Finite Group Databasebeta

← #28483892

Subgroups — #28483892

Order 15104.

476 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 109 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#21434104C211yesmaximal, characteristic
2#21433036C211yesmaximal, characteristic, complemented
3#21422754C211yesmaximal, characteristic, complemented
4#21422521C211yesmaximal, characteristic
5#21421993C211yesmaximal, characteristic, complemented
6#21421941C211yesmaximal, characteristic, complemented
7C59×#1955C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732C412yes—
10#16837692—21no—
11#16837607—21no—
12#16837553C412yes—
13#16837530—22no—
14#16837530—22no—
15C4×#11745670—22no—
16#16837455C2211yescharacteristic
17#16837455C2211yescharacteristic
18#16837428—21no—
19#16837426C2211yescharacteristic
20#16837357—21no—
21#16837150—21no—
22#16837147—21no—
23D4⋊(C59⋊C8)—22no—
24C2×C4×D4×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C2×C22⋊C8×C59—21no—
27C2×C22⋊C8×C59—21no—
28C2×#97×C59C2211yescharacteristic
29C2×#11745670—22no—
30C2×#11745664—21no—
31#14273727D411yescharacteristic
32#14273727D411yescharacteristic
33#14273719—21no—
34#14273706—21no—
35#14273706C2×C412yes—
36C4×Dic118—21no—
37C4×Dic118D412yes—
38#14273686—21no—
39#14273658—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)D412yes—
44Dic59⋊C8—24no—
45#14273650—22no—
46#14273648—22no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)C2×C412yes—
49#14273637—22no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C8×Dic59—22no—
53C22×D4×C59C2×C411yescharacteristic
54C23×C4×C59—21no—
55C22×C8×C59—21no—
56C22×C8×C59D411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C2×C4⋊C4×C59C2×C411yescharacteristic
63C2×C22⋊C4×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C22⋊C8×C59—42no—
68C22⋊C8×C59—42no—
69#97×C59—21no—
70#97×C59—21no—
71C24×C59—21no—
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—42no—
75C22×C4×C59—21no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59#4111yescharacteristic
85C2×Dic118—21no—
86C2×Dic118C22⋊C412yes—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—21no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—41no—
95C2×C8×C59SD1612yescomplemented
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59#4111yescharacteristic
104C22⋊Dic59—22no—
105#11745670—44no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59C22⋊C412yes—
110Dic59⋊C4—22no—
111Dic59⋊C4—22no—
112C4×Dic59—22no—
113#11745664—41no—
114#11745664—41no—
115C2×C59⋊C8—22no—
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×C59C4×D411yescharacteristic
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×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×C4×C59C2×SD1611yescharacteristic
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—22no—
150C2×Dic59—22no—
151Dic118—42no—
152Dic118—81no—
153C8×C59—22no—
154C8×C59—41no—
155C8×C59—81no—
156C59⋊C8—42no—
157C59⋊C8—42no—
158C59⋊C8—81no—
159#9050—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—42no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59#38611yescharacteristic
170C22×C59C4×SD1611yescharacteristic
171C22×C59#44011yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—81no—
177C4×C59#44811yescharacteristic
178C4×C59#44911yescharacteristic
179C4×C59#46011yescharacteristic
180C4×C59#46411yescharacteristic
181Dic59—42no—
182Dic59—42no—
183#3101—591no—
184#2961—591no—
185#1955D5911yescharacteristic
186#1542—591no—
187#1486—591no—
188#1301—591no—
189#1281—591no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—82no—
193C2×C59#188011yescharacteristic
194C2×C59#188111yescharacteristic
195C2×C59#189111yescharacteristic
196#533—591no—
197C2×C4×D4C2×D5911yescharacteristic
198#483—592no—
199#476—1181no—
200#461—1181no—
201#450—592no—
202#446—1182no—
203#446—1182no—
204C4×SD16—1182no—
205#431—591no—
206#431—591no—
207#424—1181no—
208#423—591no—
209C2×C4⋊C8C2×D5911yescharacteristic
210C2×C22⋊C8—21no—
211C2×C22⋊C8—21no—
212#405—1181no—
213C2×#97C2×D5911yescharacteristic
214#333—1181no—
215#332—1181no—
216D4⋊C8—1182no—
217C59#905011yescharacteristic, complemented, Sylow
218C22×D4C4×D5911yescharacteristic
219C23×C4—21no—
220C2×SD16—1182no—
221C2×M16—1181no—
222C22×C8—21no—
223C22×C8C22⋊D5911yescharacteristic
224C22×C8C22⋊D5911yescharacteristic
225C4⋊Q8—591no—
226C4⋊Q8—591no—
227#125—1181no—
228C22⋊Q8—592no—
229C22⋊Q8—1181no—
230C4×Q8—592no—
231C4×Q8—1181no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—1181no—
237C2×C4⋊C4C4×D5911yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C22×D5911yescharacteristic
240#106—1182no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—592no—
244C4⋊C8—1184no—
245C4⋊C8—1181no—
246C4⋊C8—1181no—
247C4⋊C8—1181no—
248#103—1182no—
249#102—1182no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—592no—
253C22⋊C8—1181no—
254#97—21no—
255#97—21no—
256#97—1182no—
257C4×C8—1182no—
258C4×C8—1181no—
259C4×C8—1181no—
260C24—21no—
261C2×Q8—592no—
262C2×Q8—1181no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—42no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4#1174568511yescharacteristic
275C22×C4C2×C22⋊D5911yescharacteristic
276SD16—2364no—
277M16—2361no—
278M16—2361no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—1182no—
288C2×C8—1182no—
289C2×C8—2361no—
290C2×C8#1174567012yescomplemented
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—592no—
294C4⋊C4—1182no—
295C4⋊C4—1181no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—1182no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—1182no—
302C42—1182no—
303C42D4×D5911yescharacteristic
304C42D4×D5911yescharacteristic
305C42D4×D5911yescharacteristic
306C42#1174568711yescharacteristic
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C4×C22⋊D5911yescharacteristic
315Q8—4721no—
316Q8—2362no—
317D4—22no—
318D4—22no—
319D4—41no—
320D4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—1182no—
331C2×C4—1182no—
332C2×C4C22⋊C4×D5911yescharacteristic
333C2×C4(C22⋊D59)⋊C411yescharacteristic
334C2×C4#1427369511yescharacteristic
335C2×C4#1427369811yescharacteristic
336C2×C4#1427370011yescharacteristic
337C2×C4#1427370911yescharacteristic
338C2×C4C2×#1174567011yescharacteristic
339C2×C4#1427374711yescharacteristic
340C2×C4#1427375311yescharacteristic
341C2×C4#1427375711yescharacteristic
342C8—22no—
343C8—41no—
344C8—81no—
345C8—4721no—
346C8—2362no—
347C8—2362no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1683727811yescharacteristic
358C22C4×#1174567011yescharacteristic
359C22#1683750211yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—2362no—
366C4—2362no—
367C4#1683754011yescharacteristic
368C4#1683754511yescharacteristic
369C4#1683759911yescharacteristic
370C4#1683762111yescharacteristic
371C2—21no—
372C2—21no—
373C2—82no—
374C2#2142445411yescharacteristic
375C2#2142446111yescharacteristic
376C2#2142451111yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1955C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1955D59
hypocenterC59#9050
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #21433036C2semidirect
3. #21422754C2semidirect
5. #21421993C2semidirect
6. #21421941C2semidirect
95. C2×C8×C59SD16semidirect
217. C59#9050semidirect
290. C2×C8#11745670semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214330362
#214227542
#214225212
#214219932
#214219412
C59×#19552
#905059