
66C2×C16×C59—21no—
67C2×C16×C59D411yescharacteristic
68C2×C16×C59D411yescharacteristic
69C23×D59—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×C59D811yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—41no—
80C2×D236—22no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—41no—
86M16×C59—21no—
87M16×C59—22no—
88M16×C59—22no—
89M16×C59C22⋊C412yes—
90C2×C8×C59—21no—
91C2×C8×C59—22no—
92C2×C8×C59—22no—
93C2×C8×C59C22×C411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C2×C8×C59#4111yescharacteristic
99#11745671—44no—
100#11745669—44no—
101C4⋊Dic59—41no—
102C4⋊Dic59—41no—
103#11745664—22no—
104#11745664—41no—
105#11745664—41no—
106#11745664—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109D472—22no—
110D472—22no—
111D472—41no—
112D472—41no—
113C16×C59—21no—
114C16×C59—22no—
115C16×C59—21no—
116C23×C59—21no—
117C23×C59#13511yescharacteristic
118C22×D59—42no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—161no—
122C22×D59—161no—
123Q8×C59—21no—
124Q8×C59#10212yes—
125D4×C59—22no—
126D4×C59—22no—
127D4×C59—21no—
128D4×C59#10212yes—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59C2×C22⋊C411yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59C2×D811yescharacteristic
136C2×C4×C59C2×SD1611yescharacteristic
137C2×C4×C59#13511yescharacteristic
138C2×Dic59—81no—
139C2×Dic59—161no—
140D236—42no—
141D236—42no—
142D236—81no—
143D236—81no—
144C8×C59—22no—
145C8×C59C4×D411yescharacteristic
146C8×C59C4×D411yescharacteristic
147C8×C59#12011yescharacteristic
148C8×C59#12311yescharacteristic
149C59⋊C8—42no—
150C59⋊C8—81no—
151#13073—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59#41611yescharacteristic
158C22×C59#46511yescharacteristic
159C22×C59#46611yescharacteristic
160C2×D59—82no—
161C2×D59—162no—
162C2×D59—161no—
163C2×D59—161no—
164C2×D59—162no—
165C4×C59—22no—
166C4×C59#38611yescharacteristic
167C4×C59C2×#10211yescharacteristic
168C4×C59#46511yescharacteristic
169C4×C59#46611yescharacteristic
170Dic59—161no—
171C2×#504D5911yescharacteristic, complemented
172C2×#472—591no—
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#189311yescharacteristic
182C2×C59#222111yescharacteristic
183C2×C59#222311yescharacteristic
184D59—162no—
185C2×AGL(1,ℤ/8)—1181no—
186C22×D8—591no—
187C2×#131C2×D5911yescharacteristic
188#504—22no—
189#504—22no—
190C2×M32—21no—
191C2×M32C2×D5911yescharacteristic
192C22×C16—21no—
193C22×C16C2×D5911yescharacteristic
194#472—594no—
195#472—1182no—
196C2×#108—591no—
197C2×#108—591no—
198C2×#107—591no—
199#417—1181no—
200C2×#100—1181no—
201#361—1182no—
202#357—1181no—
203#357—1181no—
204#343—1181no—
205C59#1307311yescharacteristic, complemented, Sylow
206C2×#41D23611yescharacteristic
207C22×D4—1181no—
208AGL(1,ℤ/8)—1184no—
209AGL(1,ℤ/8)—2362no—
210C2×SD16—2361no—
211C2×D8—592no—
212C2×D8—1181no—
213C2×D8—1181no—
214C2×D8—1181no—
215C2×D8—1181no—
216C2×D8—1181no—
217C2×D8—2361no—
218#131—21no—
219#131—21no—
220#131C4×D5912yes—
221#131C22⋊D5912yes—
222C2×M16—21no—
223C2×M16—1181no—
224C2×M16—1181no—
225C2×M16D23611yescharacteristic
226C22×C8—21no—
227C22×C8C22×D5911yescharacteristic
228C2×C4⋊C4—1181no—
229M32—24no—
230M32—21no—
