Finite Group Databasebeta

← #28486540

Subgroups — #28486540

Order 15104.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#21433176C211yesmaximal, characteristic, complemented
2#21429005C211yesmaximal, characteristic
3#21424089C211yesmaximal, characteristic, complemented
4#21422521C211yesmaximal, characteristic
5#21421956C211yesmaximal, characteristic, complemented
6#21421955C211yesmaximal, characteristic, complemented
7C59×#1486C211yesmaximal, characteristic
8C2×#14273706—21no—
9C2×C4×Dic118C2211yescharacteristic
10C8×Dic118—22no—
11#16837526—22no—
12#16837455C2211yescharacteristic
13#16837452—22no—
14#16837426C2211yescharacteristic
15C2×#14273645—21no—
16#16837333C412yes—
17#16837316C412yes—
18#16837290—22no—
19#16837268C2211yescharacteristic
20#16837262—21no—
21#16837170—21no—
22D4⋊(C59⋊C8)—22no—
23C2×C4×D4×C59C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×D4⋊C8—22no—
27C22×Dic118D411yescharacteristic
28C2×#11745664—21no—
29#14273706—42no—
30#14273706—42no—
31C4×Dic118—21no—
32C4×Dic118—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59D411yescharacteristic
37C2×Dic59⋊C4—22no—
38C2×Dic59⋊C4C2×C412yes—
39C2×C22⋊Dic59—22no—
40C2×C22⋊Dic59—22no—
41C2×C4×Dic59D412yes—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45Dic59⋊C8—24no—
46#14273645—42no—
47C22⋊(C59⋊C8)—21no—
48C4×C59⋊C8D412yes—
49C8×Dic59—24no—
50C59⋊#95—21no—
51C59⋊#95—22no—
52C59⋊#95C2×C412yes—
53C22×D4×C59C2×C411yescharacteristic
54C23×C4×C59—21no—
55C2×M16×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C2×C4⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59—21no—
62C2×C42×C59C2311yescharacteristic
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66C22⋊C8×C59—21no—
67C4×C8×C59D412yes—
68C24×C59—21no—
69C2×D4×C59—21no—
70C2×D4×C59—21no—
71C2×D4×C59—42no—
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59—41no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×Dic59—22no—
83C22×Dic59C22⋊C412yes—
84C22×Dic59C22⋊C412yes—
85C2×Dic118—21no—
86C2×Dic118—21no—
87C2×Dic118—21no—
88C2×Dic118—21no—
89C2×Dic118D812yes—
90C2×Dic118SD1612yes—
91M16×C59—41no—
92M16×C59—41no—
93C2×C8×C59—22no—
94C2×C8×C59—41no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C42×C59C2×D411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59#4111yescharacteristic
102C42×C59#4111yescharacteristic
103C22⋊Dic59—44no—
104C22⋊Dic59—44no—
105C4⋊Dic59—21no—
106C4⋊Dic59—21no—
107C4⋊Dic59—41no—
108C4⋊Dic59—41no—
109Dic59⋊C4—22no—
110Dic59⋊C4—22no—
111Dic59⋊C4—44no—
112C4×Dic59—22no—
113C4×Dic59—22no—
114#11745664—41no—
115#11745664—41no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8—41no—
118C23×C59—22no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59C2×C22⋊C411yescharacteristic
126D4×C59—22no—
127D4×C59—22no—
128D4×C59—41no—
129D4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59#13511yescharacteristic
149C2×Dic59—22no—
150C2×Dic59—22no—
151C2×Dic59—22no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59#10214yes—
157Dic118—22no—
158Dic118—22no—
159Dic118—22no—
160Dic118—22no—
161C8×C59—42no—
162C8×C59—81no—
163C59⋊C8—42no—
164C59⋊C8—81no—
165#9486—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59#38611yescharacteristic
176C22×C59C2×#10211yescharacteristic
177C22×C59#41611yescharacteristic
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—81no—
183C4×C59#46511yescharacteristic
184C4×C59#46511yescharacteristic
185C4×C59#46611yescharacteristic
186C4×C59#46611yescharacteristic
187Dic59—24no—
188Dic59—42no—
189Dic59—82no—
190#2982—591no—
191#2536—591no—
192#1801—591no—
193#1486—591no—
194#1486D5911yescharacteristic
195#1286—591no—
196#1286—591no—
197C2×C59—21no—
198C2×C59—21no—
199C2×C59—82no—
200C2×C59#189111yescharacteristic
201C2×C59#189311yescharacteristic
