Finite Group Databasebeta

← #28480463

Subgroups — #28480463

Order 15104.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16838117C211yesmaximal, characteristic
2#21422782C212yesmaximal, complemented
3#21422520C212yesmaximal, complemented
4#21422417C211yesmaximal, characteristic, complemented
5C59×#1800C211yesmaximal, characteristic, complemented
6#16838117—24no—
7#16838117—24no—
8C2×#14273727C2211yescharacteristic
9C2×#14273713—22no—
10C2×#14273701C412yes—
11C2×#14273698—21no—
12C2×C4×D236C2211yescharacteristic
13C2×#14273686C412yes—
14C2×#14273685—21no—
15#16837455C2212yes—
16#16837147—24no—
17#16837129—24no—
18#16837126—22no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#423C2212yes—
21C2×#100×C59—21no—
22C59×#343—21no—
23C22×Dic118—21no—
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—22no—
26C22×C4×D59—21no—
27#14273727—22no—
28#14273727D414yes—
29#14273713—216no—
30#14273701—24no—
31#14273701—24no—
32#14273698—28no—
33C4×D236—22no—
34C4×D236D414yes—
35#14273686—24no—
36#14273686—24no—
37#14273685—28no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59C2×C412yes—
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×C22⋊Dic59—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—21no—
44C2×C4×Dic59—22no—
45C4⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—22no—
47C4×C59⋊C8—22no—
48C22×D4×C59D411yescharacteristic
49C23×C4×C59C2×C412yes—
50C2×M16×C59—22no—
51C4×D4×C59—22no—
52C4×D4×C59—22no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D412yes—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—22no—
57C4⋊C8×C59—22no—
58#100×C59—42no—
59C24×C59C22⋊C412yes—
60C23×D59—21no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—21no—
64C2×D4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—44no—
68C22×C4×C59C22⋊C412yes—
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C2×C22⋊D59—28no—
73C2×C22⋊D59—84no—
74C22×Dic59—21no—
75C22×Dic59—22no—
76C2×D236—21no—
77C2×D236—42no—
78C2×D236C22⋊C412yes—
79C2×C4×D59—24no—
80C2×C4×D59—42no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×Dic118—24no—
84C2×Dic118—82no—
85M16×C59—42no—
86M16×C59—42no—
87C2×C8×C59—42no—
88C4⋊C4×C59—22no—
89C22⋊C4×C59—24no—
90C42×C59C2×D412yes—
91C42×C59C2×D412yes—
92C22⋊Dic59—28no—
93(C2×D59)⋊C4—28no—
94(C2×D59)⋊C4—24no—
95C4⋊Dic59—21no—
96C4⋊Dic59—22no—
97C4⋊Dic59—24no—
98C4⋊Dic59C22⋊C412yes—
99C4⋊Dic59C22⋊C414yes—
100C4×Dic59—28no—
101C2×C59⋊C8—42no—
102C23×C59—22no—
103C23×C59—22no—
104C23×C59—22no—
105C23×C59—42no—
106C23×C59—42no—
107C23×C59C2×C22⋊C411yescharacteristic
108C22×D59—22no—
109C22×D59—42no—
110C22×D59—42no—
111C22×D59—41no—
112C22×D59—41no—
113D4×C59—42no—
114D4×C59—42no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—41no—
118C2×C4×C59—44no—
119C2×C4×C59—44no—
120C2×C4×C59—42no—
121C2×C4×C59C2×C22⋊C411yescharacteristic
122C2×C4×C59C2×C22⋊C411yescharacteristic
123C2×C4×C59#12012yes—
124C2×C4×C59#12012yes—
125C2×C4×C59#13512yes—
126C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C59#13711yescharacteristic
128C22⋊D59—88no—
129C2×Dic59—22no—
130C2×Dic59—24no—
131C2×Dic59—44no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134D236—24no—
135D236—42no—
136C4×D59—44no—
137C4×D59—44no—
138Dic118—84no—
139C8×C59—82no—
140C59⋊C8—82no—
141#8397—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—82no—
148C22×C59—82no—
149C22×C59#37911yescharacteristic
150C22×C59#41611yescharacteristic
151C22×C59#41711yescharacteristic
152C2×D59—42no—
153C2×D59—42no—
154C2×D59—42no—
155C2×D59—82no—
156C2×D59—82no—
157C4×C59—22no—
158C4×C59—21no—
159C4×C59—42no—
160C4×C59#41312yes—
161C4×C59#44912yes—
162C4×C59#45212yes—
163Dic59—44no—
164Dic59—42no—
165C2×#555—591no—
166#1800D5911yescharacteristic, complemented
167#1549—592no—
