Finite Group Databasebeta

← #29599163

Subgroups — #29599163

Order 15616.

540 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 75 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#21453807C211yesmaximal, characteristic, complemented
2#21445778C212yesmaximal
3#21443946C211yesmaximal, characteristic, complemented
4#21443945C212yesmaximal, complemented
5C61×#1793C211yesmaximal, characteristic
6C2×#14519704—22no—
7#16839763—22no—
8C2×#14519696C2212yes—
9#16839701—21no—
10#16839694—22no—
11#16839669C2212yes—
12#16839660—21no—
13#16839649—22no—
14C2×2−1+4×C61C2211yescharacteristic
15C2×#104×C61—21no—
16C61×#417—21no—
17C2×#101×C61C2211yescharacteristic
18C61×#391C2211yescharacteristic
19C2×#11758194—22no—
20C2×C4⋊Dic61—21no—
21C2×C4⋊Dic61—21no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61D412yes—
24C2×#11758196—22no—
25C2×C4×Dic61D412yes—
26#14519704—44no—
27#14519701—42no—
28#14519698—42no—
29#14519696—22no—
30#14519696—22no—
31#14519684—22no—
32#14519684—21no—
33#14519684—22no—
34#14519684—22no—
352−1+4×C61—21no—
362−1+4×C61—22no—
372−1+4×C61—42no—
382−1+4×C61Q812yes—
39C2×#41×C61—22no—
40C2×#41×C61—21no—
41C2×#41×C61—41no—
42C22×Q8×C61—41no—
43C22×Q8×C61C2311yescharacteristic
44C2×M16×C61—21no—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C42×C61D411yescharacteristic
47#104×C61—42no—
48#103×C61—41no—
49#102×C61—41no—
50#101×C61—22no—
51#95×C61—22no—
52#41×C61—22no—
53#41×C61—24no—
54#41×C61—24no—
55#41×C61—22no—
56#41×C61—42no—
57#41×C61—41no—
58#41×C61—42no—
59#41×C61—42no—
60#41×C61—44no—
61#41×C61—42no—
62#41×C61—41no—
63C2×Q8×C61—21no—
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—42no—
68C2×Q8×C61—42no—
69C2×Q8×C61—44no—
70C2×Q8×C61C2×Q811yescharacteristic
71C2×Q8×C61#4111yescharacteristic
72C2×Q8×C61#4112yes—
73C2×D4×C61—21no—
74C2×D4×C61—22no—
75C2×D4×C61—41no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—22no—
83C22×Dic61—22no—
84M16×C61—41no—
85M16×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—41no—
88C42×C61—41no—
89C4⋊Dic61—42no—
90C4⋊Dic61—42no—
91C4⋊Dic61—81no—
92C4⋊Dic61—81no—
93#11758196—44no—
94C4×Dic61—42no—
95#11758194—42no—
96#11758194—42no—
97C2×#92418—42no—
98C23×C61—41no—
99C23×C61#12011yescharacteristic
100Q8×C61—22no—
101Q8×C61—24no—
102Q8×C61—41no—
103Q8×C61—41no—
104Q8×C61—42no—
105Q8×C61—42no—
106Q8×C61—41no—
107D4×C61—22no—
108D4×C61—24no—
109D4×C61—42no—
110D4×C61—41no—
111D4×C61—42no—
112D4×C61—42no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—44no—
119C2×C4×C61—42no—
120C2×C4×C61—44no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61C22⋊Q812yes—
128C2×C4×C61C22⋊Q811yescharacteristic
129C2×C4×C61#12412yes—
130C2×C4×C61#12411yescharacteristic
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C8×C61—81no—
138#92418—82no—
139#13379—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—42no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61#39311yescharacteristic
146C22×C61B(4,𝔽2)11yescharacteristic
147C22×C61B(4,𝔽2)11yescharacteristic
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—42no—
153C4×C61—42no—
