Finite Group Databasebeta

← #28489395

Subgroups — #28489395

Order 15104.

536 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434898C211yesmaximal, characteristic
2C2×#16837720C211yesmaximal, characteristic
3#21424176C211yesmaximal, characteristic, complemented
4#21422460C211yesmaximal, characteristic, complemented
5#21422455C211yesmaximal, characteristic
6#21421984C211yesmaximal, characteristic, complemented
7C2×C59×#483C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9#16837735—21no—
10#16837720—22no—
11#16837720—22no—
12#16837715—21no—
13#16837650—22no—
14#16837552—21no—
15#16837541—21no—
16C2×#14273658—21no—
17C2×#14273647—21no—
18#16837366C2211yescharacteristic
19#16837366C2211yescharacteristic
20C2×C22⋊(C59⋊C8)C2211yescharacteristic
21#16837171—22no—
22#16837128—22no—
23#16837125—22no—
24C2×C22⋊Q8×C59C2211yescharacteristic
25C22×C4⋊C4×C59C2211yescharacteristic
26C59×#483—22no—
27C59×#483—22no—
28C2×#106×C59—21no—
29C2×#103×C59—21no—
30C2×C22⋊C8×C59C2211yescharacteristic
31C22×D236D411yescharacteristic
32C22×C22⋊D59—21no—
33C2×#11745671—22no—
34C2×#11745664—21no—
35C2×#11745664D412yes—
36C22×C59⋊C8—21no—
37#14273701—21no—
38#14273701—21no—
39#14273701—21no—
40#14273701—21no—
41C2×C4⋊Dic59D411yescharacteristic
42C2×(C2×D59)⋊C4—21no—
43#14273660—22no—
44#14273658—24no—
45#14273651—22no—
46#14273647—24no—
47#14273647—22no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)—21no—
52C22⋊(C59⋊C8)—21no—
53C22⋊(C59⋊C8)D412yes—
54C22×Q8×C59D411yescharacteristic
55C23×C4×C59—21no—
56C23×C4×C59C2311yescharacteristic
57C22×C8×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C4⋊C4×C59D412yes—
70C2×C22⋊C4×C59—21no—
71#106×C59—24no—
72#103×C59—24no—
73C22⋊C8×C59—21no—
74C22⋊C8×C59D412yes—
75C24×C59#4111yescharacteristic
76C23×D59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—21no—
79C2×Q8×C59—42no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—22no—
83C22×C4×C59—42no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—82no—
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C2×C22⋊D59—41no—
94C2×C22⋊D59—41no—
95C2×C22⋊D59—82no—
96C22×Dic59—21no—
97C2×D236—21no—
98C2×D236—21no—
99C2×D236—42no—
100C2×C8×C59—22no—
101C2×C8×C59—42no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112#11745671—82no—
113(C2×D59)⋊C4—41no—
114(C2×D59)⋊C4—41no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117#11745664—42no—
118#11745664—42no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—42no—
122C2×C59⋊C8—41no—
123C23×C59—22no—
124C23×C59—22no—
125C23×C59—41no—
126C23×C59#12111yescharacteristic
127C23×C59#12111yescharacteristic
128C23×C59#12111yescharacteristic
129C22×D59—41no—
130C22×D59—42no—
131C22×D59—81no—
132C22×D59—81no—
133Q8×C59—42no—
134Q8×C59—42no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—82no—
146C2×C4×C59—82no—
147C2×C4×C59—82no—
148C2×C4×C59—81no—
149C2×C4×C59#12011yescharacteristic
150C2×C4×C59#12011yescharacteristic
151C2×C4×C59#12011yescharacteristic
152C2×C4×C59#12711yescharacteristic
153C22⋊D59—82no—
154C22⋊D59—82no—
155C2×Dic59—41no—
156C2×Dic59—81no—
157D236—42no—
158D236—42no—
159C8×C59—42no—
160C59⋊C8—42no—
161C59⋊C8—42no—
162C59⋊C8—81no—
163#9993—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—44no—
172C22×C59#39211yescharacteristic
173C22×C59#47112yes—
174C22×C59#47112yes—
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—162no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—81no—
183C4×C59—82no—
184C4×C59—81no—
185C4×C59#45812yes—
186Dic59—81no—
187#3193—591no—
188C2×#481—591no—
189C2×#483D5911yescharacteristic, complemented
190#1821—591no—
191#1472—591no—
