Finite Group Databasebeta

← #28484156

Subgroups — #28484156

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273752C211yesmaximal, characteristic
2#21422483C211yesmaximal, characteristic, complemented
3C2×#16837131C211yesmaximal, characteristic, complemented
4#21422416C211yesmaximal, characteristic, complemented
5#21421945C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C8×#41×C59C211yesmaximal, characteristic, complemented
8C2×#14273752C411yescharacteristic
9C2×C4×D236C2211yescharacteristic
10#16837512—21no—
11#16837500—21no—
12C4×#11745671—21no—
13C4×#11745669—21no—
14#16837455C2211yescharacteristic
15#16837445C411yescharacteristic
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17#16837406C411yescharacteristic
18#16837403C411yescharacteristic
19C4×#11745664C2211yescharacteristic
20Q8⋊(C59⋊C8)—21no—
21#16837131—21no—
22#16837131—21no—
23D4⋊(C59⋊C8)—21no—
24#16837125—22no—
25C4×#41×C59C2211yescharacteristic
26C8×Q8×C59—21no—
27C8×D4×C59—21no—
28C8×D4×C59—22no—
29C59×#431—21no—
30C59×#431C2211yescharacteristic
31C2×C4×C8×C59—21no—
32C2×C4×C8×C59C2211yescharacteristic
33C22×D236C2×C411yescharacteristic
34C22×C4×D59—21no—
35#14273752—22no—
36#14273752C814yescomplemented
37C2×#11745671—21no—
38C2×#11745669—21no—
39C2×#11745664C2×C411yescharacteristic
40C2×#11745664C2×C411yescharacteristic
41C22×C59⋊C8—21no—
42C4×D236—21no—
43C4×D236—21no—
44C4×D236—21no—
45C4×D236—21no—
46C2×C4⋊Dic59C2×C411yescharacteristic
47C2×(C2×D59)⋊C4—21no—
48#14273660—21no—
49#14273658—21no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)—21no—
53#14273651—21no—
54#14273648—21no—
55#14273647—21no—
56#14273647—21no—
57C22⋊(C59⋊C8)—21no—
58#14273638—21no—
59C4×C59⋊C8—21no—
60C4×C59⋊C8—21no—
61C2×#41×C59C2×C411yescharacteristic
62C22×C8×C59—22no—
63C22×C8×C59—21no—
64C4×Q8×C59D411yescharacteristic
65C4×Q8×C59D411yescharacteristic
66C4×D4×C59—21no—
67C4×D4×C59—21no—
68C4×D4×C59D411yescharacteristic
69C4×D4×C59D411yescharacteristic
70#117×C59—21no—
71#117×C59C2×C411yescharacteristic
72C2×C42×C59—21no—
73C2×C42×C59C2311yescharacteristic
74C4⋊C8×C59—22no—
75C4⋊C8×C59—21no—
76C22⋊C8×C59—22no—
77C22⋊C8×C59D412yes—
78C4×C8×C59—22no—
79C4×C8×C59—21no—
80C4×C8×C59—21no—
81C4×C8×C59D411yescharacteristic
82C4×C8×C59D411yescharacteristic
83C23×D59—21no—
84#41×C59—21no—
85#41×C59—21no—
86#41×C59C2×C812yes—
87#41×C59M1612yes—
88C2×Q8×C59C22⋊C411yescharacteristic
89C2×D4×C59—21no—
90C2×D4×C59C22⋊C411yescharacteristic
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C22×Dic59—21no—
98C2×D236—21no—
99C2×D236—21no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×D236C2×C812yescomplemented
104C2×C4×D59—41no—
105C2×C4×D59—41no—
106C2×C4×D59—81no—
107C2×C4×D59—81no—
108C2×C8×C59—21no—
109C2×C8×C59—22no—
110C2×C8×C59—21no—
111C2×C8×C59—42no—
112C2×C8×C59—44no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59C22⋊C411yescharacteristic
116C4⋊C4×C59C22⋊C411yescharacteristic
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59C2×D411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59#4111yescharacteristic
127#11745671—24no—
128#11745669—24no—
129(C2×D59)⋊C4—41no—
130(C2×D59)⋊C4—41no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133#11745664—21no—
134#11745664—21no—
135#11745664—21no—
136#11745664C2×C812yescomplemented
137C2×C59⋊C8—21no—
138C2×C59⋊C8—21no—
139C2×C59⋊C8—41no—
140C2×C59⋊C8—41no—
141C2×C59⋊C8—81no—
142C23×C59—21no—
