Finite Group Databasebeta

← #28743286

Subgroups — #28743286

Order 15104.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837842C211yesmaximal, characteristic, complemented
2#21437199C211yesmaximal, characteristic, complemented
3C2×#16837677C211yesmaximal, characteristic, complemented
4#21426224C214yesmaximal, complemented
5#21426219C214yesmaximal, complemented
6#21425890C211yesmaximal, characteristic, complemented
7#21425845C211yesmaximal, characteristic, complemented
8C2×#16837217C211yesmaximal, characteristic, complemented
9C59×#3067C211yesmaximal, characteristic
10C2×#14273757—21no—
11C22×Dic236C2211yescharacteristic
12#16837842—22no—
13#16837842—22no—
14C2×#14273673C2212yescomplemented
15C2×#14273671C2212yescomplemented
16#16837819—22no—
17#16837819C2214yes—
18C2×#14273667—22no—
19C2×#14273667C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21#16837677—22no—
22#16837677C2214yes—
23C2×#14273663C2211yescharacteristic
24C2×#14273645—21no—
25#16837231C2214yescomplemented
26#16837218C2212yescomplemented
27#16837217C2214yescomplemented
28#16837215C2212yescomplemented
29C59×#575—21no—
30C2×#135×C59C2211yescharacteristic
31C2×#131×C59C2211yescharacteristic
32C59×#473C2212yes—
33C59×#472C2212yes—
34C59×#471—22no—
35C2×#108×C59C2211yescharacteristic
36C59×#413—21no—
37C22×Dic118—21no—
38#14273757—24no—
39#14273757—42no—
40C2×Dic236—21no—
41C2×Dic236—21no—
42C2×Dic236—21no—
43C2×Dic236—21no—
44C2×Dic236—21no—
45C2×Dic236C2312yes—
46C2×#11745672—41no—
47C2×#11745670—41no—
48C2×#11745664—21no—
49#14273673—24no—
50#14273671—24no—
51#14273667—21no—
52#14273667—24no—
53#14273667D412yes—
54#14273667D412yes—
55#14273667C2312yes—
56C2×C59⋊C16—21no—
57C2×C59⋊C16—24no—
58C2×C59⋊C16C2312yes—
59#14273663—21no—
60#14273663C2312yes—
61#14273645—24no—
62C2×#41×C59—21no—
63C2×#41×C59D411yescharacteristic
64#137×C59—22no—
65#137×C59—41no—
66AGL(1,ℤ/8)×C59—22no—
67AGL(1,ℤ/8)×C59—41no—
68#135×C59—21no—
69#135×C59—21no—
70#135×C59—21no—
71#135×C59—21no—
72#135×C59—42no—
73C2×Q16×C59C2311yescharacteristic
74C2×SD16×C59—21no—
75C2×D8×C59C2311yescharacteristic
76#131×C59—21no—
77#131×C59—21no—
78#131×C59D412yes—
79#131×C59C2312yes—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59D411yescharacteristic
83C22×C8×C59—21no—
84C22×C8×C59C2311yescharacteristic
85#108×C59—21no—
86#108×C59C2312yes—
87#101×C59—22no—
88#100×C59—22no—
89#41×C59—21no—
90#41×C59—21no—
91#41×C59—41no—
92#41×C59—41no—
93#41×C59—41no—
94#41×C59—41no—
95#41×C59D812yes—
96#41×C59C2×D412yes—
97C2×Q8×C59—21no—
98C2×Q8×C59D811yescharacteristic
99C2×D4×C59—21no—
100C2×D4×C59—21no—
101C2×D4×C59—41no—
102C2×D4×C59D811yescharacteristic
103C22×C4×C59—21no—
104C22×C4×C59—41no—
105C22×C4×C59C2×D411yescharacteristic
106C22×Dic59—41no—
107C2×Dic118—22no—
108C2×Dic118—41no—
109C2×Dic118—41no—
110C2×Dic118—41no—
111C2×Dic118—41no—
112C2×Dic118—41no—
113Q16×C59—21no—
114Q16×C59—21no—
115Q16×C59—42no—
116SD16×C59—42no—
117SD16×C59—42no—
118SD16×C59—41no—
119SD16×C59—41no—
120D8×C59—21no—
121D8×C59—21no—
122D8×C59—42no—
123M16×C59—21no—
124M16×C59—22no—
125M16×C59—22no—
126M16×C59—22no—
127M16×C59—41no—
128M16×C59C2×D412yes—
129C2×C8×C59—21no—
130C2×C8×C59—22no—
