Finite Group Databasebeta

← #29736682

Subgroups — #29736682

Order 15616.

1066 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 227 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×#16840132C212yesmaximal, complemented
2#21454171C211yesmaximal, characteristic
3#21446957C214yesmaximal, complemented
4#21446789C212yesmaximal, complemented
5#21446131C214yesmaximal
6#21445581C211yesmaximal, characteristic
7C2×C61×#533C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9C2×#14519756C2212yes—
10#16840132—28no—
11C2×#14519700C2218yes—
12C2×C22⋊#92418C2212yes—
13C2×#14519688C2212yes—
14#16839668C2212yes—
15#16839660—22no—
16#16839632C2211yescharacteristic
17#16839622—22no—
18#16839524C2218yes—
19#16839517C2212yes—
20C61×#533—28no—
21C2×C4⋊Q8×C61C2212yes—
22C2×#125×C61C2212yes—
23C2×C22⋊Q8×C61—22no—
24C2×C4×Q8×C61—22no—
25C2×#117×C61C2212yes—
26C22×C42×C61C2211yescharacteristic
27C22×Dic122C2312yes—
28C22×#92418C2314yes—
29#14519777—24no—
30#14519756—28no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61C2312yes—
34C2×#11758196—22no—
35C2×C22⋊Dic61—22no—
36#14519700—216no—
37C22⋊#92418—24no—
38#14519688—24no—
39#14519684—24no—
40C22×Q8×C61—22no—
41C23×C4×C61D412yes—
42C23×C4×C61C2311yescharacteristic
43C4⋊Q8×C61—28no—
44#125×C61—24no—
45#125×C61D418yes—
46C22⋊Q8×C61—216no—
47C4×Q8×C61—216no—
48#117×C61—28no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—24no—
51C2×C4⋊C4×C61C2314yes—
52C2×C22⋊C4×C61—22no—
53C2×C42×C61—22no—
54C2×C42×C61—24no—
55C2×C42×C61D412yes—
56C2×C42×C61C2311yescharacteristic
57C2×C42×C61C2311yescharacteristic
58C24×C61C2×D411yescharacteristic
59C2×Q8×C61—28no—
60C2×Q8×C61—44no—
61C2×Q8×C61—44no—
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—24no—
65C22×C4×C61—24no—
66C22×C4×C61—24no—
67C22×C4×C61—42no—
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D412yes—
71C22×C4×C61#4114yes—
72C22×C4×C61C2411yescharacteristic
73C22×Dic61—22no—
74C22×Dic61—22no—
75C2×Dic122—24no—
76C2×Dic122—24no—
77C2×Dic122—24no—
78C4⋊C4×C61—28no—
79C4⋊C4×C61—216no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61C2×D418yes—
82C22⋊C4×C61—28no—
83C42×C61—28no—
84C42×C61—24no—
85C42×C61—22no—
86C42×C61C2×D414yes—
87C22⋊Dic61—44no—
88C4⋊Dic61—24no—
89C4⋊Dic61—44no—
90C4⋊Dic61—44no—
91#11758196—44no—
92C2×#92418—24no—
93C2×#92418—24no—
94C2×#92418—24no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61#12412yes—
100C23×C61C22×D411yescharacteristic
101Q8×C61—416no—
102C2×C4×C61—21no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—28no—
109C2×C4×C61—28no—
110C2×C4×C61—48no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61#12014yes—
114C2×C4×C61#12412yes—
115C2×C4×C61#13714yes—
116C2×C4×C61C22×D412yes—
117C2×C4×C61C2×#4112yes—
118C2×C4×C612−1+412yes—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—44no—
124Dic122—48no—
125#92418—44no—
126#25910—611nomaximal, Sylow
127C22×C61—22no—
128C22×C61—22no—
129C22×C61—22no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61C2×#12011yescharacteristic
133C22×C61C2×#12411yescharacteristic
134C22×C61#53611yescharacteristic
135C22×C61C2×#13712yes—
136C22×C61#57512yes—
137C4×C61—22no—
138C4×C61—24no—
139C4×C61—24no—
140C4×C61—48no—
141C4×C61#54814yes—
142Dic61—82no—
143Dic61—82no—
144C2×#533D6111yescharacteristic
145C2×#491—612no—
