Finite Group Databasebeta

← #29469394

Subgroups — #29469394

Order 15104.

1706 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437920C212yesmaximal, complemented
2#21437066C214yesmaximal, complemented
3#21437057C214yesmaximal, complemented
4#21436968C218yesmaximal, complemented
5#21434624C212yesmaximal, complemented
6#21433952C214yesmaximal, complemented
7#21433914C212yesmaximal, complemented
8#21433030C211yesmaximal, characteristic, complemented
9#21432979C212yesmaximal
10C4×#14273749C211yesmaximal, characteristic
11C2×C59×#499C211yesmaximal, characteristic, complemented
12C2×#14273749C2211yescharacteristic
13#16838129C2218yes—
14#16838108C2218yes—
15#16838107—28no—
16#16838033C2214yes—
17#16838032—24no—
18#16837959C2211yescharacteristic
19#16837958—22no—
20#125×D59—22no—
21#16837883C2214yes—
22C4×#11745685C2212yes—
23#16837809C2218yes—
24#16837802C2214yes—
25#16837799C2214yes—
26#16837708C2218yes—
27#16837706C2218yes—
28#16837705C2218yes—
29#16837699C2218yes—
30#16837696C2218yes—
31#16837681C2218yes—
32#16837532C2214yes—
33C8×Dic118C2214yes—
34#16837513C2214yes—
35#16837508C2214yes—
36C4×Dic236C2212yes—
37C4×#11745660C2214yes—
38#16837489C2218yes—
39C4×D472C2212yes—
40#16837442C2214yes—
41#16837440C2211yescharacteristic
42#16837428C2212yes—
43#16837426C2211yescharacteristic
44#16837401C2214yes—
45#16837399C2212yes—
46#16837357C2212yes—
47C2×#125×C59C2212yes—
48C59×#499C2214yes—
49C59×#499C2214yes—
50C2×#107×C59C2212yes—
51C2×#106×C59C2212yes—
52C2×C4×C8×C59C2211yescharacteristic
53C2×#11745685C2312yes—
54#14273749Q818yescomplemented
55C2×Dic236C2311yescharacteristic
56C2×#11745660C2312yes—
57C2×D472C2311yescharacteristic
58C2×#11745664C2312yes—
59#14273727D412yes—
60#14273726—28no—
61#14273726—24no—
62#14273726C2318yes—
63#14273719D412yes—
64#14273718—24no—
65#14273718C2318yes—
66#14273717—28no—
67#14273717—28no—
68#14273707—28no—
69#14273706—28no—
70#14273706D414yes—
71#14273705—28no—
72#14273705—28no—
73C4×Dic118C2314yes—
74C4×Dic118C2314yes—
75#14273692—28no—
76#14273692C2318yes—
77C4×D236C2314yes—
78#14273689C2318yes—
79C4×C22⋊D59—24no—
80C4×C22⋊D59—28no—
81#14273686C2312yes—
82#14273686C2312yes—
83#14273685—24no—
84#14273684—22no—
85C4⋊C4×D59—24no—
86C4⋊C4×D59—28no—
87C42×D59—22no—
88#14273661C2312yes—
89#14273660C2318yes—
90#14273659C2312yes—
91#14273658C2318yes—
92C4⋊(C59⋊C8)C2314yes—
93Dic59⋊C8C2318yes—
94#14273652C2314yes—
95#14273650C2318yes—
96#14273649C2314yes—
97#14273647C2318yes—
98#14273637C2314yes—
99C8×Dic59C2314yes—
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#125×C59—22no—
103#125×C59—22no—
104#125×C59C2318yes—
105C2×C4⋊C4×C59—24no—
106C2×C4⋊C4×C59C2314yes—
107C2×C42×C59C2311yescharacteristic
108#107×C59C2314yes—
109#107×C59C2314yes—
110#106×C59C2314yes—
111#106×C59C2314yes—
112C4×C8×C59C2312yes—
113C4×C8×C59C2312yes—
114C22×C4×C59—24no—
115C22×C4×C59C2×D412yes—
116C22×C4×C59C2411yescharacteristic
117C2×C22⋊D59—22no—
118#11745685—24no—
119#11745685C2×Q818yes—
120C2×D236C2412yes—
121C2×C4×D59—22no—
122C2×C4×D59—24no—
123C2×C4×D59—28no—
124C2×Dic118—28no—
125C2×Dic118C2×D414yes—
126C2×Dic118C2412yes—
127C2×C8×C59—22no—
128C2×C8×C59C2×Q814yes—
