Finite Group Databasebeta

← #28499053

Subgroups — #28499053

Order 15104.

632 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14273644C211yesmaximal, characteristic, complemented
2#21422414C211yesmaximal, characteristic, complemented
3#21422126C212yesmaximal, complemented
4#21422099C212yesmaximal, complemented
5C59×#2388C211yesmaximal, characteristic
6C2×#14273644—21no—
7C2×C22⋊(C59⋊C8)—21no—
8C4×#11745664—21no—
9#16837171C2212yes—
10#16837128C2212yes—
11#16837126—21no—
12C22×#41×C59C411yescharacteristic
13C2×C4×D4×C59—21no—
14C2×C4×D4×C59C2211yescharacteristic
15C59×#395C2211yescharacteristic
16C59×#394—21no—
17C59×#394C2211yescharacteristic
18C59×#393C411yescharacteristic
19C59×#392—21no—
20C59×#392C411yescharacteristic
21C4×C22⋊C4×C59—21no—
22C4×C22⋊C4×C59C411yescharacteristic
23C2×#11745664—22no—
24C22×C59⋊C8—41no—
25#14273644—24no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—41no—
28#14273638—41no—
29C4×C59⋊C8—41no—
30C2×#41×C59—44no—
31C2×#41×C59—41no—
32C2×#41×C59—41no—
33C22×Q8×C59C2×C411yescharacteristic
34C22×D4×C59—21no—
35C22×D4×C59—21no—
36C22×D4×C59C2×C411yescharacteristic
37C22×D4×C59D411yescharacteristic
38C23×C4×C59—21no—
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C4×D4×C59—44no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59C2311yescharacteristic
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59—21no—
54C2×C42×C59D411yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59C2×C411yescharacteristic
58#95×C59C2×C411yescharacteristic
59C24×C59—21no—
60C24×C59C22⋊C411yescharacteristic
61C24×C59C22⋊C411yescharacteristic
62#41×C59—44no—
63#41×C59—44no—
64#41×C59—44no—
65#41×C59—44no—
66C2×Q8×C59—42no—
67C2×Q8×C59—41no—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—41no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—41no—
74C2×D4×C59—42no—
75C2×D4×C59—42no—
76C2×D4×C59—42no—
77C2×D4×C59—42no—
78C2×D4×C59—42no—
79C22×C4×C59—24no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—42no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59C22⋊C411yescharacteristic
93C22×C4×C59C22⋊C411yescharacteristic
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C4⋊C4×C59—41no—
98C4⋊C4×C59—42no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—42no—
102C22⋊C4×C59—42no—
103C22⋊C4×C59—42no—
104C22⋊C4×C59—42no—
105C22⋊C4×C59—42no—
106C42×C59—21no—
107C42×C59—21no—
108C42×C59—42no—
109#11745664—24no—
110#11745664—42no—
111C2×C59⋊C8—42no—
112C2×C59⋊C8—42no—
113C2×C59⋊C8—82no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—22no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59#10412yes—
131C23×C59C2×C22⋊C411yescharacteristic
132Q8×C59—42no—
133Q8×C59—42no—
134D4×C59—44no—
135D4×C59—42no—
136D4×C59—48no—
137D4×C59—48no—
138D4×C59—42no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—22no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—44no—
148C2×C4×C59—42no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—44no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—81no—
163C2×C4×C59#9911yescharacteristic
164C2×C4×C59#9911yescharacteristic
165C2×C4×C59#13512yes—
166C59⋊C8—44no—
167C59⋊C8—82no—
168#11733—591nomaximal, Sylow
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—22no—
172C22×C59—42no—
173C22×C59—42no—
174C22×C59—42no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—42no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59C2×#9911yescharacteristic
189C22×C59#41611yescharacteristic
190C22×C59C2×#10411yescharacteristic
191C4×C59—21no—
192C4×C59—22no—
193C4×C59—42no—
194C4×C59—41no—
195C4×C59—42no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—81no—
199C4×C59AΣL(1,𝔽16)12yes—
200#2388D5911yescharacteristic
201C4×#100—591no—
202#1459—591no—
203#1354—592no—
204#1345—592no—
205C2×C59—21no—
206C2×C59—21no—
207C2×C59—42no—
208C2×C59—81no—
209C2×C59—82no—
210C2×C59#150111yescharacteristic
211C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
212C2×C59#213211yescharacteristic
213C22×#41Dic5911yescharacteristic
214C2×C4×D4—21no—
215C2×C4×D4C2×D5911yescharacteristic
216C2×#100—1181no—
217C2×C22⋊C8—1181no—
218C4×M16—1181no—
219#395C2×D5911yescharacteristic
220#394—21no—
221#394C2×D5911yescharacteristic
222#393Dic5911yescharacteristic
223#392—21no—
224#392Dic5911yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4Dic5911yescharacteristic
227#343—592no—
228#324—592no—
229#323—1181no—
230C59#1173311yescharacteristic, complemented, Sylow
231C2×#41—44no—
232C2×#41—41no—
233C2×#41—41no—
234C22×Q8C2×Dic5911yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic5911yescharacteristic
238C22×D4C22⋊D5911yescharacteristic
239C23×C4—21no—
240C23×C4C2×Dic5911yescharacteristic
241C23×C4C22⋊D5911yescharacteristic
242C2×M16—1182no—
243C22×C8—2361no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22×D5911yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4C2×Dic5911yescharacteristic
255C2×C22⋊C4C22⋊D5911yescharacteristic
256C2×C42—21no—
257C2×C42C22⋊D5911yescharacteristic
258#100—1184no—
259C22⋊C8—1182no—
260C22⋊C8—2361no—
261#97—2361no—
262C4×C8—2361no—
263#95—21no—
264#95—21no—
265#95C2×Dic5911yescharacteristic
266#95C2×Dic5911yescharacteristic
267C24—21no—
268C24C22⋊Dic5911yescharacteristic
269C24C22⋊Dic5911yescharacteristic
270#41—44no—
271#41—44no—
272#41—44no—
273#41—44no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C22⋊Dic5911yescharacteristic
301C22×C4C22⋊Dic5911yescharacteristic
302C22×C4C22×Dic5911yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304C22×C4C2×C22⋊D5911yescharacteristic
305M16—1184no—
306M16—2362no—
307C2×C8—4722no—
308C2×C8—2362no—
309C2×C8—2362no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C42—21no—
320C42—21no—
321C42—42no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23#1427365412yes—
339C23C2×C22⋊Dic5911yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—48no—
345D4—48no—
346D4—42no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4#1427364211yescharacteristic
372C2×C4#1427364211yescharacteristic
373C2×C4#1427374712yes—
374C8—4722no—
375C8—2364no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C2×#1427364211yescharacteristic
396C22#1683739811yescharacteristic
397C22C2×#1427365411yescharacteristic
398C4—21no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—81no—
406C4#1683718912yes—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—82no—
411C2—81no—
412C2#2142258711yescharacteristic
413C2C2×#1683718911yescharacteristic
414C2#2142578111yescharacteristic
subgroupG/H
centerC2×C4#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
Fitting, prime radicalC59×#2388C2
socleC22×C59C2×#99
cosocleC2×C4⋊C4×C59C23
hypercenter#2388D59
hypocenterC59#11733
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. C4×#14273644C2semidirect
2. #21422414C2semidirect
3. #21422126C2semidirect
4. #21422099C2semidirect
230. C59#11733semidirect

Maximal subgroups

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

subgroupindex
C4×#142736442
#214224142
#214221262
#214220992
C59×#23882
#1173359