146#2857—611no—
147#2024—614no—
148#1992—612no—
149#1873—614no—
150#1749—611no—
151C2×C61—42no—
152C2×C61#243611yescharacteristic
153C2×C61#300712yes—
154C2×C61#313511yescharacteristic
155C2×C61#313911yescharacteristic
156C2×C61#323712yes—
157#533—28no—
158C2×C4⋊Q8—611no—
159C2×C4⋊Q8C2×D6112yes—
160C2×#125C2×D6112yes—
161C2×C22⋊Q8—22no—
162C2×C22⋊Q8—612no—
163C2×C4×Q8—22no—
164C2×#117C2×D6112yes—
165C22×C42C2×D6111yescharacteristic
166#491—1228no—
167C2×#103—618no—
168C2×C22⋊C8—612no—
169C2×#97—612no—
170#391—612no—
171#389—1222no—
172#384—611no—
173#380—1222no—
174#340—618no—
175#336—612no—
176C61#2591011yescharacteristic, complemented, Sylow
177C22×Q8—22no—
178C22×Q8—612no—
179C23×C4C22⋊D6112yes—
180C23×C4C22×D6111yescharacteristic
181C22×C8—614no—
182C4⋊Q8—28no—
183C4⋊Q8—1224no—
184#125—24no—
185#125C22⋊D6118yes—
186C22⋊Q8—216no—
187C22⋊Q8—1228no—
188C4×Q8—216no—
189#117—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—612no—
193C2×C4⋊C4—1222no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4C22×D6114yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1222no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42C22⋊D6112yes—
202C2×C42C22×D6111yescharacteristic
203C2×C42C22×D6111yescharacteristic
204#103—12216no—
205C22⋊C8—1224no—
206#97—1224no—
207#95—1224no—
208C24C2×C22⋊D6111yescharacteristic
209C2×Q8—28no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—1224no—
213C2×Q8—1224no—
214C2×Q8—1224no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—1222no—
222C22×C4—1222no—
223C22×C4#1175822614yes—
224C22×C4C2×C22⋊D6112yes—
225C22×C4C2×C22⋊D6111yescharacteristic
226C22×C4C2×C22⋊D6112yes—
227C22×C4C23×D6111yescharacteristic
228C2×C8—1224no—
229C2×C8—1224no—
230C2×C8—1224no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—216no—
234C4⋊C4—1224no—
235C4⋊C4—2444no—
236C4⋊C4—2444no—
237C4⋊C4—2444no—
238C4⋊C4C2×C22⋊D6118yes—
239C22⋊C4—28no—
240C22⋊C4—2444no—
241C42—28no—
242C42—22no—
243C42—24no—
244C42C2×C22⋊D6114yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23#1451976612yes—
250C23C22×C22⋊D6111yescharacteristic
251Q8—416no—
252Q8—2448no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2444no—
268C2×C4—2442no—
269C2×C4#1451974614yes—
270C2×C4#1451976612yes—
271C2×C4#1451980714yes—
272C2×C4C22×C22⋊D6112yes—
273C2×C4C2×#1175822612yes—
274C2×C4#1451982512yes—
275C8—2444no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22C2×#1451974611yescharacteristic
282C22C2×#1451976611yescharacteristic
283C22#1684033611yescharacteristic
284C22C2×#1451980712yes—
285C22#1684056012yes—
286C4—22no—
287C4—24no—
288C4—24no—
289C4—48no—
290C4—4882no—
291C4—4882no—
292C4#1684042314yes—
293C2—42no—
294C2#2144947311yescharacteristic
295C2#2145507912yes—
296C2#2145616711yescharacteristic
297C2#2145620111yescharacteristic
298C2#2145703312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#533C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#533D61
hypocenterC61#25910
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16840132C2semidirect
3. #21446957C2semidirect
4. #21446789C2semidirect
176. C61#25910semidirect

Maximal subgroups

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

subgroupindex
C2×#168401322
#214541712
#214469572
#214467892
#214461312
#214455812
C2×C61×#5332
#2591061