143C23×C59C4×D411yescharacteristic
144C22×D59—22no—
145C22×D59—41no—
146C22×D59—41no—
147C22×D59—81no—
148C22×D59—81no—
149Q8×C59—21no—
150Q8×C59C22⋊C812yes—
151D4×C59—22no—
152D4×C59—22no—
153D4×C59—21no—
154D4×C59C22⋊C812yes—
155C2×C4×C59—21no—
156C2×C4×C59—22no—
157C2×C4×C59—22no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59C2×C22⋊C411yescharacteristic
167C2×C4×C59C4×D411yescharacteristic
168C2×C4×C59#12011yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12311yescharacteristic
172C2×C4×C59C22×C811yescharacteristic
173C2×C4×C59C2×M1611yescharacteristic
174C2×C4×C59#13111yescharacteristic
175C2×C4×C59#13111yescharacteristic
176C2×Dic59—41no—
177C2×Dic59—81no—
178D236—22no—
179D236—22no—
180D236—41no—
181D236—41no—
182C4×D59—82no—
183C4×D59—81no—
184C4×D59—81no—
185C8×C59—22no—
186C8×C59—42no—
187C8×C59—41no—
188C59⋊C8—22no—
189C59⋊C8—41no—
190C59⋊C8—81no—
191C59⋊C8—81no—
192#9095—591nomaximal, Sylow
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—22no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59#38611yescharacteristic
199C22×C59C8×D411yescharacteristic
200C22×C59#43511yescharacteristic
201C2×D59—42no—
202C2×D59—82no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—82no—
206C4×C59—22no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59C2×C22⋊C811yescharacteristic
213C4×C59#40811yescharacteristic
214C4×C59C8×D411yescharacteristic
215C4×C59#43511yescharacteristic
216C4×C59#47111yescharacteristic
217C4×C59#47211yescharacteristic
218Dic59—81no—
219C8×#41D5911yescharacteristic, complemented
220C4×AGL(1,ℤ/8)—591no—
221#1477—591no—
222C2×D4⋊C8—591no—
223#1460—591no—
224#1283—591no—
225#1278—591no—
226C2×C59—21no—
227C2×C59—21no—
228C2×C59—42no—
229C2×C59#186911yescharacteristic
230C2×C59#190611yescharacteristic
231C2×C59#190711yescharacteristic
232D59—82no—
233C2×AGL(1,ℤ/8)—591no—
234C4×#41C2×D5911yescharacteristic
235C2×C4×D4—591no—
236C8×Q8—21no—
237#442—1181no—
238#440—1181no—
239C4×SD16—1181no—
240C4×D8—1181no—
241C8×D4—21no—
242C8×D4—22no—
243#431—21no—
244#431—591no—
245#431C2×D5911yescharacteristic
246#428—591no—
247C2×C4⋊C8—591no—
248#418—591no—
249#417—591no—
250C4×M16—591no—
251C2×C4×C8—21no—
252C2×C4×C8C2×D5911yescharacteristic
253Q8⋊C8—1181no—
254D4⋊C8—1181no—
255D4⋊C8—1181no—
256D4⋊C8—1181no—
257#323—1182no—
258C59#909511yescharacteristic, complemented, Sylow
259C2×#41C4×D5911yescharacteristic
260C22×D4—591no—
261C23×C4—1181no—
262AGL(1,ℤ/8)—594no—
263AGL(1,ℤ/8)—1182no—
264C2×SD16—1181no—
265C2×D8—1181no—
266C2×M16—591no—
267C2×M16—591no—
268C22×C8—22no—
269C22×C8—21no—
270C22×C8—1181no—
271C4×Q8D23611yescharacteristic
272C4×Q8C22⋊D5911yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—1181no—
276C4×D4—1181no—
277C4×D4—1181no—
278C4×D4—1181no—
279C4×D4D23611yescharacteristic
280C4×D4C22⋊D5911yescharacteristic
281#117—21no—
282#117C4×D5911yescharacteristic
283C2×C4⋊C4—591no—
284C2×C22⋊C4—1181no—
285C2×C42—21no—
286C2×C42C22×D5911yescharacteristic
287#107—1181no—
288#106—1181no—
289C4⋊C8—22no—
290C4⋊C8—21no—
291C4⋊C8—1181no—
292C4⋊C8—1181no—
293C4⋊C8—1181no—
294#103—1181no—
295#102—1181no—
296#102—1181no—
297#102—1181no—
298C22⋊C8—22no—
299C22⋊C8—1181no—
300C22⋊C8C22⋊D5912yes—
301#97—1181no—
302C4×C8—22no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—1181no—
306C4×C8—1181no—
307C4×C8C22⋊D5911yescharacteristic
308C4×C8C22⋊D5911yescharacteristic
309C24—1181no—
310#41—21no—