168#1486—592no—
169#1460—591no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—82no—
173C2×C59#178511yescharacteristic
174C2×C59#178811yescharacteristic
175C2×C59#179511yescharacteristic
176D59—82no—
177#555—1184no—
178#555—1184no—
179C2×C4⋊Q8—591no—
180C2×#124—1182no—
181C2×#121—592no—
182C2×#121—1181no—
183C2×C4×D4—591no—
184C2×C4×D4C2×D5911yescharacteristic
185C2×#117—592no—
186C2×#117—1181no—
187#431—592no—
188#423C2×D5912yes—
189C2×#100—21no—
190#343—21no—
191#332—1184no—
192D4⋊C8—1184no—
193#323—1182no—
194C59#839711yescharacteristic, complemented, Sylow
195C22×Q8—1181no—
196C22×D4—591no—
197C22×D4—1182no—
198C22×D4C22⋊D5911yescharacteristic
199C23×C4—1181no—
200C23×C4C4×D5912yes—
201C2×M16—22no—
202C4⋊Q8—594no—
203C4⋊Q8—1182no—
204#124—11816no—
205#121—1184no—
206#121—1184no—
207#121—1188no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—594no—
211C4×D4—1182no—
212#117—1188no—
213#117—1184no—
214#117—1184no—
215C2×C4⋊C4—591no—
216C2×C4⋊C4—592no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4C22⋊D5911yescharacteristic
219C2×C22⋊C4—1182no—
220C2×C22⋊C4—1182no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4D23612yes—
223C2×C42—1182no—
224C2×C42C22×D5911yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—1182no—
228#100—42no—
229C22⋊C8—1182no—
230C4×C8—1182no—
231C24—1181no—
232C24(C2×D59)⋊C412yes—
233C2×Q8—1184no—
234C2×Q8—4722no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—592no—
240C2×D4—1188no—
241C2×D4—1181no—
242C2×D4—4724no—
243C2×D4—2362no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—44no—
247C22×C4—1181no—
248C22×C4—1182no—
249C22×C4—1184no—
250C22×C4—2362no—
251C22×C4—2361no—
252C22×C4—2361no—
253C22×C4(C2×D59)⋊C412yes—
254C22×C4C2×C4×D5911yescharacteristic
255C22×C4C2×D23611yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—2362no—
261C4⋊C4—22no—
262C4⋊C4—592no—
263C4⋊C4—594no—
264C4⋊C4—1181no—
265C4⋊C4—1182no—
266C4⋊C4—1184no—
267C22⋊C4—24no—
268C22⋊C4—1188no—
269C22⋊C4—1188no—
270C22⋊C4—1184no—
271C42—1188no—
272C42D4×D5912yes—
273C42D4×D5912yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—1182no—
280C23—2362no—
281C23—2362no—
282C23—2361no—
283C23—2361no—
284C23C2×(C2×D59)⋊C411yescharacteristic
285Q8—4724no—
286D4—42no—
287D4—42no—
288D4—1184no—
289D4—4728no—
290D4—2362no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—1182no—
298C2×C4—1184no—
299C2×C4—2364no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2364no—
303C2×C4—2364no—
304C2×C4C22⋊C4×D5911yescharacteristic
305C2×C4C22⋊C4×D5911yescharacteristic
306C2×C4#1427369512yes—
307C2×C4#1427369712yes—
308C2×C4#1427374712yes—
309C2×C4#1427375311yescharacteristic
310C2×C4#1427375811yescharacteristic
311C8—82no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—4722no—
321C22—4722no—
322C22—2362no—
323C22—2362no—
324C22—2362no—
325C22#1683725711yescharacteristic
326C22#1683739611yescharacteristic
327C22#1683740111yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—2362no—
332C4—2364no—
333C4#1683738312yes—
334C4#1683754512yes—
335C4#1683756212yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—4722no—
340C2#2142402111yescharacteristic
341C2#2142403411yescharacteristic
342C2#2142406711yescharacteristic
subgroupG/H
centerC22#16837257
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#379
prime radicalC2×#16838117C2
cosocleC2×C42×C59C23
hypercenter#1800D59
hypocenterC59#8397
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21422782C2semidirect
3. #21422520C2semidirect
4. #21422417C2semidirect
5. C59×#1800C2semidirect
166. #1800D59semidirect
194. C59#8397semidirect

Maximal subgroups

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

subgroupindex
C2×#168381172
#214227822
#214225202
#214224172
C59×#18002
#839759