Finite Group Databasebeta

← #28480433

Subgroups — #28480433

Order 15104.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic118C211yesmaximal, characteristic
2#21422778C212yesmaximal, complemented
3#21422524C212yesmaximal, complemented
4#21422417C211yesmaximal, characteristic
5C59×#1800C211yesmaximal, characteristic
6C23×Dic118C412yes—
7D4×Dic118—24no—
8D4×Dic118—24no—
9C2×#14273727C2211yescharacteristic
10C2×#14273726—21no—
11C2×#14273706C412yes—
12C2×#14273704—22no—
13C2×C4×Dic118C2211yescharacteristic
14C2×D4×Dic59—21no—
15#16837455C2212yes—
16#16837150—24no—
17#16837132—24no—
18#16837126—22no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#423C2212yes—
21C2×#100×C59—21no—
22C59×#343—21no—
23C22×Dic118—22no—
24C22×Dic118—24no—
25C22×Dic118—24no—
26C22×Dic118—48no—
27C22×Dic118C2×C411yescharacteristic
28C22×Dic118C2×C412yes—
29C23×Dic59—22no—
30#14273727—22no—
31#14273727D414yes—
32#14273726—28no—
33#14273706—24no—
34#14273706—24no—
35#14273704—216no—
36C4×Dic118—22no—
37C4×Dic118D414yes—
38D4×Dic59—28no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59C2×C411yescharacteristic
42C2×Dic59⋊C4—22no—
43C2×Dic59⋊C4—21no—
44C2×C22⋊Dic59—22no—
45C2×C4×Dic59—21no—
46C4⋊(C59⋊C8)—22no—
47C22⋊(C59⋊C8)—22no—
48C4×C59⋊C8—22no—
49C22×D4×C59D411yescharacteristic
50C23×C4×C59C2×C412yes—
51C2×M16×C59—22no—
52C4×D4×C59—22no—
53C4×D4×C59—22no—
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59D412yes—
56C2×C42×C59C2311yescharacteristic
57C4⋊C8×C59—22no—
58C4⋊C8×C59—22no—
59#100×C59—42no—
60C24×C59C22⋊C412yes—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59SD1612yes—
64C2×D4×C59Q1612yes—
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
72C22×Dic59—21no—
73C22×Dic59—28no—
74C22×Dic59—21no—
75C22×Dic59—22no—
76C22×Dic59—48no—
77C2×Dic118—21no—
78C2×Dic118—24no—
79C2×Dic118—28no—
80C2×Dic118—22no—
81C2×Dic118—416no—
82C2×Dic118—48no—
83C2×Dic118—48no—
84C2×Dic118—48no—
85C2×Dic118—44no—
86C2×Dic118C22⋊C412yes—
87C2×Dic118C22⋊C414yes—
88M16×C59—42no—
89M16×C59—42no—
90C2×C8×C59—42no—
91C4⋊C4×C59—22no—
92C22⋊C4×C59—24no—
93C42×C59C2×D412yes—
94C42×C59C2×D412yes—
95C22⋊Dic59—28no—
96C4⋊Dic59—21no—
97C4⋊Dic59—24no—
98C4⋊Dic59—24no—
99C4⋊Dic59C22⋊C412yes—
100Dic59⋊C4—28no—
101Dic59⋊C4—24no—
102C4×Dic59—24no—
103C2×C59⋊C8—42no—
104C23×C59—22no—
105C23×C59—22no—
106C23×C59—42no—
107C23×C59—42no—
108C23×C59#10314yes—
109C23×C59C2×C22⋊C411yescharacteristic
110D4×C59—24no—
111D4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—41no—
115C2×C4×C59—44no—
116C2×C4×C59—44no—
117C2×C4×C59—42no—
118C2×C4×C59C2×C22⋊C411yescharacteristic
119C2×C4×C59C2×C22⋊C411yescharacteristic
120C2×C4×C59#12012yes—
121C2×C4×C59#12012yes—
122C2×C4×C59C2×SD1611yescharacteristic
123C2×C4×C59C2×Q1611yescharacteristic
124C2×C4×C59#13712yes—
125C2×Dic59—24no—
126C2×Dic59—22no—
127C2×Dic59—22no—
128C2×Dic59—24no—
129C2×Dic59—416no—
130C2×Dic59—44no—
131C2×Dic59—42no—
132Dic118—24no—
133Dic118—22no—
134Dic118—416no—
135Dic118—44no—
136Dic118—44no—
137Dic118#10014yescomplemented
138C8×C59—82no—
139C59⋊C8—82no—
140#8391—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—24no—
144C22×C59—44no—
145C22×C59—44no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59#37911yescharacteristic
149C22×C59C2×#10311yescharacteristic
150C22×C59#41911yescharacteristic
151C4×C59—22no—
152C4×C59—21no—
153C4×C59—42no—
154C4×C59C2×#10012yes—
155C4×C59#44812yes—
156C4×C59#45112yes—
157Dic59—24no—
158Dic59—42no—
159Dic59—44no—
160C2×D4×Q8—591no—
161#1800D5911yescharacteristic
162#1548—592no—