311#41—21no—
312#41C8×D5912yes—
313#41#1174565912yes—
314C2×Q8(C2×D59)⋊C411yescharacteristic
315C2×D4—21no—
316C2×D4—592no—
317C2×D4—1181no—
318C2×D4—1181no—
319C2×D4—1181no—
320C2×D4—1181no—
321C2×D4—1181no—
322C2×D4(C2×D59)⋊C411yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—1181no—
327C22×C4—4721no—
328C22×C4—4721no—
329C22×C4—2361no—
330C22×C4—2361no—
331C22×C4C2×C4×D5911yescharacteristic
332C22×C4#1174568511yescharacteristic
333C22×C4C2×C22⋊D5911yescharacteristic
334SD16—1184no—
335D8—1184no—
336M16—592no—
337M16—1181no—
338M16—1181no—
339M16—1181no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—42no—
344C2×C8—44no—
345C2×C8—1181no—
346C2×C8—1181no—
347C2×C8—4721no—
348C2×C8—2361no—
349C2×C8—2361no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—1181no—
353C4⋊C4—1181no—
354C4⋊C4(C2×D59)⋊C411yescharacteristic
355C4⋊C4(C2×D59)⋊C411yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—2361no—
360C22⋊C4—2361no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42C2×D23611yescharacteristic
365C42#1174568511yescharacteristic
366C42C2×C22⋊D5911yescharacteristic
367C42C2×C22⋊D5911yescharacteristic
368C23—21no—
369C23—1182no—
370C23—4721no—
371C23—4721no—
372C23—2361no—
373C23—2361no—
374C23C4×C22⋊D5911yescharacteristic
375Q8—21no—
376Q8(C2×D59)⋊C812yes—
377D4—22no—
378D4—22no—
379D4—21no—
380D4—1182no—
381D4—1182no—
382D4—2361no—
383D4—2361no—
384D4(C2×D59)⋊C812yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—4722no—
397C2×C4—4721no—
398C2×C4—4721no—
399C2×C4—4721no—
400C2×C4—2361no—
401C2×C4C2×(C2×D59)⋊C411yescharacteristic
402C2×C4C4×C22⋊D5911yescharacteristic
403C2×C4#1427369611yescharacteristic
404C2×C4#1427370111yescharacteristic
405C2×C4#1427370111yescharacteristic
406C2×C4#1427371011yescharacteristic
407C2×C4C2×C8×D5911yescharacteristic
408C2×C4C2×#1174565911yescharacteristic
409C2×C4#1427373311yescharacteristic
410C2×C4#1427373311yescharacteristic
411C8—22no—
412C8—42no—
413C8—41no—
414C8—1182no—
415C8—4721no—
416C8—4721no—
417C8—2361no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—4722no—
424C22—4722no—
425C22—4721no—
426C22—4721no—
427C22—2362no—
428C22#1683728111yescharacteristic
429C22C8×C22⋊D5911yescharacteristic
430C22#1683747311yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—41no—
436C4—41no—
437C4—4721no—
438C4C2×(C2×D59)⋊C811yescharacteristic
439C4#1683736811yescharacteristic
440C4C8×C22⋊D5911yescharacteristic
441C4#1683747311yescharacteristic
442C4#1683766511yescharacteristic
443C4#1683767211yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—4722no—
448C2#2142438811yescharacteristic
449C2#2142459711yescharacteristic
450C2#2142460411yescharacteristic
subgroupG/H
centerC2×C4#14273701
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC8×#41×C59C2
socleC22×C59#386
prime radicalC2×#14273752C4
cosocleC2×C42×C59C23
hypercenterC8×#41D59
hypocenterC59#9095
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #21422483C2semidirect
3. C2×#16837131C2semidirect
4. #21422416C2semidirect
5. #21421945C2semidirect
6. #21421933C2semidirect
7. C8×#41×C59C2semidirect
36. #14273752C8semidirect
103. C2×D236C2×C8semidirect
136. #11745664C2×C8semidirect
219. C8×#41D59semidirect
258. C59#9095semidirect

Maximal subgroups

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

subgroupindex
C4×#142737522
#214224832
C2×#168371312
#214224162
#214219452
#214219332
C8×#41×C592
#909559