Finite Group Databasebeta

← #29592753

Subgroups — #29592753

Order 15616.

500 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 59 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#21453843C211yesmaximal, characteristic
2#21444742C212yesmaximal, complemented
3#21443815C212yesmaximal, complemented
4#21443749C211yesmaximal, characteristic, complemented
5C61×#2030C211yesmaximal, characteristic, complemented
6C2×#14519751—21no—
7#16839912—22no—
8#16839689—22no—
9#16839672—22no—
10#16839671—21no—
11#16839671C2212yes—
12#16839499—22no—
13#16839486C2212yes—
14C61×#543C2211yescharacteristic
15C61×#381—21no—
16C61×#381C2211yescharacteristic
17C61×#342—21no—
18C61×#342C2211yescharacteristic
19C22×D244—21no—
20C22×D244—21no—
21C22×D244D412yes—
22C22×C22⋊D61—22no—
23C2×#11758201—42no—
24#14519751—42no—
25#14519751—42no—
26C2×C4⋊Dic61—21no—
27C2×#11758198—22no—
28C2×#11758198—22no—
29C2×#11758198—22no—
30#14519697—42no—
31C22⋊#92418—22no—
32#14519684—21no—
33#14519684D412yes—
34C23×C4×C61D411yescharacteristic
35C22⋊Q8×C61—22no—
36C22⋊Q8×C61—42no—
37C2×C22⋊C4×C61—21no—
38C2×C22⋊C4×C61—21no—
39C2×C22⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61C2311yescharacteristic
41#95×C61—22no—
42#95×C61—42no—
43C24×C61C2×D411yescharacteristic
44C23×D61—22no—
45C2×Q8×C61—22no—
46C22×C4×C61—21no—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—22no—
50C22×C4×C61—41no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61SD1612yescomplemented
54C22×C4×C61C2×D412yes—
55C2×C22⋊D61—42no—
56C2×C22⋊D61—42no—
57C2×C22⋊D61—82no—
58C2×C22⋊D61—82no—
59C22×Dic61—22no—
60C2×D244—42no—
61C2×D244—42no—
62C2×D244—82no—
63C2×D244—81no—
64C2×D244—81no—
65C2×D244—82no—
66C2×D244—81no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—42no—
69C22⋊C4×C61—41no—
70C22⋊C4×C61—41no—
71C22⋊C4×C61—41no—
72C22⋊C4×C61—41no—
73C22⋊C4×C61—41no—
74C22⋊C4×C61—41no—
75#11758201—84no—
76#11758198—44no—
77#11758198—82no—
78#11758198—82no—
79C4⋊Dic61—42no—
80C2×#92418—42no—
81C23×C61—21no—
82C23×C61—21no—
83C23×C61—42no—
84C23×C61#12011yescharacteristic
85C23×C61C2×SD1611yescharacteristic
86C23×C61AGL(1,ℤ/8)11yescharacteristic
87C22×D61—42no—
88C22×D61—42no—
89C22×D61—42no—
90C22×D61—42no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61—42no—
94Q8×C61—82no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61—81no—
105C2×C4×C61—82no—
106C2×C4×C61—81no—
107C2×C4×C61—82no—
108C22⋊D61—84no—
109C22⋊D61—84no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113D244—44no—
114D244—82no—
115D244—84no—
116D244—84no—
117#92418—82no—
118#12206—611nomaximal, Sylow
119C22×C61—22no—
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61—42no—
125C22×C61—81no—
126C22×C61#45011yescharacteristic
127C22×C61#45511yescharacteristic
128C22×C61B(4,𝔽2)11yescharacteristic
129C2×D61—82no—
130C2×D61—82no—
131C2×D61—82no—
132C2×D61—82no—
133C2×D61—82no—
134C2×D61—82no—
135C2×D61—82no—
136C4×C61—42no—
137C4×C61—42no—
138C4×C61—81no—
139C4×C61—82no—
140C4×C61—161no—
141Dic61—82no—
142#2815—611no—
143#2030D6111yescharacteristic, complemented
144#1596—612no—
145#1343—612no—
146#1317—611no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—41no—
150C2×C61—41no—
151C2×C61#160111yescharacteristic
152C2×C61#220012yes—
153D61—162no—
154#543C2×D6111yescharacteristic
155C2×#121—1221no—
156#450—1222no—
157#397—1222no—
158#392—612no—
159#392—1222no—
160#392—1221no—
161#381—21no—
162#381C2×D6111yescharacteristic
163#342—21no—
164#342C2×D6111yescharacteristic
165#328—1222no—
166#323—612no—
167C61#1220611yescharacteristic, complemented, Sylow
168C22×D4—612no—
169C22×D4—1221no—
170C22×D4—1221no—
171C22×D4—1222no—
172C23×C4C22⋊D6111yescharacteristic
173C2×SD16—2442no—
174C22⋊Q8—22no—
175C22⋊Q8—42no—
176#121—2442no—
177#121—2442no—
178C2×C4⋊C4—1221no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—1222no—
182C2×C22⋊C4—1222no—
183C2×C22⋊C4—1222no—
184C2×C22⋊C4C22⋊D6111yescharacteristic
185C2×C22⋊C4C22×D6111yescharacteristic
186#102—2442no—
187C22⋊C8—1222no—
188#95—22no—
189#95—42no—
190#95—612no—
191#95—1221no—
192C24—1222no—
193C24C2×C22⋊D6111yescharacteristic
194C2×Q8—22no—
195C2×D4—4881no—
196C2×D4—4881no—
197C2×D4—4882no—
198C2×D4—4881no—
199C2×D4—4882no—
200C2×D4—4882no—
201C2×D4—4882no—
202C2×D4—2442no—
203C2×D4—2442no—
204C2×D4—2442no—
205C2×D4—2442no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—1222no—
214C22×C4#1175820112yescomplemented
215C22×C4D4×D6112yes—
216SD16—4884no—
217C2×C8—2442no—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—2442no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—4882no—
228C22⋊C4—4882no—
229C22⋊C4—2444no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—2442no—
234C23—2442no—
235C23—2442no—
236C23—2442no—
237C23—2442no—
238C23—2442no—
239C23—2442no—
240C23#1451974511yescharacteristic
241C23C2×#1175820111yescharacteristic
242C23#1451980211yescharacteristic
243Q8—82no—
244D4—4882no—
245D4—4884no—
246D4—4884no—
247D4—4884no—
248D4—4884no—
249D4—2444no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C8—4882no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—4882no—
275C22—4882no—
276C22—4882no—
277C22—4882no—
278C22—4882no—
279C22—4882no—
280C22—4882no—
281C22#1683991211yescharacteristic
282C22#1683993911yescharacteristic
283C22#1683994811yescharacteristic
284C4—42no—
285C4—42no—
286C4—81no—
287C4—82no—
288C4—161no—
289C4—4882no—
290C2—21no—
291C2—21no—
292C2—41no—
293C2—41no—
294C2—9762no—
295C2#2144477011yescharacteristic
296C2#2144782312yes—
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2030C2
socleC22×C61B(4,𝔽2)
prime radical#21453843C2
hypercenter#2030D61
hypocenterC61#12206
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
2. #21444742C2semidirect
3. #21443815C2semidirect
4. #21443749C2semidirect
5. C61×#2030C2semidirect
53. C22×C4×C61SD16semidirect
143. #2030D61semidirect
167. C61#12206semidirect
214. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214538432
#214447422
#214438152
#214437492
C61×#20302
#1220661