202C2×C59#191211yescharacteristic
203C2×C22⋊Q8—1181no—
204C2×C4×Q8—591no—
205C2×C4×D4C2×D5911yescharacteristic
206C8×Q8—1182no—
207C8×Q8—1182no—
208#431—591no—
209#431—1182no—
210#431C2×D5911yescharacteristic
211#423—591no—
212#423C2×D5911yescharacteristic
213C2×#101—1181no—
214#398—592no—
215#394—592no—
216#388—1182no—
217#382—591no—
218#380—1181no—
219#343—1181no—
220D4⋊C8—22no—
221D4⋊C8—1182no—
222C59#948611yescharacteristic, complemented, Sylow
223C22×Q8—591no—
224C22×D4C4×D5911yescharacteristic
225C23×C4—21no—
226C2×M16—21no—
227C2×M16—1181no—
228C22⋊Q8—2362no—
229C22⋊Q8—2362no—
230C4×Q8—1181no—
231C4×Q8—1181no—
232C4×Q8—1181no—
233C4×Q8—1181no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—592no—
239C2×C4⋊C4—591no—
240C2×C4⋊C4—1182no—
241C2×C4⋊C4—1181no—
242C2×C4⋊C4C4×D5911yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1182no—
245C2×C22⋊C4—1182no—
246C2×C42—592no—
247C2×C42C22×D5911yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1184no—
252C4⋊C8—1181no—
253C4⋊C8—1181no—
254C4⋊C8—1181no—
255#101—2362no—
256C22⋊C8—21no—
257C22⋊C8—1181no—
258C4×C8—592no—
259C4×C8—1184no—
260C4×C8C22⋊D5912yes—
261#95—592no—
262#95—1182no—
263#95—1181no—
264C24—21no—
265C2×Q8—592no—
266C2×Q8—592no—
267C2×Q8—1181no—
268C2×Q8—1181no—
269C2×Q8—1181no—
270C2×Q8—1181no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—592no—
282C22×C4—592no—
283C22×C4—1182no—
284C22×C4C2×C4×D5911yescharacteristic
285C22×C4D4×D5911yescharacteristic
286C22×C4D4×D5911yescharacteristic
287M16—41no—
288M16—41no—
289M16—2361no—
290M16—2361no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—1182no—
294C2×C8—2361no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—1182no—
298C4⋊C4—1182no—
299C4⋊C4—1181no—
300C4⋊C4—1181no—
301C4⋊C4—2364no—
302C4⋊C4—2361no—
303C4⋊C4—2361no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—2364no—
307C22⋊C4—2364no—
308C42—1182no—
309C42—1182no—
310C42#1174568511yescharacteristic
311C42D4×D5911yescharacteristic
312C42#1174568711yescharacteristic
313C42C2×C22⋊D5911yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22⋊C4×D5911yescharacteristic
322Q8—1182no—
323Q8—1182no—
324Q8—1182no—
325Q8—1182no—
326D4—22no—
327D4—22no—
328D4—41no—
329D4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—594no—
340C2×C4—1182no—
341C2×C4—1182no—
342C2×C4—1182no—
343C2×C4—2362no—
344C2×C4—2362no—
345C2×C4—2362no—
346C2×C4—2362no—
347C2×C4(C22⋊D59)⋊C411yescharacteristic
348C2×C4C4×C22⋊D5911yescharacteristic
349C2×C4#1427369511yescharacteristic
350C2×C4#1427369811yescharacteristic
351C2×C4#1427370011yescharacteristic
352C2×C4#1427370911yescharacteristic
353C2×C4D8×D5911yescharacteristic
354C2×C4SD16×D5911yescharacteristic
355C2×C4#1427374611yescharacteristic
356C2×C4#1427374811yescharacteristic
357C8—42no—
358C8—81no—
359C8—4721no—
360C8—2362no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1683727811yescharacteristic
371C22#102×D5911yescharacteristic
372C22#1683739511yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—1184no—
379C4—4722no—
380C4—2362no—
381C4#1683762811yescharacteristic
382C4#1683763011yescharacteristic
383C4#1683763511yescharacteristic
384C4#1683763711yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#2142451111yescharacteristic
389C2#2142452111yescharacteristic
390C2#2142463011yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1486C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1486D59
hypocenterC59#9486
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #21433176C2semidirect
3. #21424089C2semidirect
5. #21421956C2semidirect
6. #21421955C2semidirect
222. C59#9486semidirect

Maximal subgroups

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

subgroupindex
#214331762
#214290052
#214240892
#214225212
#214219562
#214219552
C59×#14862
#948659