Finite Group Databasebeta

← #28562605

Subgroups — #28562605

Order 15104.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434849C212yesmaximal
2#21429838C211yesmaximal, characteristic
3#21423104C214yesmaximal, complemented
4#21423036C214yesmaximal, complemented
5#21422668C212yesmaximal, complemented
6C2×#16837128C211yesmaximal, characteristic, complemented
7C2×C59×#544C211yesmaximal, characteristic, complemented
8C2×#14273710—21no—
9C2×#14273701C2211yescharacteristic
10#16837730C2214yes—
11#16837720—22no—
12#16837633—24no—
13#16837552C2214yes—
14#16837541—22no—
15C2×C22⋊(C59⋊C8)C2212yes—
16#16837332C2212yes—
17#16837324—22no—
18#16837319C2212yes—
19#16837281—21no—
20#16837262C2211yescharacteristic
21#16837139C2218yes—
22#16837128C2214yes—
23C59×#544—24no—
24C59×#544—22no—
25C59×#544C2214yes—
26C2×C4⋊Q8×C59—22no—
27C2×#125×C59—22no—
28C2×C22⋊Q8×C59—21no—
29C2×C22⋊Q8×C59C2212yes—
30C2×#117×C59—21no—
31C22×C4⋊C4×C59C2211yescharacteristic
32C22×D236C2311yescharacteristic
33C22×C22⋊D59—21no—
34C2×#11745671—24no—
35C22×C59⋊C8—22no—
36#14273710—44no—
37#14273701—22no—
38#14273701—22no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59C2311yescharacteristic
41C2×Dic59⋊C4—22no—
42C2×C22⋊Dic59—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—22no—
45C2×(C2×D59)⋊C4—22no—
46#14273658—24no—
47#14273651—24no—
48#14273647—24no—
49C22⋊(C59⋊C8)—22no—
50C22⋊(C59⋊C8)C2314yes—
51C59⋊#95—22no—
52C59⋊#95C2312yes—
53C22×Q8×C59—21no—
54C22×Q8×C59D412yes—
55C23×C4×C59D411yescharacteristic
56C23×C4×C59D411yescharacteristic
57C23×C4×C59C2311yescharacteristic
58C4⋊Q8×C59—28no—
59C4⋊Q8×C59—28no—
60#125×C59—28no—
61#125×C59—28no—
62C22⋊Q8×C59—24no—
63C22⋊Q8×C59—24no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—22no—
66C22⋊Q8×C59D414yes—
67C22⋊Q8×C59C2314yes—
68#117×C59—24no—
69#117×C59—24no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59—22no—
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59—22no—
76C2×C4⋊C4×C59—22no—
77C2×C4⋊C4×C59D412yes—
78C2×C4⋊C4×C59D412yes—
79C2×C4⋊C4×C59C2312yes—
80C2×C22⋊C4×C59—22no—
81C2×C42×C59—22no—
82C24×C59C2×D411yescharacteristic
83C23×D59—21no—
84C2×Q8×C59—24no—
85C2×Q8×C59—22no—
86C2×Q8×C59—82no—
87C2×Q8×C59—82no—
88C2×Q8×C59C2×D414yes—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—22no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—81no—
97C22×C4×C59—81no—
98C22×C4×C59SD1614yescomplemented
99C22×C4×C59C2×D412yes—
100C22×C4×C59C2×D412yes—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D412yes—
103C22×C4×C59C2×D412yes—
104C22×C4×C59C2411yescharacteristic
105C2×C22⋊D59—41no—
106C2×C22⋊D59—41no—
107C2×C22⋊D59—42no—
108C2×C22⋊D59—42no—
109C22×Dic59—21no—
110C22×Dic59—22no—
111C2×D236—22no—
112C2×D236—81no—
113C4⋊C4×C59—24no—
114C4⋊C4×C59—24no—
115C4⋊C4×C59—24no—
116C4⋊C4×C59—28no—
117C4⋊C4×C59—24no—
118C4⋊C4×C59—24no—
119C4⋊C4×C59—22no—
120C4⋊C4×C59—24no—
121C4⋊C4×C59—24no—
122C4⋊C4×C59C2×D414yes—
