Finite Group Databasebeta

← #28492402

Subgroups — #28492402

Order 15104.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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#21433893C211yesmaximal, characteristic
2#21422854C211yesmaximal, characteristic, complemented
3#21422787C211yesmaximal, characteristic, complemented
4#21422550C211yesmaximal, characteristic, complemented
5C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic
6#21421970C211yesmaximal, characteristic, complemented
7C59×#1800C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837702—22no—
10#16837692—21no—
11#16837682—21no—
12#16837442—21no—
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837407—21no—
16C2×C4×C59⋊C8C2211yescharacteristic
17#16837154—22no—
18#16837150—21no—
19#16837147—21no—
20#16837132—21no—
21#16837132—21no—
22D4⋊(C59⋊C8)—21no—
23D4⋊(C59⋊C8)—21no—
24C2×C4×D4×C59C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#423C2211yescharacteristic
27C2×#100×C59—21no—
28C59×#343—21no—
29C2×#11745664—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273727D411yescharacteristic
33#14273727D411yescharacteristic
34#14273719—21no—
35#14273706—21no—
36#14273706—21no—
37C4×Dic118—21no—
38C4×Dic118—21no—
39#14273686—21no—
40#14273660—22no—
41#14273658—22no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D412yes—
46#14273648—22no—
47#14273648—22no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8D411yescharacteristic
50C4×C59⋊C8D411yescharacteristic
51C22×D4×C59D411yescharacteristic
52C23×C4×C59—21no—
53C2×M16×C59—21no—
54C2×M16×C59—21no—
55C4×D4×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C2×C4⋊C4×C59D411yescharacteristic
60C2×C22⋊C4×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66#100×C59—42no—
67C24×C59—21no—
68C2×D4×C59—21no—
69C2×D4×C59—21no—
70C2×D4×C59—21no—
71C2×D4×C59—21no—
72C2×D4×C59—21no—
73C2×D4×C59Q1612yes—
74C22×C4×C59—21no—
75C22×C4×C59—41no—
76C22×C4×C59—41no—
77C22×C4×C59—81no—
78C22×C4×C59—81no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C2×Dic118—21no—
83C2×Dic118—21no—
84M16×C59—41no—
85M16×C59—41no—
86M16×C59—41no—
87M16×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C42×C59C2×D411yescharacteristic
95C42×C59C2×D411yescharacteristic
96C42×C59C2×D411yescharacteristic
97C42×C59#4111yescharacteristic
98C22⋊Dic59—41no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103Dic59⋊C4—41no—
104Dic59⋊C4—41no—
105C4×Dic59—41no—
106#11745664—41no—
107#11745664—41no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—42no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—81no—
114C23×C59—22no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—81no—
118C23×C59—81no—
119C23×C59#12311yescharacteristic
120D4×C59—22no—
121D4×C59—22no—
122D4×C59—41no—
123D4×C59—41no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—82no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59—81no—
132C2×C4×C59#12011yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12311yescharacteristic
137C2×C4×C59#12411yescharacteristic
138C2×C4×C59C2×Q1611yescharacteristic
139C2×C4×C59#13511yescharacteristic
140C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C59#13711yescharacteristic
142C2×Dic59—41no—
143C2×Dic59—41no—
144Dic118—41no—
145Dic118—41no—
146Dic118—81no—
147C8×C59—81no—
148C8×C59—81no—
149C59⋊C8—22no—
150C59⋊C8—41no—
151C59⋊C8—42no—
152C59⋊C8—81no—
153#10461—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—42no—
157C22×C59—82no—
158C22×C59—82no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59#39411yescharacteristic
162C22×C59#48411yescharacteristic
163C22×C59#48511yescharacteristic
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—81no—
168C4×C59#44911yescharacteristic
169C4×C59#45111yescharacteristic
170C4×C59#46711yescharacteristic
171C4×C59#46811yescharacteristic
172C4×C59#47111yescharacteristic
173C4×C59#47211yescharacteristic
174Dic59—81no—
175Dic59—81no—
176#3077—591no—
177#1800D5911yescharacteristic
178#1567—591no—
179#1550—591no—
180#1493—591no—
