Finite Group Databasebeta

← #29469436

Subgroups — #29469436

Order 15104.

1674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437906C212yesmaximal
2#21437064C218yesmaximal, complemented
3#21437029C218yesmaximal, complemented
4#21434624C212yesmaximal
5#21434620C211yesmaximal, characteristic
6#21433934C214yesmaximal
7#21433901C212yesmaximal
8#21433036C211yesmaximal, characteristic
9#21432964C212yesmaximal
10C2×C59×#441C211yesmaximal, characteristic
11Q8×Dic118—22no—
12Q8×Dic118C2214yes—
13#16838131—22no—
14#16838131C2214yes—
15#16838128—24no—
16#16838128C2218yes—
17#16838033—24no—
18#16838033C2214yes—
19#16837959—24no—
20#16837959C2212yes—
21#16837959C2211yescharacteristic
22#16837880C2214yes—
23#16837811C2214yes—
24#16837811C2214yes—
25#16837809C2218yes—
26#16837802C2214yes—
27#16837799C2214yes—
28#16837711C2218yes—
29#16837709C2218yes—
30#16837707C2218yes—
31#16837697C2214yes—
32#16837695C2218yes—
33#16837687C2214yes—
34#16837685C2218yes—
35#16837530C2218yes—
36#16837509C2214yes—
37#16837509C2214yes—
38Q16×Dic59C2218yes—
39#16837442C2214yes—
40#16837440C2212yes—
41#16837440C2212yes—
42#16837428C2212yes—
43#16837426C2211yescharacteristic
44#16837411C2214yes—
45#16837357C2212yes—
46C22×Q16×C59C2211yescharacteristic
47C2×C4×Q8×C59C2212yes—
48C59×#441C2218yes—
49C2×#106×C59C2211yescharacteristic
50C2×#103×C59C2212yes—
51C2×#97×C59C2211yescharacteristic
52C2×#11745664C2312yes—
53#14273727—24no—
54#14273727—24no—
55#14273727D412yes—
56#14273727C2312yes—
57#14273727C2312yes—
58#14273726—28no—
59#14273726—24no—
60#14273726—28no—
61#14273726C2318yes—
62#14273719—28no—
63#14273719D412yes—
64#14273719C2314yes—
65#14273718—28no—
66#14273718—24no—
67#14273718—28no—
68#14273718C2318yes—
69#14273706—22no—
70#14273706—22no—
71#14273706—28no—
72#14273706D414yes—
73Q8×Dic59—24no—
74Q8×Dic59—24no—
75Q8×Dic59C2318yes—
76Q8×Dic59C2318yes—
77C4×Dic118—24no—
78C4×Dic118—22no—
79C4×Dic118—22no—
80C4×Dic118—24no—
81C4×Dic118—28no—
82C4×Dic118C2314yes—
83#14273686—28no—
84#14273686C2314yes—
85#14273686C2312yes—
86#14273661C2314yes—
87#14273661C2314yes—
88#14273660C2318yes—
89#14273659C2312yes—
90#14273659C2312yes—
91#14273659C2312yes—
92#14273659C2312yes—
93#14273658C2318yes—
94C4⋊(C59⋊C8)C2314yes—
95Dic59⋊C8C2318yes—
96#14273651C2318yes—
97#14273651C2318yes—
98#14273637C2314yes—
99C8×Dic59C2314yes—
100C22×Q8×C59C2312yes—
101C2×Q16×C59Q818yescomplemented
102C2×Q16×C59C2314yes—
103C22×C8×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C4×Q8×C59—22no—
106C4×Q8×C59—22no—
107C4×Q8×C59C2314yes—
108C4×Q8×C59C2314yes—
109C2×C4⋊C4×C59—22no—
110C2×C4⋊C4×C59C2312yes—
111C2×C42×C59—22no—
112C2×C42×C59C2311yescharacteristic
113#106×C59C2314yes—
114#103×C59C2318yes—
115#97×C59C2314yes—
116C2×Q8×C59—22no—
117C2×Q8×C59—24no—
118C2×Q8×C59C2×Q818yes—
119C2×Q8×C59C2414yes—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59C2×D412yes—
123C22×C4×C59C2411yescharacteristic
124C2×Dic118—22no—
125C2×Dic118—22no—
126C2×Dic118—28no—
127C2×Dic118C2×D414yes—
128Q16×C59C2×Q8116yes—