129C2×C8×C59C2412yes—
130C2×C8×C59C2411yescharacteristic
131C2×C8×C59C2411yescharacteristic
132C4⋊C4×C59—28no—
133C4⋊C4×C59—28no—
134C4⋊C4×C59—24no—
135C4⋊C4×C59C2418yes—
136C42×C59C2×D411yescharacteristic
137C42×C59C2×D411yescharacteristic
138C42×C59C2412yes—
139C22⋊Dic59—22no—
140C22⋊Dic59C2×D414yes—
141(C2×D59)⋊C4—24no—
142(C2×D59)⋊C4—28no—
143C4⋊Dic59—28no—
144C4⋊Dic59C2×D414yes—
145C4⋊Dic59C2×D414yes—
146C4⋊Dic59C2414yes—
147Dic59⋊C4—24no—
148Dic59⋊C4—28no—
149Dic59⋊C4—28no—
150Dic59⋊C4—28no—
151Dic59⋊C4C2×D418yes—
152C4×Dic59—22no—
153C4×Dic59—24no—
154C4×Dic59—28no—
155C4×Dic59C2414yes—
156#11745664—24no—
157C2×C59⋊C8C2414yes—
158Dic236C2×Q814yes—
159D472C2×Q814yes—
160#11745660C2×Q818yes—
161C23×C59C22×D411yescharacteristic
162C22×D59—22no—
163C2×C4×C59—28no—
164C2×C4×C59—42no—
165C2×C4×C59—84no—
166C2×C4×C59C22×D412yes—
167C2×C4×C59C22×D412yes—
168C2×C4×C59C22×D411yescharacteristic
169C2×C4×C59C22×Q812yes—
170C2×C4×C592−1+412yes—
171C2×C4×C59C2511yescharacteristic
172C22⋊D59—28no—
173C2×Dic59—22no—
174C2×Dic59—24no—
175C2×Dic59C22×D418yes—
176D236—22no—
177D236C22×Q814yes—
178C4×D59—28no—
179C4×D59—88no—
180C4×D59—84no—
181Dic118—22no—
182Dic118—44no—
183Dic118—88no—
184Dic118C22×Q814yes—
185C8×C59—22no—
186C8×C59C22×Q812yes—
187C8×C59C22×Q812yes—
188C59⋊C8—24no—
189#61375—591nomaximal, Sylow
190C22×C59—22no—
191C22×C59D4×Q812yes—
192C22×C59C23×D411yescharacteristic
193C2×D59—24no—
194C2×D59—44no—
195C4×C59—22no—
196C4×C59—22no—
197C4×C59—48no—
198C4×C59D4×Q812yes—
199C4×C59C23×Q811yescharacteristic
200C4×C59C2×2−1+411yescharacteristic
201Dic59—28no—
202Dic59—24no—
203Dic59—44no—
204#3504—592no—
205#3390—594no—
206#3389—594no—
207#3381—598no—
208#3163—592no—
209C2×#499D5911yescharacteristic, complemented
210#3083—594no—
211#3080—592no—
212#2960—591no—
213#2952—592no—
214C4×#135—591no—
215C2×C59—22no—
216C2×C59C2×D4×Q811yescharacteristic
217C2×C59#358412yes—
218D59—44no—
219C2×#135—591no—
220#557—598no—
221#554—598no—
222#554—1188no—
223#544—594no—
224#544—1184no—
225#533—591no—
226#533—1182no—
227C2×#125—1182no—
228C2×#125C2×D5912yes—
229#520—594no—
230C4×#41—592no—
231#501—598no—
232#499—594no—
233#499C2×D5914yes—
234#499C2×D5914yes—
235#498—594no—
236#479—598no—
237#479—598no—
238#478—598no—
239#478—598no—
240#477—598no—
241#474—598no—
242#446—594no—
243C8×Q8—594no—
244#442—594no—
245#441—594no—
246C4×Q16—592no—
247C4×SD16—598no—
248C4×SD16—594no—
249C4×D8—592no—
250#428—594no—
251#427—591no—
252C2×#107C2×D5912yes—
253C2×#106C2×D5912yes—
254#424—592no—
255#423—591no—
256#417—594no—
257#416—592no—
258#405—592no—
259C2×C4×C8C2×D5911yescharacteristic
260C59#6137511yescharacteristic, complemented, Sylow
261C2×#41—592no—
262#135—598no—
263C2×Q16—591no—
264C2×SD16—592no—
265C2×D8—591no—
266C2×M16—592no—
267C22×C8C22×D5911yescharacteristic
268C22×C8C22×D5911yescharacteristic
269C4⋊Q8—598no—
270C4⋊Q8—592no—
271C4⋊Q8—1184no—
272C4⋊Q8—1188no—
273#125—22no—
274#125—22no—
275#125—598no—
276#125—592no—
277#125—1184no—
278#125—1188no—
279#125—1188no—
280#125C22×D5918yes—
281C22⋊Q8—594no—
282C22⋊Q8—1188no—
283C22⋊Q8—1188no—
284C22⋊Q8—1188no—
285C22⋊Q8—1188no—
286C4×Q8—594no—
287C4×Q8—598no—
288C4×Q8—594no—