192#1471—591no—
193#1298—591no—
194C2×C59—22no—
195C2×C59—22no—
196C2×C59—41no—
197C2×C59#201911yescharacteristic
198C2×C59#201911yescharacteristic
199C2×C59#202211yescharacteristic
200D59—161no—
201C2×C22⋊Q8C2×D5911yescharacteristic
202C2×#121—591no—
203C22×C4⋊C4C2×D5911yescharacteristic
204#484—1181no—
205#483—22no—
206#483—22no—
207#481—1182no—
208#481—1182no—
209#480—1181no—
210#468—1182no—
211#450—1181no—
212#448—1181no—
213C2×#106—21no—
214C2×#106—1181no—
215C2×#103—21no—
216C2×#102—1181no—
217#407—591no—
218#407—591no—
219C2×C22⋊C8—591no—
220C2×C22⋊C8C2×D5911yescharacteristic
221#343—1182no—
222#324—1182no—
223#323—1182no—
224C59#999311yescharacteristic, complemented, Sylow
225C22×Q8C22⋊D5911yescharacteristic
226C22×D4—591no—
227C22×D4—1181no—
228C23×C4—21no—
229C23×C4C22×D5911yescharacteristic
230C2×SD16—1182no—
231C2×M16—592no—
232C2×M16—1181no—
233C22×C8—21no—
234C22×C8—1181no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—1181no—
240#121—1181no—
241#121—1181no—
242#121—1181no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4C22⋊D5912yes—
251C2×C4⋊C4C22⋊D5911yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1181no—
254#107—1182no—
255#106—24no—
256#106—1184no—
257#103—24no—
258#103—1182no—
259#102—1184no—
260#102—1182no—
261C22⋊C8—21no—
262C22⋊C8—592no—
263C22⋊C8—1181no—
264C22⋊C8—1181no—
265C22⋊C8—1181no—
266C22⋊C8—1181no—
267C22⋊C8—1181no—
268C22⋊C8C22⋊D5912yes—
269C24—1181no—
270C24#1174568711yescharacteristic
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×D4—1181no—
275C2×D4—1181no—
276C2×D4—4722no—
277C2×D4—2362no—
278C2×D4—2361no—
279C2×D4—2361no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—1181no—
288C22×C4D4×D5911yescharacteristic
289C22×C4D4×D5911yescharacteristic
290C22×C4D4×D5911yescharacteristic
291C22×C4C2×C22⋊D5911yescharacteristic
292C22×C4C2×C22⋊D5911yescharacteristic
293C22×C4C2×C22⋊D5911yescharacteristic
294SD16—4722no—
295M16—2362no—
296M16—2362no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—1182no—
300C2×C8—1182no—
301C2×C8—2361no—
302C2×C8—2362no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—4721no—
321C23—4721no—
322C23—2362no—
323C23—2361no—
324C23#1427369811yescharacteristic
325C23#1427369911yescharacteristic
326C23#1427369911yescharacteristic
327Q8—42no—
328Q8—42no—
329D4—4722no—
330D4—4722no—
331D4—2362no—
332D4—2362no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—4721no—
348C2×C4—2361no—
349C2×C4#1427369511yescharacteristic
350C2×C4#1427369511yescharacteristic
351C2×C4#1427369611yescharacteristic
352C2×C4#1427372311yescharacteristic
353C8—42no—
354C8—4721no—
355C8—2362no—
356C8—2362no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—9442no—
366C22—4721no—
367C22—4721no—
368C22—4721no—
369C22#1683731011yescharacteristic
370C22#1683766212yes—
371C22#1683766712yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—82no—
377C4—81no—
378C4—4721no—
379C4#1683759112yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—9441no—
384C2#2142520711yescharacteristic
385C2#2142520811yescharacteristic
386C2#2142522411yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC2×C59×#483C2
socleC22×C59#392
prime radicalC2×#14273701C22
hypercenterC2×#483D59
hypocenterC59#9993
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #21424176C2semidirect
4. #21422460C2semidirect
6. #21421984C2semidirect
7. C2×C59×#483C2semidirect
189. C2×#483D59semidirect
224. C59#9993semidirect

Maximal subgroups

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

subgroupindex
#214348982
C2×#168377202
#214241762
#214224602
#214224552
#214219842
C2×C59×#4832
#999359