Finite Group Databasebeta

← #29592526

Subgroups — #29592526

Order 15616.

332 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 55 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#21457273C211yesmaximal, characteristic, complemented
2#21447770C212yesmaximal
3#21443806C212yesmaximal, complemented
4#21443772C211yesmaximal, characteristic, complemented
5C61×#2053C211yesmaximal, characteristic
6#16840374—21no—
7#16840202—22no—
8#16840070—22no—
9#16840020C2212yes—
10#16840003—21no—
11#16839973—22no—
12#16839572—22no—
13#16839544C2212yes—
14C61×#575C2211yescharacteristic
15C61×#424C2211yescharacteristic
16C61×#416—21no—
17C61×#410—21no—
18C61×H(3,ℤ/4)C2211yescharacteristic
19C2×Dic244—21no—
20C2×Dic244D412yes—
21#14519777—21no—
22#14519770—42no—
23#14519769—21no—
24#14519767—42no—
25#14519756—22no—
26C4×Dic122—22no—
27#14519723—42no—
28#14519721—42no—
29#14519717—22no—
30#14519711—21no—
31#14519709D412yes—
32#14519702—22no—
33#14519702—22no—
34C2×#41×C61—21no—
35#137×C61—41no—
36AGL(1,ℤ/8)×C61—41no—
37#135×C61—22no—
38C2×M16×C61C2311yescharacteristic
39C22×C8×C61D411yescharacteristic
40#117×C61—21no—
41#117×C61D411yescharacteristic
42C4⋊C8×C61—22no—
43#103×C61—41no—
44#102×C61—41no—
45#99×C61—42no—
46#41×C61—41no—
47#41×C61—41no—
48#41×C61—41no—
49#41×C61—41no—
50C2×Q8×C61—21no—
51C2×D4×C61—21no—
52C2×D4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61C2×D411yescharacteristic
55C2×Dic122—22no—
56Q16×C61—22no—
57SD16×C61—42no—
58D8×C61—22no—
59M16×C61—41no—
60C2×C8×C61—21no—
61C2×C8×C61—41no—
62C2×C8×C61C2×D412yes—
63C4⋊C4×C61—21no—
64C4⋊C4×C61—41no—
65C22⋊C4×C61—41no—
66C22⋊C4×C61D812yes—
67C42×C61—21no—
68C42×C61—41no—
69C22⋊Dic61—42no—
70C4⋊Dic61—22no—
71C4⋊Dic61—42no—
72#11758196—42no—
73#11758196—42no—
74#11758196—42no—
75C4×Dic61—42no—
76Dic244—42no—
77Dic244—81no—
78#11758185—42no—
79C23×C61—41no—
80C23×C61C2×D811yescharacteristic
81Q8×C61—41no—
82Q8×C61—41no—
83D4×C61—41no—
84D4×C61—41no—
85D4×C61—81no—
86D4×C61—81no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—22no—
90C2×C4×C61—41no—
91C2×C4×C61—81no—
92C2×C4×C61—81no—
93C2×C4×C61—81no—
94C2×C4×C61#12011yescharacteristic
95C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
96C2×Dic61—42no—
97C2×Dic61—42no—
98Dic122—42no—
99Dic122—42no—
100C8×C61—22no—
101C8×C61—41no—
102#12171—611nomaximal, Sylow
103C22×C61—21no—
104C22×C61—41no—
105C22×C61—81no—
106C22×C61—161no—
107C22×C61#44711yescharacteristic
108C4×C61—21no—
109C4×C61—42no—
110C4×C61—81no—
111C4×C61—161no—
112C4×C61#45411yescharacteristic
113C4×C61B(4,𝔽2)11yescharacteristic
114Dic61—82no—
115Dic61—82no—
116#3261—611no—
117#2191—612no—
118#2053D6111yescharacteristic
119#1340—612no—
120#1328—611no—
121C2×C61—21no—
122C2×C61—41no—
123C2×C61—161no—
124C2×C61#159811yescharacteristic
125#575C2×D6111yescharacteristic
126#541—1221no—
127#510—1222no—
128#479—1222no—
129#470—612no—
130#467—1221no—
131#462—1222no—
132#424C2×D6111yescharacteristic
133#416—21no—
134#410—21no—
135#360—1222no—
136#349—612no—
137H(3,ℤ/4)C2×D6111yescharacteristic
138C61#1217111yescharacteristic, complemented, Sylow
139C2×#41—21no—
140#137—41no—
141AGL(1,ℤ/8)—41no—
142#135—22no—
143C2×Q16—612no—
144C2×Q16—1221no—
145C2×M16C22×D6111yescharacteristic
146C22×C8C22⋊D6111yescharacteristic
147C4⋊Q8—1221no—
148#126—2442no—
149#125—1221no—
150#125—2442no—
151C22⋊Q8—1222no—
152C4×Q8—1222no—
153#117—21no—
154#117C22⋊D6111yescharacteristic
155Q32—2442no—
156SD32—2442no—
157M32—1222no—
158#107—1221no—
159#106—612no—
160C4⋊C8—22no—
161#103—41no—
162#103—1222no—
163#103—1222no—
164#102—41no—
165#99—42no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—1222no—
172C2×D4—21no—
173C2×D4—41no—
174C22×C4—41no—
175C22×C4C2×C22⋊D6111yescharacteristic
176Q16—22no—
177Q16—4881no—
178Q16—2442no—
179SD16—42no—
180D8—22no—
181M16—41no—
182C2×C8—21no—
183C2×C8—41no—
184C2×C8D4×D6112yes—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—1222no—
188C4⋊C4—2442no—
189C4⋊C4—2442no—
190C4⋊C4—2442no—
191C4⋊C4—2442no—
192C22⋊C4—41no—
193C22⋊C4—2442no—
194C22⋊C4#1175819912yes—
195C42—21no—
196C42—41no—
197C42—2442no—
198C16—2442no—
199C23—41no—
200C23C2×#1175819911yescharacteristic
201Q8—41no—
202Q8—41no—
203Q8—2442no—
204Q8—2442no—
205D4—41no—
206D4—41no—
207D4—81no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—2442no—
217C2×C4—2442no—
218C2×C4#1451974511yescharacteristic
219C2×C4#1451980211yescharacteristic
220C8—22no—
221C8—41no—
222C22—21no—
223C22—41no—
224C22—81no—
225C22—161no—
226C22#1683989511yescharacteristic
227C4—21no—
228C4—42no—
229C4—81no—
230C4—161no—
231C4—4882no—
232C4—4882no—
233C4#1683993411yescharacteristic
234C4#1683994811yescharacteristic
235C2—21no—
236C2—41no—
237C2—161no—
238C2#2144475311yescharacteristic
subgroupG/H
centerC2#21444753
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#2053C2
socleC2×C61#1598
prime radicalC61×#575C22
hypercenter#2053D61
hypocenterC61#12171
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2053

Complements

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

NH ≅ G/Ntype
1. #21457273C2semidirect
3. #21443806C2semidirect
4. #21443772C2semidirect
138. C61#12171semidirect

Maximal subgroups

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

subgroupindex
#214572732
#214477702
#214438062
#214437722
C61×#20532
#1217161