289C4×Q8—1188no—
290C4×D4—598no—
291C4×D4—594no—
292C4×D4—1188no—
293C4×D4—1184no—
294#117—592no—
295#117—592no—
296#117—1184no—
297#117—1182no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—1184no—
300C2×C4⋊C4—1188no—
301C2×C4⋊C4C22×D5914yes—
302C2×C42—1182no—
303C2×C42C22×D5911yescharacteristic
304#107—592no—
305#107—598no—
306#107C22×D5914yes—
307#107C22×D5914yes—
308#106—592no—
309#106—598no—
310#106C22×D5914yes—
311#106C22×D5914yes—
312C4⋊C8—598no—
313C4⋊C8—594no—
314#103—594no—
315#103—598no—
316#102—594no—
317#102—598no—
318#97—594no—
319C4×C8—594no—
320C4×C8C22×D5912yes—
321C4×C8C22×D5912yes—
322#41—598no—
323#41—1184no—
324C2×Q8—594no—
325C2×Q8—592no—
326C2×Q8—1188no—
327C2×D4—592no—
328C2×D4—1182no—
329C22×C4—24no—
330C22×C4—1188no—
331C22×C4—1182no—
332C22×C4—1184no—
333C22×C4D4×D5912yes—
334C22×C4C23×D5911yescharacteristic
335Q16—594no—
336SD16—598no—
337D8—594no—
338M16—1184no—
339C2×C8—22no—
340C2×C8—594no—
341C2×C8Q8×D5914yes—
342C2×C8C23×D5912yes—
343C2×C8C23×D5911yescharacteristic
344C2×C8C23×D5911yescharacteristic
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—594no—
349C4⋊C4—598no—
350C4⋊C4—594no—
351C4⋊C4—594no—
352C4⋊C4—1188no—
353C4⋊C4—1188no—
354C4⋊C4—1188no—
355C4⋊C4—1188no—
356C4⋊C4—1184no—
357C4⋊C4C23×D5918yes—
358C22⋊C4—594no—
359C22⋊C4—1188no—
360C22⋊C4—1182no—
361C22⋊C4—1184no—
362C42—594no—
363C42—1188no—
364C42—1182no—
365C42—1184no—
366C42D4×D5911yescharacteristic
367C42D4×D5911yescharacteristic
368C42C23×D5912yes—
369C23—1182no—
370C23C2×D4×D5911yescharacteristic
371Q8—594no—
372Q8—1182no—
373Q8—4728no—
374Q8—2364no—
375D4—594no—
376D4—1188no—
377D4—1182no—
378C2×C4—28no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—598no—
382C2×C4—1182no—
383C2×C4—1188no—
384C2×C4—1184no—
385C2×C4—4724no—
386C2×C4—4728no—
387C2×C4C2×D4×D5912yes—
388C2×C4C2×D4×D5912yes—
389C2×C4C2×D4×D5911yescharacteristic
390C2×C4C2×Q8×D5912yes—
391C2×C4#1427377512yes—
392C2×C4C24×D5911yescharacteristic
393C8—22no—
394C8—1184no—
395C8C2×Q8×D5912yes—
396C8C2×Q8×D5912yes—
397C22—22no—
398C22—1184no—
399C22—2364no—
400C22#1683806712yes—
401C22C22×D4×D5911yescharacteristic
402C4—22no—
403C4—22no—
404C4—48no—
405C4—1184no—
406C4—1188no—
407C4—2364no—
408C4#1683806712yes—
409C4C22×Q8×D5911yescharacteristic
410C4C2×#1427377511yescharacteristic
411C2—22no—
412C2—2364no—
413C2C2×#1683806711yescharacteristic
414C2#2143886512yes—
subgroupG/H
centerC22C22×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C59×#499C2
socleC22×C59C23×D4
prime radicalC2×#14273749C22
hypercenterC2×#499D59
hypocenterC59#61375
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
1. #21437920C2semidirect
2. #21437066C2semidirect
3. #21437057C2semidirect
4. #21436968C2semidirect
5. #21434624C2semidirect
6. #21433952C2semidirect
7. #21433914C2semidirect
8. #21433030C2semidirect
11. C2×C59×#499C2semidirect
54. #14273749Q8semidirect
209. C2×#499D59semidirect
260. C59#61375semidirect

Maximal subgroups

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

subgroupindex
#214379202
#214370662
#214370572
#214369682
#214346242
#214339522
#214339142
#214330302
#214329792
C4×#142737492
C2×C59×#4992
#6137559