163#1487—592no—
164#1460—591no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—44no—
168C2×C59#178311yescharacteristic
169C2×C59#178711yescharacteristic
170C2×C59#179611yescharacteristic
171C23×Q8—592no—
172D4×Q8—1184no—
173D4×Q8—1184no—
174C2×C4⋊Q8—591no—
175C2×C4⋊Q8—1181no—
176C2×C22⋊Q8—592no—
177C2×C22⋊Q8—1182no—
178C2×C4×Q8—591no—
179C2×C4×D4—1181no—
180C2×C4×D4C2×D5911yescharacteristic
181#431—592no—
182#423C2×D5912yes—
183C2×#100—21no—
184#343—21no—
185#333—1184no—
186Q8⋊C8—1184no—
187#323—1182no—
188C59#839111yescharacteristic, complemented, Sylow
189C22×Q8—591no—
190C22×Q8—592no—
191C22×Q8—1184no—
192C22×Q8—1184no—
193C22×Q8—1182no—
194C22×Q8—2368no—
195C22×D4C22⋊D5911yescharacteristic
196C23×C4—1182no—
197C23×C4C4×D5912yes—
198C2×M16—22no—
199C4⋊Q8—594no—
200C4⋊Q8—1188no—
201C4⋊Q8—1182no—
202C22⋊Q8—11816no—
203C22⋊Q8—1184no—
204C22⋊Q8—1184no—
205C4×Q8—594no—
206C4×Q8—1182no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—1188no—
210C2×C4⋊C4—591no—
211C2×C4⋊C4—1182no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C22⋊C4—1182no—
217C2×C22⋊C4D23612yes—
218C2×C42—1181no—
219C2×C42C22×D5911yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—1182no—
223#100—42no—
224C22⋊C8—1182no—
225C4×C8—1182no—
226C24(C2×D59)⋊C412yes—
227C2×Q8—592no—
228C2×Q8—594no—
229C2×Q8—1184no—
230C2×Q8—1188no—
231C2×Q8—1181no—
232C2×Q8—1182no—
233C2×Q8—2364no—
234C2×Q8—2368no—
235C2×Q8—2368no—
236C2×Q8—23616no—
237C2×Q8—2368no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4#1174567012yes—
241C2×D4#1174567212yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—1181no—
246C22×C4—1188no—
247C22×C4—1181no—
248C22×C4—1182no—
249C22×C4—2368no—
250C22×C4(C2×D59)⋊C412yes—
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254M16—42no—
255M16—42no—
256C2×C8—42no—
257C2×C8—2362no—
258C4⋊C4—22no—
259C4⋊C4—592no—
260C4⋊C4—1188no—
261C4⋊C4—1181no—
262C4⋊C4—1184no—
263C4⋊C4—1184no—
264C4⋊C4—1184no—
265C22⋊C4—24no—
266C22⋊C4—1188no—
267C42—1184no—
268C42D4×D5912yes—
269C42D4×D5912yes—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23#1427365014yes—
275C23C2×(C2×D59)⋊C411yescharacteristic
276Q8—594no—
277Q8—1184no—
278Q8—1182no—
279Q8—2364no—
280Q8—23616no—
281Q8—2364no—
282D4—24no—
283D4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—1184no—
291C2×C4—1182no—
292C2×C4—1182no—
293C2×C4—1184no—
294C2×C4—23616no—
295C2×C4—2362no—
296C2×C4—2364no—
297C2×C4C22⋊C4×D5911yescharacteristic
298C2×C4C22⋊C4×D5911yescharacteristic
299C2×C4#1427369512yes—
300C2×C4#1427369712yes—
301C2×C4C2×#1174567011yescharacteristic
302C2×C4C2×#1174567211yescharacteristic
303C2×C4#1427375712yes—
304C8—82no—
305C8—4722no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22#1683725711yescharacteristic
314C22C2×#1427365011yescharacteristic
315C22#1683741011yescharacteristic
316C4—21no—
317C4—22no—
318C4—42no—
319C4—1184no—
320C4—2364no—
321C4—2362no—
322C4#100×D5912yes—
323C4#1683754012yes—
324C4#1683755712yes—
325C2—21no—
326C2—21no—
327C2—44no—
328C2#2142401511yescharacteristic
329C2#2142403011yescharacteristic
330C2#2142407211yescharacteristic
subgroupG/H
centerC22#16837257
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#379
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#1800D59
hypocenterC59#8391
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. #21422778C2semidirect
3. #21422524C2semidirect
137. Dic118#100semidirect
188. C59#8391semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1182
#214227782
#214225242
#214224172
C59×#18002
#839159