129C2×C8×C59—22no—
130C2×C8×C59C2×Q814yes—
131C2×C8×C59C2412yes—
132C2×C8×C59C2412yes—
133C4⋊C4×C59—24no—
134C4⋊C4×C59—24no—
135C4⋊C4×C59—22no—
136C4⋊C4×C59C2414yes—
137C42×C59—24no—
138C42×C59—24no—
139C42×C59C2×D411yescharacteristic
140C42×C59C2×D411yescharacteristic
141C42×C59C2412yes—
142C22⋊Dic59—24no—
143C22⋊Dic59C2×D414yes—
144C4⋊Dic59—22no—
145C4⋊Dic59—22no—
146C4⋊Dic59—28no—
147C4⋊Dic59—28no—
148C4⋊Dic59—28no—
149C4⋊Dic59C2×D414yes—
150C4⋊Dic59C2×D414yes—
151C4⋊Dic59C2414yes—
152C4⋊Dic59C2414yes—
153Dic59⋊C4—24no—
154Dic59⋊C4—24no—
155Dic59⋊C4—28no—
156Dic59⋊C4—28no—
157Dic59⋊C4C2×D418yes—
158C4×Dic59—24no—
159C4×Dic59—28no—
160C4×Dic59—28no—
161C4×Dic59C2414yes—
162#11745664—24no—
163C2×C59⋊C8C2414yes—
164C23×C59C22×D411yescharacteristic
165Q8×C59—28no—
166Q8×C59—24no—
167Q8×C59C22×Q818yes—
168C2×C4×C59—28no—
169C2×C4×C59—24no—
170C2×C4×C59—24no—
171C2×C4×C59—42no—
172C2×C4×C59—82no—
173C2×C4×C59C22×D412yes—
174C2×C4×C59C22×D412yes—
175C2×C4×C59C22×D411yescharacteristic
176C2×C4×C59C22×Q812yes—
177C2×C4×C592−1+412yes—
178C2×C4×C59C2511yescharacteristic
179C2×Dic59—24no—
180C2×Dic59—24no—
181C2×Dic59C22×D418yes—
182Dic118—44no—
183Dic118—44no—
184Dic118—88no—
185Dic118—82no—
186C8×C59—22no—
187C8×C59C22×Q814yes—
188C59⋊C8—24no—
189#61379—591nomaximal, Sylow
190C22×C59—22no—
191C22×C59D4×Q812yes—
192C22×C59C23×D411yescharacteristic
193C4×C59—22no—
194C4×C59—22no—
195C4×C59—28no—
196C4×C59—44no—
197C4×C59D4×Q812yes—
198C4×C59C23×Q811yescharacteristic
199C4×C59C2×2−1+411yescharacteristic
200Dic59—28no—
201Dic59—44no—
202Dic59—44no—
203#3502—592no—
204#3390—598no—
205#3387—598no—
206#3163—592no—
207#3162—591no—
208#3082—594no—
209#3078—592no—
210#2961—591no—
211#2950—592no—
212C2×#441D5911yescharacteristic
213C2×C59—22no—
214C2×C59C2×D4×Q811yescharacteristic
215C2×C59#358612yes—
216C22×Q16C2×D5911yescharacteristic
217Q82—594no—
218Q82—1182no—
219#557—598no—
220#557—594no—
221#557—1184no—
222#557—1182no—
223#544—594no—
224#544—1184no—
225#533—591no—
226#533—592no—
227#533—1184no—
228#519—594no—
229C2×C4×Q8C2×D5912yes—
230#501—598no—
231#501—594no—
232#501—594no—
233#499—594no—
234#498—594no—
235#479—598no—
236#479—598no—
237#479—598no—
238#477—594no—
239#477—598no—
240#475—594no—
241#475—598no—
242#446—598no—
243#441—594no—
244#441—594no—
245#441C2×D5918yes—
246C4×Q16—598no—
247#428—594no—
248#427—592no—
249#427—592no—
250C2×#106C2×D5911yescharacteristic
251#424—592no—
252#423—591no—
253#419—594no—
254C2×#103C2×D5912yes—
255#405—592no—
256C2×#97C2×D5911yescharacteristic
257C59#6137911yescharacteristic, complemented, Sylow
258C22×Q8C22×D5912yes—
259C2×Q16Dic11818yescomplemented
260C2×Q16C22×D5914yes—
261C2×M16—592no—
262C22×C8C22×D5911yescharacteristic
263C22×C8C22×D5911yescharacteristic
264C4⋊Q8—598no—
265C4⋊Q8—592no—
266C4⋊Q8—592no—
267C4⋊Q8—592no—
268C4⋊Q8—1184no—
269C4⋊Q8—1184no—
270C4⋊Q8—1184no—
271C4⋊Q8—1188no—
272C4⋊Q8—1188no—
273#125—598no—
274#125—592no—
275#125—594no—
276#125—1188no—
277#125—1184no—
278#125—1188no—
279#125—1188no—
