Finite Group Databasebeta

← C61×#10334 #29513767

Subgroups — C61×#10334 #29513767

Order 15616.

754 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C61×#3195C211yesmaximal, characteristic
2C61×#2434C212yesmaximal
3C61×#1784C211yesmaximal, characteristic
4C61×#1508C212yesmaximal
5C61×#1297C211yesmaximal, characteristic
6C23×Q8×C61C2211yescharacteristic
7C2×C4⋊Q8×C61—22no—
8C2×C22⋊Q8×C61C2212yes—
9C61×#483—22no—
10C61×#470—22no—
11C61×#451—22no—
12C2×#101×C61—22no—
13C61×#407C2211yescharacteristic
14C61×#407C2211yescharacteristic
15C61×#395—24no—
16C61×#391—22no—
17C61×#385—24no—
18C4×C22⋊C4×C61C2212yes—
19C61×#328—24no—
20C22×Q8×C61—22no—
21C22×Q8×C61—21no—
22C22×Q8×C61—22no—
23C22×Q8×C61—21no—
24C22×Q8×C61—44no—
25C22×Q8×C61D412yes—
26C23×C4×C61D412yes—
27C23×C4×C61C2311yescharacteristic
28C2×Q16×C61—22no—
29C2×M16×C61—21no—
30C2×M16×C61D412yes—
31C4⋊Q8×C61—48no—
32C22⋊Q8×C61—22no—
33C22⋊Q8×C61—22no—
34C22⋊Q8×C61—22no—
35C22⋊Q8×C61—22no—
36C2×C4⋊C4×C61—28no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61D412yes—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—22no—
41C2×C42×C61—22no—
42#106×C61—22no—
43#103×C61—24no—
44#101×C61—44no—
45C22⋊C8×C61—22no—
46C22⋊C8×C61—22no—
47#95×C61—28no—
48#95×C61D414yes—
49C24×C61C2×D411yescharacteristic
50C2×Q8×C61—24no—
51C2×Q8×C61—28no—
52C2×Q8×C61—22no—
53C2×Q8×C61—22no—
54C2×Q8×C61—44no—
55C2×Q8×C61—42no—
56C2×Q8×C61—48no—
57C2×Q8×C61—44no—
58C2×Q8×C61—44no—
59C2×Q8×C61—41no—
60C2×Q8×C61—42no—
61C2×Q8×C61—41no—
62C22×C4×C61—24no—
63C22×C4×C61—24no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—44no—
68C22×C4×C61—42no—
69C22×C4×C61C2×D412yes—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61#4111yescharacteristic
73Q16×C61—82no—
74M16×C61—42no—
75M16×C61—42no—
76C2×C8×C61—22no—
77C2×C8×C61—41no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—416no—
81C4⋊C4×C61—42no—
82C4⋊C4×C61—42no—
83C22⋊C4×C61—28no—
84C22⋊C4×C61—42no—
85C22⋊C4×C61—42no—
86C42×C61—44no—
87C23×C61—24no—
88C23×C61—41no—
89C23×C61#12012yes—
90C23×C61#12111yescharacteristic
91Q8×C61—44no—
92Q8×C61—48no—
93Q8×C61—44no—
94Q8×C61—44no—
95Q8×C61—82no—
96C2×C4×C61—24no—
97C2×C4×C61—22no—
98C2×C4×C61—28no—
99C2×C4×C61—28no—
100C2×C4×C61—24no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—44no—
104C2×C4×C61—44no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—44no—
108C2×C4×C61—44no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—44no—
112C2×C4×C61#12011yescharacteristic
113C2×C4×C61#12112yes—
114C2×C4×C61#12711yescharacteristic
115C8×C61—42no—
116C8×C61—81no—
117#10334C6111yesmaximal, characteristic, direct product factor, Sylow
118C22×C61—22no—
119C22×C61—22no—
120C22×C61—22no—
121C22×C61—21no—
122C22×C61—44no—
123C22×C61#39211yescharacteristic
124C22×C61#45614yes—
125C4×C61—21no—
126C4×C61—22no—
127C4×C61—48no—
128C4×C61—44no—
129C4×C61—82no—
130C4×C61—82no—
131C4×C61B(4,𝔽2)12yes—
132#3195C2×C6111yescharacteristic
133#2434C2×C6112yes—
134#1784C2×C6111yescharacteristic
135#1508C2×C6112yes—
136#1297C2×C6111yescharacteristic
137C2×C61—24no—
138C2×C61—41no—
139C2×C61#200512yes—
140C2×C61#202111yescharacteristic
141C23×Q8C22×C6111yescharacteristic
142C2×C4⋊Q8—22no—
143C2×C22⋊Q8C22×C6112yes—
144#483—22no—
145#470—22no—
146#451—22no—
147C2×#101—22no—
148#407C22×C6111yescharacteristic
149#407C22×C6111yescharacteristic
150#395—24no—
151#391—22no—
152#385—24no—
153C4×C22⋊C4C22×C6112yes—
154#328—24no—
155C61#1033411yescharacteristic, direct product factor, Sylow
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—44no—
161C22×Q8D4×C6112yes—
162C23×C4D4×C6112yes—
163C23×C4C23×C6111yescharacteristic
164C2×Q16—22no—
165C2×M16—21no—
166C2×M16D4×C6112yes—
167C4⋊Q8—48no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4D4×C6112yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178#106—22no—
179#103—24no—
180#101—44no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183#95—28no—
184#95D4×C6114yes—
185C24C2×D4×C6111yescharacteristic
186C2×Q8—24no—
187C2×Q8—28no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—44no—
191C2×Q8—42no—
192C2×Q8—48no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—41no—
196C2×Q8—42no—
197C2×Q8—41no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4C2×D4×C6112yes—
206C22×C4C2×D4×C6111yescharacteristic
207C22×C4C2×D4×C6112yes—
208C22×C4#41×C6111yescharacteristic
209Q16—82no—
210M16—42no—
211M16—42no—
212C2×C8—22no—
213C2×C8—41no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—416no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C22⋊C4—28no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—44no—
223C23—24no—
224C23—41no—
225C23#120×C6112yes—
226C23#121×C6111yescharacteristic
227Q8—44no—
228Q8—48no—
229Q8—44no—
230Q8—44no—
231Q8—82no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—28no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4#120×C6111yescharacteristic
249C2×C4#121×C6112yes—
250C2×C4#127×C6111yescharacteristic
251C8—42no—
252C8—81no—
253C22—22no—
254C22—22no—
255C22—21no—
256C22—22no—
257C22—44no—
258C22C61×#39211yescharacteristic
259C22C61×#45614yes—
260C4—21no—
261C4—22no—
262C4—48no—
263C4—44no—
264C4—82no—
265C4—82no—
266C4C61×B(4,𝔽2)12yes—
267C2—24no—
268C2—41no—
269C2C61×#200512yes—
270C2C61×#202111yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC23×C4C23×C61
prime radicalC24×C61C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
117. #10334C61direct (N × H)
155. C61#10334direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31952
C61×#24342
C61×#17842
C61×#15082
C61×#12972
#1033461