154C4×C61—81no—
155Dic61—82no—
156Dic61—82no—
157#2809—611no—
158#1793—612no—
159#1793D6111yescharacteristic
160#1394—612no—
161#1394—611no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—82no—
165C2×C61#202911yescharacteristic
166C2×C61#220311yescharacteristic
167C2×C61#220411yescharacteristic
168C2×2−1+4C2×D6111yescharacteristic
169C2×#104—21no—
170C2×#104—1222no—
171#417—21no—
172#417—1222no—
173C2×#101—612no—
174C2×#101C2×D6111yescharacteristic
175#400—1221no—
176#398—1222no—
177#391—612no—
178#391C2×D6111yescharacteristic
179#389—1221no—
180#387—1222no—
181C61#1337911yescharacteristic, complemented, Sylow
1822−1+4—21no—
1832−1+4—22no—
1842−1+4—42no—
1852−1+4Dic12212yes—
186C2×#41—22no—
187C2×#41—21no—
188C2×#41—41no—
189C22×Q8—41no—
190C22×Q8C22×D6111yescharacteristic
191C2×M16—21no—
192C2×M16—1222no—
193C2×C4⋊C4—612no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4C22⋊D6111yescharacteristic
199C2×C42—612no—
200C2×C42C22⋊D6111yescharacteristic
201#104—42no—
202#104—2444no—
203#103—41no—
204#103—2442no—
205#102—41no—
206#102—2442no—
207#101—22no—
208#101—1222no—
209#101—1222no—
210#95—22no—
211#95—1221no—
212#95—1222no—
213#95—1222no—
214#95—1222no—
215#41—22no—
216#41—24no—
217#41—24no—
218#41—22no—
219#41—41no—
220#41—42no—
221#41—42no—
222#41—42no—
223#41—44no—
224#41—42no—
225#41—41no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8C2×Dic12211yescharacteristic
234C2×Q8#1175822211yescharacteristic
235C2×Q8#1175822412yes—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4D4×D6112yes—
246C22×C4C2×C22⋊D6111yescharacteristic
247M16—41no—
248M16—41no—
249M16—2442no—
250M16—2442no—
251C2×C8—41no—
252C2×C8—2442no—
253C4⋊C4—41no—
254C4⋊C4—4881no—
255C4⋊C4—4881no—
256C4⋊C4—2442no—
257C4⋊C4—2444no—
258C4⋊C4—2442no—
259C42—41no—
260C42—2442no—
261C23—41no—
262C23#1451974511yescharacteristic
263Q8—24no—
264Q8—22no—
265Q8—41no—
266Q8—41no—
267Q8—42no—
268Q8—41no—
269Q8—42no—
270D4—24no—
271D4—22no—
272D4—41no—
273D4—42no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4#1451975412yes—
297C2×C4#1451975611yescharacteristic
298C2×C4#1451976312yes—
299C2×C4#1451976411yescharacteristic
300C8—81no—
301C8—4882no—
302C22—21no—
303C22—21no—
304C22—42no—
305C22—81no—
306C22—81no—
307C22#1683967411yescharacteristic
308C22#1683994811yescharacteristic
309C22#1683994811yescharacteristic
310C4—21no—
311C4—22no—
312C4—22no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—4882no—
318C4—4882no—
319C2—21no—
320C2—21no—
321C2—82no—
322C2#2144698311yescharacteristic
323C2#2144783911yescharacteristic
324C2#2144784411yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC22×Q8×C61C23
FrattiniC22×Q8C22×D61
FittingC61×#1793C2
socleC22×C61B(4,𝔽2)
prime radicalC2×2−1+4×C61C22
hypercenter#1793D61
hypocenterC61#13379
LCSG ▷ C22×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #21453807C2semidirect
3. #21443946C2semidirect
4. #21443945C2semidirect
181. C61#13379semidirect

Maximal subgroups

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

subgroupindex
#214538072
#214457782
#214439462
#214439452
C61×#17932
#1337961