231M32C22⋊D5912yes—
232C2×C16—24no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16C22⋊D5911yescharacteristic
237C2×C16C22⋊D5911yescharacteristic
238#108—592no—
239#108—1181no—
240#108—1181no—
241#108—1181no—
242#107—1181no—
243#107—1181no—
244#103—2361no—
245#102—2361no—
246#100—1184no—
247C24—2361no—
248#41—21no—
249#41—21no—
250#41#1174566012yes—
251#41(C2×D59)⋊C412yes—
252C2×Q8D47211yescharacteristic
253C2×D4—21no—
254C2×D4—1182no—
255C2×D4—2361no—
256C2×D4—2361no—
257C2×D4—2361no—
258C2×D4—2361no—
259C2×D4—2361no—
260C2×D4D47211yescharacteristic
261C22×C4—21no—
262C22×C4—2361no—
263C22×C4C2×D23611yescharacteristic
264SD16—2364no—
265D8—1182no—
266D8—1182no—
267D8—2364no—
268D8—2361no—
269D8—2361no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—1182no—
274M16—2361no—
275M16—2361no—
276M16—2361no—
277M16(C2×D59)⋊C412yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—2361no—
282C2×C8—2361no—
283C2×C8C2×C4×D5911yescharacteristic
284C2×C8#1174568511yescharacteristic
285C2×C8#1174568511yescharacteristic
286C2×C8C2×C22⋊D5911yescharacteristic
287C2×C8C2×C22⋊D5911yescharacteristic
288C2×C8C2×C22⋊D5911yescharacteristic
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C16—21no—
292C16—22no—
293C16—21no—
294C23—21no—
295C23—9441no—
296C23—9441no—
297C23—4721no—
298C23—4721no—
299C23—2362no—
300C23#1427374911yescharacteristic
301Q8—21no—
302Q8#1427364912yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—4721no—
307D4—4721no—
308D4—2362no—
309D4—2362no—
310D4#1427364912yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—9441no—
315C2×C4—4721no—
316C2×C4C2×(C2×D59)⋊C411yescharacteristic
317C2×C4#1427370111yescharacteristic
318C2×C4#1427370111yescharacteristic
319C2×C4C2×D47211yescharacteristic
320C2×C4C2×#1174566011yescharacteristic
321C2×C4#1427374911yescharacteristic
322C8—22no—
323C8—4721no—
324C8—2362no—
325C8C4×C22⋊D5911yescharacteristic
326C8C4×C22⋊D5911yescharacteristic
327C8#1427369611yescharacteristic
328C8#1427371011yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—9442no—
335C22—9442no—
336C22—9441no—
337C22—9441no—
338C22—4722no—
339C22#1683739911yescharacteristic
340C22#1683763111yescharacteristic
341C22#1683763811yescharacteristic
342C4—22no—
343C4—9441no—
344C4#1683728111yescharacteristic
345C4C2×#1427364911yescharacteristic
346C4#1683763111yescharacteristic
347C4#1683763811yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—9442no—
352C2#2142452411yescharacteristic
353C2#2142621611yescharacteristic
354C2#2142622611yescharacteristic
subgroupG/H
centerC22#16837638
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#504C2
socleC22×C59#466
prime radicalC2×#16837672C2
cosocleC22×C8×C59C23
hypercenterC2×#504D59
hypocenterC59#13073
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. C2×#16837216C2semidirect
3. C2×#16837204C2semidirect
4. #21424160C2semidirect
5. #21422205C2semidirect
6. #21422190C2semidirect
7. C2×C59×#504C2semidirect
11. #16837672C4semidirect
171. C2×#504D59semidirect
205. C59#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#168376722
C2×#168372162
C2×#168372042
#214241602
#214222052
#214221902
C2×C59×#5042
#1307359