Finite Group Databasebeta

← #28489826

Subgroups — #28489826

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic
2#21429092C211yesmaximal, characteristic
3#21425045C211yesmaximal, characteristic, complemented
4C2×#16837148C211yesmaximal, characteristic, complemented
5C2×#16837131C211yesmaximal, characteristic
6#21421936C211yesmaximal, characteristic, complemented
7C2×C59×#487C211yesmaximal, characteristic, complemented
8C2×#14273716—21no—
9C2×C4×D236C2211yescharacteristic
10#16837689—22no—
11#16837689—22no—
12C2×#14273658—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273647—21no—
16#16837328—22no—
17#16837309—21no—
18#16837259—22no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837164—22no—
21#16837148—22no—
22#16837131—22no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×C4⋊Q8×C59C2211yescharacteristic
25C59×#487—22no—
26C59×#487—22no—
27C2×#103×C59—22no—
28C2×C4×C8×C59C2211yescharacteristic
29C22×D236D411yescharacteristic
30C22×C4×D59—21no—
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33#14273716—42no—
34#14273716—42no—
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59D412yes—
40C2×C4⋊Dic59D411yescharacteristic
41C2×(C2×D59)⋊C4—24no—
42C2×(C2×D59)⋊C4—21no—
43C2×C4×Dic59—21no—
44#14273658—24no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)D412yes—
49#14273647—24no—
50C59⋊#95—24no—
51C59⋊#95—21no—
52C22×Q8×C59—21no—
53C22×Q8×C59—21no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59D412yes—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59D412yes—
67C2×C42×C59C2311yescharacteristic
68#103×C59—28no—
69C4×C8×C59—21no—
70C4×C8×C59D412yes—
71C23×D59—21no—
72C2×Q8×C59—22no—
73C2×Q8×C59—44no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—41no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C22×C4×C59—22no—
82C22×C4×C59—41no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×Dic59—21no—
87C22×Dic59—22no—
88C2×D236—21no—
89C2×D236—21no—
90C2×D236—21no—
91C2×D236—21no—
92C2×D236—21no—
93C2×D236SD1612yes—
94C2×C4×D59—41no—
95C2×C4×D59—41no—
96C2×C4×D59—42no—
97C2×C4×D59—42no—
98C2×C8×C59—22no—
99C2×C8×C59—42no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112(C2×D59)⋊C4—48no—
113(C2×D59)⋊C4—41no—
114(C2×D59)⋊C4—41no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59—22no—
118C4⋊Dic59—22no—
119C4×Dic59—41no—
120C4×Dic59—41no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—42no—
126C23×C59#12011yescharacteristic
127C22×D59—22no—
128C22×D59—41no—
129C22×D59—42no—
130C22×D59—41no—
131C22×D59—42no—
132Q8×C59—44no—
133Q8×C59—82no—
134Q8×C59—81no—
135Q8×C59—81no—
136Q8×C59—81no—
137Q8×C59—81no—
138C2×C4×C59—22no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—82no—
144C2×C4×C59#12011yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12111yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12711yescharacteristic
150C2×C4×C59C2×SD1611yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59#13512yes—
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59#13711yescharacteristic
155C2×C4×C59#13712yes—
156C2×Dic59—41no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59—42no—
161D236—22no—
162D236—22no—
163D236—41no—
164D236—41no—
165C4×D59—82no—
166C4×D59—81no—
167C4×D59—81no—
168C8×C59—42no—
169C59⋊C8—42no—
170C59⋊C8—81no—
171#10058—591nomaximal, Sylow
172C22×C59#39211yescharacteristic
173C22×C59#45011yescharacteristic
174C22×C59#45211yescharacteristic
175C22×C59#45212yes—
176C22×C59#45412yes—
177C2×D59—44no—
178C2×D59—42no—
179C2×D59—82no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—82no—
185C4×C59—82no—
186C4×C59#44811yescharacteristic
187C4×C59#44911yescharacteristic
188C4×C59#46412yes—
189C4×C59#46511yescharacteristic
190C4×C59#46811yescharacteristic
191C4×C59#49511yescharacteristic
192C4×C59#49711yescharacteristic
193Dic59—82no—
194Dic59—81no—
195C2×#487D5911yescharacteristic, complemented
196C2×#476—591no—
197#2547—591no—
198#1987—591no—
199C2×#332—591no—
200C2×D4⋊C8—591no—
201#1279—591no—
202C2×C59#163512yes—
203C2×C59#163812yes—