123C22⋊C4×C59—24no—
124C22⋊C4×C59—24no—
125C42×C59—28no—
126C22⋊Dic59—42no—
127#11745671—84no—
128(C2×D59)⋊C4—42no—
129(C2×D59)⋊C4—44no—
130(C2×D59)⋊C4—82no—
131C4⋊Dic59—22no—
132C4⋊Dic59—81no—
133Dic59⋊C4—44no—
134C2×C59⋊C8—24no—
135C2×C59⋊C8—82no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59#12011yescharacteristic
139C23×C59#12011yescharacteristic
140C23×C59C2×SD1612yes—
141C23×C59#13712yes—
142C23×C59C22×D411yescharacteristic
143C22×D59—42no—
144C22×D59—42no—
145C22×D59—41no—
146C22×D59—41no—
147C22×D59—41no—
148Q8×C59—44no—
149Q8×C59—82no—
150Q8×C59—84no—
151C2×C4×C59—22no—
152C2×C4×C59—22no—
153C2×C4×C59—24no—
154C2×C4×C59—24no—
155C2×C4×C59—24no—
156C2×C4×C59—24no—
157C2×C4×C59—22no—
158C2×C4×C59—42no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—42no—
162C2×C4×C59—44no—
163C2×C4×C59—42no—
164C2×C4×C59—44no—
165C2×C4×C59—42no—
166C2×C4×C59—81no—
167C2×C4×C59#12012yes—
168C2×C4×C59C2×SD1614yes—
169C2×C4×C59C22×D412yes—
170C22⋊D59—82no—
171C22⋊D59—82no—
172C2×Dic59—42no—
173C2×Dic59—42no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—42no—
177C2×Dic59—41no—
178D236—44no—
179C59⋊C8—44no—
180#18294—591nomaximal, Sylow
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—42no—
188C22×C59—42no—
189C22×C59#44812yes—
190C22×C59#44812yes—
191C22×C59C2×#12011yescharacteristic
192C22×C59C22×SD1611yescharacteristic
193C22×C59C2×#13711yescharacteristic
194C2×D59—82no—
195C2×D59—82no—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C4×C59—24no—
200C4×C59—22no—
201C4×C59—22no—
202C4×C59—44no—
203C4×C59—42no—
204C4×C59—44no—
205Dic59—82no—
206Dic59—81no—
207C2×#544D5911yescharacteristic, complemented
208#3187—592no—
209#2618—591no—
210#1611—594no—
211#1599—594no—
212#1520—592no—
213C2×#324—591no—
214C2×C59—21no—
215C2×C59—21no—
216C2×C59—22no—
217C2×C59—22no—
218C2×C59C2×#44811yescharacteristic
219C2×C59#302011yescharacteristic
220C2×C59#302611yescharacteristic
221D59—161no—
222#544—24no—
223#544—22no—
224#544C2×D5914yes—
225C2×C4⋊Q8—22no—
226C2×#125—22no—
227C2×#123—1181no—
228C2×C22⋊Q8—21no—
229C2×C22⋊Q8C2×D5912yes—
230C2×#121—591no—
231C2×#117—21no—
232C22×C4⋊C4C2×D5911yescharacteristic
233#483—594no—
234#481—1182no—
235#465—1184no—
236#450—594no—
237#448—1182no—
238C2×C22⋊C8—592no—
239#398—592no—
240#396—1182no—
241#394—592no—
242#386—1181no—
243#380—591no—
244#328—598no—
245#324—594no—
246C59#1829411yescharacteristic, complemented, Sylow
247C22×Q8—21no—
248C22×Q8C22⋊D5912yes—
249C22×D4—591no—
250C22×D4—1181no—
251C23×C4C22⋊D5911yescharacteristic
252C23×C4C22⋊D5911yescharacteristic
253C23×C4C22×D5911yescharacteristic
254C2×SD16—1184no—
255C22×C8—1182no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258#125—28no—
259#125—28no—
260#123—2364no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8C22⋊D5914yes—
266C22⋊Q8C22×D5914yes—
267#121—1182no—
268#121—1182no—
269#117—24no—
270#117—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—591no—