131C2×C8×C59—22no—
132C2×C8×C59—41no—
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59#4111yescharacteristic
137C2×C8×C59#4111yescharacteristic
138C2×C8×C59C2411yescharacteristic
139#11745672—44no—
140#11745670—44no—
141#11745664—22no—
142#11745664—41no—
143C2×C59⋊C8—41no—
144Dic236—22no—
145Dic236—22no—
146Dic236—41no—
147Dic236—41no—
148C59⋊C16—22no—
149C59⋊C16C2×D414yes—
150C23×C59—21no—
151C23×C59—41no—
152C23×C59#13511yescharacteristic
153Q8×C59—21no—
154Q8×C59—41no—
155Q8×C59—41no—
156Q8×C59C2×D812yes—
157D4×C59—22no—
158D4×C59—22no—
159D4×C59—21no—
160D4×C59—41no—
161D4×C59—41no—
162D4×C59—81no—
163D4×C59—81no—
164D4×C59C2×D812yes—
165C2×C4×C59—21no—
166C2×C4×C59—22no—
167C2×C4×C59—22no—
168C2×C4×C59—41no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59C2×D811yescharacteristic
174C2×C4×C59C2×D811yescharacteristic
175C2×C4×C59#13511yescharacteristic
176C2×C4×C59C22×D411yescharacteristic
177C2×Dic59—42no—
178C2×Dic59—81no—
179C2×Dic59—81no—
180Dic118—42no—
181Dic118—42no—
182Dic118—81no—
183Dic118—81no—
184C8×C59—22no—
185C8×C59—42no—
186C8×C59#12111yescharacteristic
187C8×C59#12111yescharacteristic
188C8×C59C22×D411yescharacteristic
189C8×C59C2×#4111yescharacteristic
190C59⋊C8—42no—
191#33619—591nomaximal, Sylow
192C22×C59—22no—
193C22×C59—41no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59—161no—
197C22×C59#46611yescharacteristic
198C22×C59#46611yescharacteristic
199C22×C59C2×#13511yescharacteristic
200C4×C59—22no—
201C4×C59—81no—
202C4×C59#46611yescharacteristic
203C4×C59#46611yescharacteristic
204C4×C59C2×#12111yescharacteristic
205C4×C59C22×D811yescharacteristic
206Dic59—82no—
207C2×#510—591no—
208#3407—591no—
209#3067D5911yescharacteristic
210C2×#473—591no—
211#2223—594no—
212#2222—594no—
213#2161—591no—
214#2148—591no—
215C2×#362—591no—
216C2×C59—21no—
217C2×C59—21no—
218C2×C59—21no—
219C2×C59—42no—
220C2×C59—161no—
221C2×C59C2×#46611yescharacteristic
222#575—21no—
223C2×#137—1181no—
224C2×#135C2×D5911yescharacteristic
225C22×Q16—591no—
226C2×#131C2×D5911yescharacteristic
227#510—1182no—
228#510—1182no—
229C2×Q32—592no—
230C2×SD32—592no—
231#504—594no—
232#504—1182no—
233C2×M32—591no—
234C2×M32—591no—
235C2×M32—1182no—
236#473—594no—
237#473—1182no—
238#473C2×D5912yes—
239#472C2×D5912yes—
240#471—22no—
241C2×#108—591no—
242C2×#108C2×D5911yescharacteristic
243#413—21no—
244C2×#101—1181no—
245#368—594no—
246#362—592no—
247#362—594no—
248#361—592no—
249C59#3361911yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41C22⋊D5911yescharacteristic
252C22×Q8—1181no—
253#137—22no—
254#137—41no—
255#137—1184no—
256#137—2362no—
257AGL(1,ℤ/8)—22no—
258AGL(1,ℤ/8)—41no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—42no—
264C2×Q16—592no—
265C2×Q16—1181no—
266C2×Q16—1181no—
267C2×Q16—1181no—
268C2×Q16—1181no—
269C2×Q16—1181no—
270C2×Q16—2361no—
271C2×Q16C22×D5911yescharacteristic
272C2×SD16—21no—
273C2×SD16—2361no—
274C2×D8C22×D5911yescharacteristic
275#131—21no—
276#131—21no—
277#131C22⋊D5912yes—
278#131C22×D5912yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—1181no—
282C2×M16C22⋊D5911yescharacteristic
283C22×C8—21no—
284C22×C8C22×D5911yescharacteristic
285Q32—1184no—
286SD32—1184no—