280C22⋊Q8—594no—
281C22⋊Q8—1188no—
282C22⋊Q8—1182no—
283C22⋊Q8—1182no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—594no—
287C4×Q8—598no—
288C4×Q8—598no—
289C4×Q8—1188no—
290C4×Q8—1184no—
291C4×Q8—1184no—
292C4×Q8—1184no—
293C4×Q8—1184no—
294C4×Q8—1182no—
295C4×Q8—1182no—
296C4×Q8C22×D5914yes—
297C4×Q8C22×D5914yes—
298#117—592no—
299#117—594no—
300#117—1188no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C22×D5912yes—
303C2×C42—22no—
304C2×C42C22×D5911yescharacteristic
305#107—594no—
306#107—594no—
307#107—598no—
308#106—592no—
309#106—592no—
310#106—592no—
311#106—592no—
312#106—598no—
313#106C22×D5914yes—
314C4⋊C8—598no—
315C4⋊C8—594no—
316#103—598no—
317#103—598no—
318#103C22×D5918yes—
319#97—594no—
320#97C22×D5914yes—
321C4×C8—594no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—594no—
325C2×Q8—1188no—
326C2×Q8—1182no—
327C2×Q8—1182no—
328C2×Q8C2×Dic11818yes—
329C2×Q8C23×D5914yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4D4×D5912yes—
333C22×C4C23×D5911yescharacteristic
334Q16C2×Dic118116yes—
335M16—1184no—
336C2×C8—22no—
337C2×C8—594no—
338C2×C8C2×Dic11814yes—
339C2×C8C23×D5912yes—
340C2×C8C23×D5912yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—594no—
345C4⋊C4—598no—
346C4⋊C4—594no—
347C4⋊C4—594no—
348C4⋊C4—594no—
349C4⋊C4—1188no—
350C4⋊C4—1188no—
351C4⋊C4—1188no—
352C4⋊C4—1188no—
353C4⋊C4—1188no—
354C4⋊C4—1184no—
355C4⋊C4—1182no—
356C4⋊C4—1184no—
357C4⋊C4—1182no—
358C4⋊C4C23×D5914yes—
359C22⋊C4—594no—
360C22⋊C4—1184no—
361C42—24no—
362C42—24no—
363C42—594no—
364C42—1188no—
365C42—1188no—
366C42—1184no—
367C42D4×D5911yescharacteristic
368C42D4×D5911yescharacteristic
369C42C23×D5912yes—
370C23C2×D4×D5911yescharacteristic
371Q8—24no—
372Q8—28no—
373Q8—4722no—
374Q8—4728no—
375Q8—2364no—
376Q8—2364no—
377Q8C22×Dic11818yes—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—598no—
384C2×C4—1184no—
385C2×C4—1184no—
386C2×C4C2×D4×D5912yes—
387C2×C4C2×D4×D5912yes—
388C2×C4C2×D4×D5911yescharacteristic
389C2×C4C22×Dic11812yes—
390C2×C4#1427377412yes—
391C2×C4C24×D5911yescharacteristic
392C8—22no—
393C8—1184no—
394C8C22×Dic11814yes—
395C22—22no—
396C22D4×Dic11812yes—
397C22C22×D4×D5911yescharacteristic
398C4—28no—
399C4—22no—
400C4—22no—
401C4—44no—
402C4—1188no—
403C4—2364no—
404C4—2364no—
405C4D4×Dic11812yes—
406C4C23×Dic11811yescharacteristic
407C4C2×#1427377411yescharacteristic
408C2—22no—
409C2C2×D4×Dic11811yescharacteristic
410C2#2143887912yes—
subgroupG/H
centerC22C22×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C59×#441C2
socleC22×C59C23×D4
prime radicalC23×C59C22×D4
hypercenterC2×#441D59
hypocenterC59#61379
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. #21437064C2semidirect
3. #21437029C2semidirect
101. C2×Q16×C59Q8semidirect
257. C59#61379semidirect
259. C2×Q16Dic118semidirect

Maximal subgroups

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

subgroupindex
#214379062
#214370642
#214370292
#214346242
#214346202
#214339342
#214339012
#214330362
#214329642
C2×C59×#4412
#6137959