279C2×C4⋊C4—1181no—
280C2×C4⋊C4—1182no—
281C2×C4⋊C4C22⋊D5912yes—
282C2×C4⋊C4C22⋊D5912yes—
283C2×C4⋊C4C22×D5912yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—1182no—
286C2×C22⋊C4—1181no—
287C2×C22⋊C4—1181no—
288C2×C22⋊C4—1182no—
289C2×C42—22no—
290#106—1184no—
291#103—1184no—
292#102—1184no—
293C22⋊C8—594no—
294C22⋊C8—1182no—
295#95—592no—
296#95—1182no—
297C24—1181no—
298C24C2×C22⋊D5911yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8C2×C22⋊D5914yes—
304C2×D4—1182no—
305C2×D4—4721no—
306C2×D4—2361no—
307C2×D4—2361no—
308C2×D4—2362no—
309C2×D4—2362no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—1182no—
320C22×C4—1181no—
321C22×C4#1174567114yescomplemented
322C22×C4C2×C22⋊D5912yes—
323C22×C4C2×C22⋊D5912yes—
324C22×C4C2×C22⋊D5911yescharacteristic
325C22×C4C2×C22⋊D5912yes—
326C22×C4C2×C22⋊D5912yes—
327C22×C4C23×D5911yescharacteristic
328SD16—4724no—
329C2×C8—1184no—
330C2×C8—4722no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—1182no—
341C4⋊C4—4721no—
342C4⋊C4—2364no—
343C4⋊C4C2×C22⋊D5914yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—4722no—
347C22⋊C4—2362no—
348C22⋊C4—2362no—
349C22⋊C4—2364no—
350C42—28no—
351C23—41no—
352C23—41no—
353C23—2362no—
354C23—2362no—
355C23—2361no—
356C23—2361no—
357C23—2361no—
358C23#1427369611yescharacteristic
359C23#1427369611yescharacteristic
360C23C2×#1174567112yes—
361C23#1427375812yes—
362C23C22×C22⋊D5911yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366D4—4722no—
367D4—4722no—
368D4—2364no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—2362no—
386C2×C4—2362no—
387C2×C4—2361no—
388C2×C4—2362no—
389C2×C4—2361no—
390C2×C4—2361no—
391C2×C4#1427369612yes—
392C2×C4C2×#1174567114yes—
393C2×C4C22×C22⋊D5912yes—
394C8—2364no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—4722no—
404C22—4722no—
405C22—4721no—
406C22—4721no—
407C22—4721no—
408C22#1683754112yes—
409C22#1683754112yes—
410C22C2×#1427369611yescharacteristic
411C22C22×#1174567111yescharacteristic
412C22C2×#1427375811yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—44no—
417C4—42no—
418C4—44no—
419C4—4722no—
420C4—4721no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—9441no—
426C2C2×#1683754111yescharacteristic
427C2#2143346011yescharacteristic
428C2#2143349311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#544C2
socleC22×C59C2×#120
prime radicalC2×#14273701C22
hypercenterC2×#544D59
hypocenterC59#18294
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21423104C2semidirect
4. #21423036C2semidirect
5. #21422668C2semidirect
6. C2×#16837128C2semidirect
7. C2×C59×#544C2semidirect
98. C22×C4×C59SD16semidirect
207. C2×#544D59semidirect
246. C59#18294semidirect
321. C22×C4#11745671semidirect

Maximal subgroups

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

subgroupindex
#214348492
#214298382
#214231042
#214230362
#214226682
C2×#168371282
C2×C59×#5442
#1829459