287M32—592no—
288M32—592no—
289M32—592no—
290M32—1184no—
291M32—1181no—
292C2×C16—592no—
293C2×C16—1184no—
294C2×C16—1181no—
295#108—21no—
296#108—592no—
297#108—1181no—
298#108C22×D5912yes—
299#101—22no—
300#101—1184no—
301#100—22no—
302#41—21no—
303#41—21no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41#1174566912yes—
309#41C2×C22⋊D5912yes—
310C2×Q8—21no—
311C2×Q8—1182no—
312C2×Q8—2361no—
313C2×Q8—2361no—
314C2×Q8—2361no—
315C2×Q8—2361no—
316C2×Q8—2361no—
317C2×Q8#1174566911yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4#1174566911yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—2361no—
325C22×C4C2×C22⋊D5911yescharacteristic
326Q16—21no—
327Q16—21no—
328Q16—42no—
329Q16—1182no—
330Q16—1182no—
331Q16—2364no—
332Q16—2361no—
333Q16—2361no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—2364no—
339D8—21no—
340D8—21no—
341D8—42no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—1182no—
348M16—2361no—
349M16C2×C22⋊D5912yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—2361no—
355C2×C8D4×D5911yescharacteristic
356C2×C8D4×D5911yescharacteristic
357C2×C8#1174568711yescharacteristic
358C2×C8#1174568711yescharacteristic
359C2×C8C2×C22⋊D5911yescharacteristic
360C2×C8C23×D5911yescharacteristic
361C16—594no—
362C16—1182no—
363C23—21no—
364C23—41no—
365C23#1427374711yescharacteristic
366Q8—21no—
367Q8—41no—
368Q8—41no—
369Q8—4721no—
370Q8—4721no—
371Q8—2362no—
372Q8—2362no—
373Q8C2×#1174566912yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4C2×#1174566912yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—4721no—
389C2×C4—4721no—
390C2×C4—2362no—
391C2×C4#1427369911yescharacteristic
392C2×C4#1427369911yescharacteristic
393C2×C4C2×#1174566911yescharacteristic
394C2×C4C2×#1174566911yescharacteristic
395C2×C4#1427374711yescharacteristic
396C2×C4C22×C22⋊D5911yescharacteristic
397C8—22no—
398C8—42no—
399C8—2362no—
400C8#1427369911yescharacteristic
401C8#1427369911yescharacteristic
402C8C2×D4×D5911yescharacteristic
403C8C2×#1174568711yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—161no—
409C22#1683763611yescharacteristic
410C22#1683763611yescharacteristic
411C22C2×#1427374711yescharacteristic
412C4—22no—
413C4—81no—
414C4—4722no—
415C4#1683763611yescharacteristic
416C4#1683763611yescharacteristic
417C4C2×#1427369911yescharacteristic
418C4C22×#1174566911yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#1683763611yescharacteristic
subgroupG/H
centerC2C2×#16837636
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3067C2
socleC2×C59C2×#466
hypercenter#3067D59
hypocenterC59#33619
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. C2×#16837842C2semidirect
2. #21437199C2semidirect
3. C2×#16837677C2semidirect
4. #21426224C2semidirect
5. #21426219C2semidirect
6. #21425890C2semidirect
7. #21425845C2semidirect
8. C2×#16837217C2semidirect
14. C2×#14273673C22semidirect
15. C2×#14273671C22semidirect
25. #16837231C22semidirect
26. #16837218C22semidirect
27. #16837217C22semidirect
28. #16837215C22semidirect
249. C59#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#168378422
#214371992
C2×#168376772
#214262242
#214262192
#214258902
#214258452
C2×#168372172
C59×#30672
#3361959