204C2×C59#200011yescharacteristic
205C2×C59#200311yescharacteristic
206C2×C59#201511yescharacteristic
207D59—82no—
208C2×C4⋊Q8C2×D5911yescharacteristic
209C2×C4⋊Q8C2×D5911yescharacteristic
210C2×#124—1181no—
211C2×C4×D4—591no—
212#487—22no—
213#487—22no—
214#476—1182no—
215#476—1182no—
216C2×#106—1181no—
217C2×C4⋊C8—591no—
218C2×C4⋊C8—591no—
219C2×#103—22no—
220C2×#102—1181no—
221C2×C4×C8C2×D5911yescharacteristic
222#397—1182no—
223#391—1181no—
224#380—1182no—
225C4×C4⋊C4—591no—
226#340—1182no—
227#332—1182no—
228D4⋊C8—1182no—
229C59#1005811yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×Q8—21no—
232C22×D4—591no—
233C23×C4—1181no—
234C22×C8—21no—
235C22×C8—1181no—
236C22×C8—1181no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D5912yes—
245#124—2362no—
246#124—2362no—
247C4×D4—1181no—
248C4×D4—1181no—
249C4×D4—1181no—
250C4×D4—1181no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—591no—
255C2×C4⋊C4—592no—
256C2×C4⋊C4C22⋊D5912yes—
257C2×C22⋊C4—1184no—
258C2×C22⋊C4—1181no—
259C2×C42—1181no—
260C2×C42C22×D5911yescharacteristic
261#106—1184no—
262C4⋊C8—592no—
263C4⋊C8—1181no—
264C4⋊C8—1181no—
265C4⋊C8—1181no—
266#103—28no—
267#102—1184no—
268C4×C8—21no—
269C4×C8C22⋊D5912yes—
270#95—1184no—
271#95—1181no—
272C24—1181no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—592no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—1181no—
291C22×C4—1182no—
292C22×C4—2361no—
293C22×C4—2361no—
294C22×C4—2362no—
295C22×C4—2362no—
296C22×C4D4×D5911yescharacteristic
297C22×C4D4×D5911yescharacteristic
298C22×C4C2×C22⋊D5911yescharacteristic
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—1182no—
302C2×C8—2361no—
303C2×C8—2362no—
304C2×C8—2361no—
305C2×C8—2361no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—1181no—
315C4⋊C4—1181no—
316C4⋊C4—1182no—
317C4⋊C4—1182no—
318C22⋊C4—2368no—
319C22⋊C4—2361no—
320C22⋊C4—2361no—
321C42—2361no—
322C42—2361no—
323C42D4×D5911yescharacteristic
324C42#1174568711yescharacteristic
325C42C2×C22⋊D5911yescharacteristic
326C42C2×C22⋊D5911yescharacteristic
327C23—1182no—
328C23—2361no—
329C23—2362no—
330C23—2361no—
331C23—2362no—
332C23#1427369511yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—1182no—
340D4—1182no—
341D4—2361no—
342D4—2361no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—4721no—
350C2×C4—4721no—
351C2×C4—4722no—
352C2×C4—2362no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2361no—
356C2×C4—2362no—
357C2×C4#1427369511yescharacteristic
358C2×C4#1427369611yescharacteristic
359C2×C4#1427369811yescharacteristic
360C2×C4#1427369911yescharacteristic
361C2×C4#1427369911yescharacteristic
362C2×C4#1427372311yescharacteristic
363C2×C4SD16×D5911yescharacteristic
364C2×C4#1427374811yescharacteristic
365C2×C4#1427375012yes—
366C2×C4#1427375411yescharacteristic
367C2×C4#1427375611yescharacteristic
368C2×C4#1427375912yes—
369C8—42no—
370C8—4721no—
371C8—2362no—
372C22—4721no—
373C22—4722no—
374C22—4721no—
375C22—2364no—
376C22—2362no—
377C22#1683731011yescharacteristic
378C22#1683755111yescharacteristic
379C22#1683756111yescharacteristic
380C22#1683756312yes—
381C22#1683757412yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—4721no—
387C4—4722no—
388C4#1683753911yescharacteristic
389C4#1683754611yescharacteristic
390C4#1683762612yes—
391C4#1683762811yescharacteristic
392C4#1683764511yescharacteristic
393C4#1683778711yescharacteristic
394C4#1683779511yescharacteristic
395C2—4722no—
396C2#2142324412yes—
397C2#2142326512yes—
398C2#2142509611yescharacteristic
399C2#2142511811yescharacteristic
400C2#2142518711yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#487C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenterC2×#487D59
hypocenterC59#10058
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #21425045C2semidirect
4. C2×#16837148C2semidirect
6. #21421936C2semidirect
7. C2×C59×#487C2semidirect
195. C2×#487D59semidirect
229. C59#10058semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
#214290922
#214250452
C2×#168371482
C2×#168371312
#214219362
C2×C59×#4872
#1005859