Finite Group Databasebeta

← #28491900

Subgroups — #28491900

Order 15104.

404 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 75 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#21430269C211yesmaximal, characteristic
2#21422629C211yesmaximal, characteristic
3#21422616C211yesmaximal, characteristic
4#21422002C211yesmaximal, characteristic
5#21421982C211yesmaximal, characteristic
6#21421949C211yesmaximal, characteristic
7C59×#1996C211yesmaximal, characteristic
8#16837366C2211yescharacteristic
9#16837366C2211yescharacteristic
10#16837341—21no—
11#16837325—21no—
12#16837322—21no—
13#16837314—21no—
14#16837306—21no—
15#16837289—21no—
16#16837265C2211yescharacteristic
17C4×C22⋊Dic59C2211yescharacteristic
18#16837139—21no—
19#16837139—21no—
20#16837139—21no—
21#16837139—21no—
22C2×C22⋊Q8×C59C2211yescharacteristic
23C59×#380C2211yescharacteristic
24C4×C22⋊C4×C59C2211yescharacteristic
25C59×#342—21no—
26C59×#342—21no—
27C22×Dic118—21no—
28C2×#11745664—21no—
29C2×#11745664—21no—
30C2×C4⋊Dic59—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C22⋊(C59⋊C8)—21no—
37C22⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95D411yescharacteristic
46C59⋊#95D411yescharacteristic
47C59⋊#95D411yescharacteristic
48C59⋊#95D411yescharacteristic
49C22×Q8×C59Q811yescharacteristic, complemented
50C23×C4×C59C2311yescharacteristic
51C22⋊Q8×C59—21no—
52C22⋊Q8×C59—21no—
53C22⋊Q8×C59—21no—
54C22⋊Q8×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59Q811yescharacteristic, complemented
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C22⋊C4×C59D411yescharacteristic
62C2×C42×C59—21no—
63#95×C59—21no—
64#95×C59—21no—
65C24×C59C2×D411yescharacteristic
66C2×Q8×C59—21no—
67C2×Q8×C59—21no—
68C2×Q8×C59—81no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59—81no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×Q811yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C2×Dic118—42no—
86C2×Dic118—81no—
87C2×Dic118—81no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—81no—
93C22⋊C4×C59—22no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C42×C59—81no—
100C22⋊Dic59—42no—
101C22⋊Dic59—81no—
102C4⋊Dic59—81no—
103Dic59⋊C4—42no—
104Dic59⋊C4—81no—
105C4×Dic59—81no—
106#11745664—81no—
107#11745664—81no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59#12011yescharacteristic
115C23×C59C22⋊Q811yescharacteristic
116C23×C59#12411yescharacteristic
117Q8×C59—81no—
118Q8×C59—81no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59C22⋊Q811yescharacteristic
139C2×C4×C59C22⋊Q811yescharacteristic
140C2×C4×C59#12411yescharacteristic
141C2×C4×C59#12411yescharacteristic
142C2×Dic59—22no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154Dic118—84no—
155C59⋊C8—81no—
156C59⋊C8—81no—
157#10383—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59#39311yescharacteristic
169C22×C59#45511yescharacteristic
170C22×C59#45611yescharacteristic
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—81no—
176C4×C59—81no—
177C4×C59—82no—
178C4×C59—81no—
179C4×C59—81no—
180Dic59—42no—
181Dic59—81no—
182Dic59—81no—
183Dic59—81no—
184#2665—591no—
185#1996D5911yescharacteristic
186#1511—591no—
187#1508—591no—
188#1305—591no—
189#1297—591no—
190#1284—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—41no—
194C2×C59—41no—
195C2×C59#202811yescharacteristic
196C2×C59#203111yescharacteristic
197C2×C59#203211yescharacteristic
198C2×C22⋊Q8C2×D5911yescharacteristic
199#407—591no—
200#407—591no—
201#400—1181no—
202#396—1181no—
203#395—1181no—
204#393—1181no—
205#391—1181no—
206#387—1181no—
207#381—591no—
208#380C2×D5911yescharacteristic
209C4×C22⋊C4—591no—
210C4×C22⋊C4C2×D5911yescharacteristic
211#342—21no—
212#342—21no—
213#328—1181no—
214#328—1181no—
215#328—1181no—
216#328—1181no—
217C59#1038311yescharacteristic, complemented, Sylow
218C22×Q8—1181no—
219C22×Q8Dic11811yescharacteristic, complemented
220C23×C4C22×D5911yescharacteristic
221C2×M16—1181no—
222C2×M16—1181no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4Dic11811yescharacteristic, complemented
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4C22⋊D5911yescharacteristic
238C2×C22⋊C4C22⋊D5911yescharacteristic
239C2×C42—21no—
240C2×C42—1181no—
241C22⋊C8—1181no—
242C22⋊C8—1181no—
243C22⋊C8—1181no—
244C22⋊C8—1181no—
245#95—21no—
246#95—21no—
247#95—591no—
248#95—591no—
249#95—591no—
250#95—591no—
251#95—1181no—
252#95—1181no—
253#95—1181no—
254#95—1181no—
255#95—1181no—
256C24C2×C22⋊D5911yescharacteristic
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—81no—
260C2×Q8—4721no—
261C2×Q8—4721no—
262C2×Q8—2362no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—81no—
269C22×C4—1181no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4C2×Dic11811yescharacteristic
274C22×C4#1174568511yescharacteristic
275C22×C4D4×D5911yescharacteristic
276C22×C4D4×D5911yescharacteristic
277C22×C4#1174568711yescharacteristic
278C22×C4#1174568711yescharacteristic
279M16—4721no—
280M16—4721no—
281C2×C8—2361no—
282C2×C8—2361no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—81no—
288C4⋊C4—4721no—
289C4⋊C4—4721no—
290C4⋊C4—2362no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—4721no—
298C22⋊C4—2362no—
299C42—81no—
300C42—4721no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23#1427369511yescharacteristic
306C23#1427370611yescharacteristic
307C23#1427371411yescharacteristic
308Q8—81no—
309Q8—81no—
310Q8—4724no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—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—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4#1427370411yescharacteristic
343C2×C4#1427370411yescharacteristic
344C2×C4#1427371311yescharacteristic
345C2×C4#1427371311yescharacteristic
346C8—4721no—
347C8—4721no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#1683731411yescharacteristic
359C22#1683757911yescharacteristic
360C22#1683758411yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4—81no—
369C4—81no—
370C4—4721no—
371C4—4721no—
372C4—4721no—
373C4—2362no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2142525711yescharacteristic
379C2#2142527011yescharacteristic
380C2#2142527311yescharacteristic
subgroupG/H
centerC22#16837314
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1996C2
socleC22×C59#393
prime radicalC24×C59C2×D4
hypercenter#1996D59
hypocenterC59#10383
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
49. C22×Q8×C59Q8semidirect
57. C2×C4⋊C4×C59Q8semidirect
217. C59#10383semidirect
219. C22×Q8Dic118semidirect
232. C2×C4⋊C4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214302692
#214226292
#214226162
#214220022
#214219822
#214219492
C59×#19962
#1038359