Finite Group Databasebeta

← #29567308

Subgroups — #29567308

Order 15616.

620 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#21445740C211yesmaximal, characteristic
2#21445734C211yesmaximal, characteristic
3C61×#1781C211yesmaximal, characteristic
4C2×#14519694—21no—
5C2×#14519694—21no—
6C2×#14519692—21no—
7C2×#14519692C412yescomplemented
8#16839726C412yescomplemented
9#16839618—21no—
10C23×D4×C61C2211yescharacteristic
11C2×#99×C61—21no—
12C2×#99×C61C412yescomplemented
13C61×#379—21no—
14C2×#11758194—22no—
15C2×C22⋊Dic61—22no—
16C2×C22⋊Dic61—42no—
17#14519694—42no—
18#14519694—42no—
19#14519692—42no—
20#14519692—42no—
21C22⋊#92418—42no—
22C25×C61D411yescharacteristic
23C25×C61Q811yescharacteristic
24C22×D4×C61—42no—
25C22×D4×C61—42no—
26C22×D4×C61—42no—
27C22×D4×C61C2×C411yescharacteristic
28C22×D4×C61C2×C411yescharacteristic
29C22×D4×C61D411yescharacteristic
30C22×D4×C61D411yescharacteristic
31C23×C4×C61C2×C411yescharacteristic
32C2×C22⋊C4×C61—22no—
33C2×C22⋊C4×C61—42no—
34#99×C61—42no—
35#99×C61—42no—
36C24×C61—22no—
37C24×C61—22no—
38C24×C61—41no—
39C24×C61—41no—
40C24×C61—41no—
41C24×C61—41no—
42C24×C61—82no—
43C24×C61—82no—
44C24×C61C22⋊C411yescharacteristic
45C24×C61C22⋊C411yescharacteristic
46C24×C61C4⋊C411yescharacteristic
47C24×C61C4⋊C411yescharacteristic
48C24×C61C4⋊C411yescharacteristic
49C2×D4×C61—22no—
50C2×D4×C61—22no—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C2×D4×C61—41no—
61C2×D4×C61—416no—
62C22×C4×C61—22no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—42no—
66C22×C4×C61C4211yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×Dic61—42no—
69C22⋊C4×C61—42no—
70C22⋊C4×C61—42no—
71C22⋊C4×C61—84no—
72C22⋊Dic61—42no—
73C22⋊Dic61—42no—
74C22⋊Dic61—84no—
75#11758194—42no—
76#11758194—42no—
77C2×#92418—42no—
78C23×C61—21no—
79C23×C61—21no—
80C23×C61—21no—
81C23×C61—21no—
82C23×C61—21no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—42no—
92C23×C61—44no—
93C23×C61—44no—
94C23×C61—44no—
95C23×C61—44no—
96C23×C61—82no—
97C23×C61—82no—
98C23×C61—82no—
99C23×C61—82no—
100C23×C61—81no—
101C23×C61—81no—
102C23×C61—82no—
103C23×C61—82no—
104C23×C61—82no—
105C23×C61—82no—
106C23×C61—82no—
107C23×C61—82no—
108C23×C61—81no—
109C23×C61—81no—
110C23×C61#9511yescharacteristic
111C23×C61#9912yes—
112C23×C61#9912yes—
113D4×C61—44no—
114D4×C61—44no—
115D4×C61—44no—
116D4×C61—44no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—44no—
123C2×C4×C61—82no—
124C2×C4×C61—84no—
125C2×C4×C61#9912yes—
126C2×Dic61—82no—
127C2×Dic61—84no—
128#92418—82no—
129#7314—611nomaximal, Sylow
130C22×C61—22no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—41no—
136C22×C61—42no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—81no—
147C22×C61—81no—
148C22×C61—81no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61—88no—
155C22×C61—88no—
156C22×C61#34211yescharacteristic
157C22×C61#34211yescharacteristic
158C22×C61#34211yescharacteristic
159C4×C61—22no—
160C4×C61—41no—
161C4×C61—162no—
162Dic61—162no—
163#1783—611no—
164#1781—611no—
165#1781D6111yescharacteristic
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—42no—
170C2×C61—42no—
171C2×C61—42no—
172C2×C61—81no—
173C2×C61—81no—
174C2×C61#130411yescharacteristic
175C23×D4C2×D6111yescharacteristic
176C2×#100—1221no—
177C2×#100—1221no—
178C2×#99—21no—
179C2×#99—612no—
180C2×#99—1221no—
181C2×#99Dic6112yescomplemented
182#407—612no—
183#379—21no—
184#379—1221no—
185C61#731411yescharacteristic, complemented, Sylow
186C25Dic12211yescharacteristic
187C25C22⋊D6111yescharacteristic
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4C4×D6111yescharacteristic
192C22×D4D24411yescharacteristic
193C22×D4C2×Dic6111yescharacteristic
194C22×D4C22⋊D6111yescharacteristic
195C23×C4C4×D6111yescharacteristic
196C2×M16—1222no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4—2442no—
201#100—2442no—
202#100—2442no—
203#99—42no—
204#99—42no—
205#99—2442no—
206#99—2442no—
207C22⋊C8—2442no—
208C24—22no—
209C24—22no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—82no—
215C24—82no—
216C24#1175819611yescharacteristic
217C24#1175819611yescharacteristic
218C24C4⋊Dic6111yescharacteristic
219C24#1175819811yescharacteristic
220C24C22⋊Dic6111yescharacteristic
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—416no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—2442no—
239C22×C4C4×Dic6111yescharacteristic
240C22×C4#1175819811yescharacteristic
241M16—2442no—
242M16—2442no—
243C2×C8—2442no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—84no—
247C22⋊C4—4884no—
248C22⋊C4—2442no—
249C22⋊C4—2442no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—42no—
264C23—44no—
265C23—44no—
266C23—44no—
267C23—44no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—81no—
273C23—81no—
274C23—82no—
275C23—82no—
276C23—81no—
277C23—81no—
278C23—82no—
279C23—82no—
280C23—82no—
281C23—82no—
282C23#1451968411yescharacteristic
283C23#1451969112yes—
284C23#1451969212yes—
285D4—44no—
286D4—44no—
287D4—44no—
288D4—44no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—4882no—
298C2×C4—4884no—
299C2×C4#1451969112yes—
300C8—4882no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—88no—
326C22—88no—
327C22#1683952811yescharacteristic
328C22#1683952811yescharacteristic
329C22#1683952911yescharacteristic
330C4—22no—
331C4—41no—
332C4—162no—
333C4—9762no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2—42no—
339C2—42no—
340C2—81no—
341C2—81no—
342C2#2144372311yescharacteristic
subgroupG/H
centerC2#21443723
commutatorC22×C4×C61C42
FrattiniC23×D4C2×D61
FittingC61×#1781C2
socleC2×C61#1304
prime radical, cosocleC23×D4×C61C22
hypercenter#1781D61
hypocenterC61#7314
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #1781

Complements

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

NH ≅ G/Ntype
7. C2×#14519692C4semidirect
8. #16839726C4semidirect
12. C2×#99×C61C4semidirect
181. C2×#99Dic61semidirect
185. C61#7314semidirect

Maximal subgroups

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

subgroupindex
#214457402
#214457342
C61×#17812
#731461