181C2×D4⋊C8—591no—
182#1292—591no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—82no—
186C2×C59#203611yescharacteristic
187C2×C59#205011yescharacteristic
188C2×C59#205111yescharacteristic
189#533—591no—
190C2×C4×D4C2×D5911yescharacteristic
191#478—1182no—
192#476—1181no—
193#474—1181no—
194#428—1181no—
195#423—591no—
196#423C2×D5911yescharacteristic
197#423C2×D5911yescharacteristic
198C2×C4⋊C8—591no—
199#418—1181no—
200C2×#100—21no—
201C2×C4×C8—591no—
202#343—21no—
203#335—1182no—
204#333—1181no—
205#332—1181no—
206Q8⋊C8—1181no—
207Q8⋊C8—1181no—
208D4⋊C8—1181no—
209D4⋊C8—1181no—
210C59#1046111yescharacteristic, complemented, Sylow
211C22×D4C22⋊D5911yescharacteristic
212C23×C4—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—1181no—
216C22×C8—1181no—
217C22×C8—1181no—
218C4⋊Q8—591no—
219C4⋊Q8—591no—
220#125—1181no—
221C22⋊Q8—1181no—
222C22⋊Q8—1181no—
223C4×Q8—1181no—
224C4×Q8—1181no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229#117—1181no—
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C22⋊C4—21no—
232C2×C42C22×D5911yescharacteristic
233#107—1182no—
234#106—1182no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—592no—
240C4⋊C8—1181no—
241C4⋊C8—1181no—
242C4⋊C8—1181no—
243#102—1182no—
244#102—1182no—
245#100—42no—
246C4×C8—591no—
247C4×C8—591no—
248C4×C8—1181no—
249C24—21no—
250C2×Q8—1181no—
251C2×Q8—1181no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4#1174567212yes—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4#1174568511yescharacteristic
264C22×C4#1174568511yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266M16—41no—
267M16—41no—
268M16—41no—
269M16—41no—
270M16—2361no—
271M16—2361no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—1181no—
275C2×C8—1182no—
276C2×C8—1181no—
277C2×C8—4721no—
278C2×C8—2362no—
279C2×C8—2361no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—1181no—
283C4⋊C4—1181no—
284C4⋊C4—1181no—
285C4⋊C4—1181no—
286C4⋊C4—2361no—
287C4⋊C4—2361no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—2361no—
291C42—2361no—
292C42D4×D5911yescharacteristic
293C42D4×D5911yescharacteristic
294C42D4×D5911yescharacteristic
295C42#1174568711yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23—81no—
300C23—81no—
301C23#1427371011yescharacteristic
302Q8—4721no—
303Q8—2361no—
304Q8—2361no—
305D4—22no—
306D4—22no—
307D4—41no—
308D4—41no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4#1427369511yescharacteristic
320C2×C4#1427369911yescharacteristic
321C2×C4#1427370011yescharacteristic
322C2×C4#1427370011yescharacteristic
323C2×C4#1427370911yescharacteristic
324C2×C4#1427371311yescharacteristic
325C2×C4C2×#1174567211yescharacteristic
326C2×C4#1427374711yescharacteristic
327C2×C4#1427375311yescharacteristic
328C2×C4#1427375711yescharacteristic
329C8—81no—
330C8—81no—
331C8—1182no—
332C8—4721no—
333C8—2361no—
334C8—2362no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—82no—
339C22—82no—
340C22—81no—
341C22—81no—
342C22#1683731711yescharacteristic
343C22#1683773711yescharacteristic
344C22#1683774311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—81no—
349C4—4721no—
350C4—4721no—
351C4#1683754511yescharacteristic
352C4#1683755711yescharacteristic
353C4#1683764111yescharacteristic
354C4#1683764611yescharacteristic
355C4#1683766411yescharacteristic
356C4#1683767111yescharacteristic
357C2—21no—
358C2—21no—
359C2—82no—
360C2#2142529911yescharacteristic
361C2#2142538111yescharacteristic
362C2#2142538811yescharacteristic
subgroupG/H
centerC22#16837317
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#394
prime radicalC22×D4×C59D4
hypercenter#1800D59
hypocenterC59#10461
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21422854C2semidirect
3. #21422787C2semidirect
4. #21422550C2semidirect
6. #21421970C2semidirect
210. C59#10461semidirect

Maximal subgroups

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

subgroupindex
#214338932
#214228542
#214227872
#214225502
C2×D4⋊(C59⋊C8)2
#214219702
C59×#18002
#1046159