Finite Group Databasebeta

← #29599683

Subgroups — #29599683

Order 15616.

540 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 79 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#21452191C211yesmaximal, characteristic
2#21446100C212yesmaximal
3#21443944C211yesmaximal, characteristic
4#21443938C212yesmaximal
5C61×#1856C211yesmaximal, characteristic
6C2×#14519754—22no—
7C2×#14519692C2212yes—
8#16839685—22no—
9#16839682—21no—
10#16839660—21no—
11#16839642C2212yes—
12#16839634—22no—
13#16839529—22no—
14C61×#518C2211yescharacteristic
15C61×#428—21no—
16C2×#100×C61C2211yescharacteristic
17C61×#390C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122—21no—
20C23×Dic61D412yes—
21#14519754—48no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—21no—
24C2×#11758196—22no—
25C2×#11758196—22no—
26C2×#11758196—42no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61D412yes—
30#14519692—22no—
31#14519692Q814yescomplemented
32#14519684—21no—
33#14519684—22no—
34#14519684—42no—
35C22×D4×C61C2311yescharacteristic
36C2×M16×C61—21no—
37C4×D4×C61—41no—
38C4×D4×C61—41no—
39#117×C61—21no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C42×C61D411yescharacteristic
44#107×C61—41no—
45#106×C61—41no—
46#100×C61—21no—
47#100×C61Q812yescomplemented
48C24×C61C2×D412yes—
49C2×D4×C61—41no—
50C2×D4×C61—41no—
51C2×D4×C61C2×Q811yescharacteristic
52C2×D4×C61C2×Q812yes—
53C2×D4×C61#4111yescharacteristic
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61C2×D411yescharacteristic
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—44no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C2×Dic122—44no—
65C2×Dic122—81no—
66M16×C61—22no—
67M16×C61—41no—
68C2×C8×C61—41no—
69C4⋊C4×C61—21no—
70C4⋊C4×C61—21no—
71C4⋊C4×C61—41no—
72C22⋊C4×C61—44no—
73C22⋊C4×C61—42no—
74C22⋊C4×C61—41no—
75C42×C61—41no—
76C42×C61—41no—
77C22⋊Dic61—24no—
78C22⋊Dic61—44no—
79C22⋊Dic61—42no—
80C22⋊Dic61—42no—
81C4⋊Dic61—24no—
82C4⋊Dic61—81no—
83#11758196—44no—
84#11758196—44no—
85#11758196—84no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—42no—
89C23×C61—42no—
90C23×C61#12011yescharacteristic
91C23×C61C22⋊Q812yes—
92C23×C61C4⋊Q812yes—
93D4×C61—41no—
94D4×C61—42no—
95D4×C61—41no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—82no—
106C2×C4×C61—82no—
107C2×C4×C61C22⋊Q811yescharacteristic
108C2×C4×C61C4⋊Q811yescharacteristic
109C2×Dic61—24no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—44no—
113C2×Dic61—42no—
114C2×Dic61—44no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—82no—
118C2×Dic61—82no—
119C2×Dic61—82no—
120Dic122—84no—
121C8×C61—42no—
122#13461—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—42no—
131C22×C61—82no—
132C22×C61#39511yescharacteristic
133C22×C61B(4,𝔽2)11yescharacteristic
134C22×C61#45811yescharacteristic
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—42no—
138C4×C61—81no—
139C4×C61—81no—
140C4×C61—81no—
141Dic61—44no—
142Dic61—82no—
143Dic61—84no—
144#2683—611no—
145#1867—612no—
146#1856D6111yescharacteristic
147#1393—611no—
148#1390—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—42no—
152C2×C61—42no—
153C2×C61#206211yescharacteristic
154C2×C61#219611yescharacteristic
155C2×C61#220111yescharacteristic
156C2×C22⋊Q8—1222no—
157#518C2×D6111yescharacteristic
158#428—21no—
159C2×#100C2×D6111yescharacteristic
160C2×#99—612no—
161#396—1222no—
162#395—1221no—
163#390C2×D6111yescharacteristic
164#389—1221no—
165#386—612no—
166#385—1222no—
167#342—1222no—
168#339—21no—
169C61#1346111yescharacteristic, complemented, Sylow
170C22×Q8—1221no—
171C22×D4C22×D6111yescharacteristic
172C23×C4—612no—
173C2×M16—21no—
174C22⋊Q8—2448no—
175C4×D4—41no—
176C4×D4—41no—
177#117—21no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—1221no—
180C2×C4⋊C4—1222no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—2442no—
183C2×C4⋊C4C22⋊D6111yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—612no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C42C22⋊D6111yescharacteristic
190#107—41no—
191#106—41no—
192#100—21no—
193#100Dic12212yescomplemented
194#99—614no—
195#99—1222no—
196#95—1221no—
197#95—1222no—
198#95—2442no—
199C24D4×D6112yes—
200C2×Q8—4881no—
201C2×Q8—2444no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic12211yescharacteristic
205C2×D4#1175822211yescharacteristic
206C2×D4Q8×D6112yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4—2444no—
214C22×C4—2442no—
215C22×C4—2442no—
216C22×C4C2×C22⋊D6111yescharacteristic
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—1224no—
224C4⋊C4—4881no—
225C4⋊C4—4884no—
226C4⋊C4—2444no—
227C4⋊C4—2444no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C22⋊C4—41no—
231C22⋊C4—1224no—
232C22⋊C4—2444no—
233C22⋊C4—2442no—
234C22⋊C4—2442no—
235C42—41no—
236C42—41no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1451974511yescharacteristic
242C23#1451975512yes—
243C23#1451977612yes—
244Q8—4884no—
245D4—41no—
246D4—41no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—1224no—
260C2×C4—4882no—
261C2×C4—4882no—
262C2×C4—4882no—
263C2×C4—2444no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4—2442no—
269C2×C4—2444no—
270C2×C4#1451975611yescharacteristic
271C2×C4#1451977511yescharacteristic
272C8—42no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1683968211yescharacteristic
283C22#1683994811yescharacteristic
284C22#1683995111yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—4882no—
292C4—4884no—
293C4—2444no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2144717211yescharacteristic
299C2#2144780211yescharacteristic
300C2#2144783111yescharacteristic
subgroupG/H
centerC22#16839948
commutator, prime radical, cosocleC22×D4×C61C23
FrattiniC22×D4C22×D61
FittingC61×#1856C2
socleC22×C61B(4,𝔽2)
hypercenter#1856D61
hypocenterC61#13461
LCSG ▷ C22×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1856

Complements

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

NH ≅ G/Ntype
31. #14519692Q8semidirect
47. #100×C61Q8semidirect
169. C61#13461semidirect
193. #100Dic122semidirect

Maximal subgroups

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

subgroupindex
#214521912
#214461002
#214439442
#214439382
C61×#18562
#1346161