Finite Group Databasebeta

← C61×#10017 #29513450

Subgroups — C61×#10017 #29513450

Order 15616.

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3104C212yesmaximal
2C61×#2399C211yesmaximal, characteristic
3C61×#1995C211yesmaximal, characteristic
4C61×#1487C212yesmaximal
5C61×#1274C211yesmaximal, characteristic
6C61×#533C2211yescharacteristic
7C2×C4×Q8×C61C2212yes—
8C61×#484—22no—
9C61×#477—22no—
10C61×#462—22no—
11C61×#451—22no—
12C4×Q16×C61—24no—
13C61×#431C2212yes—
14C61×#423C2211yescharacteristic
15C61×#398—22no—
16C61×#395—22no—
17C4×C4⋊C4×C61—22no—
18C61×#376C2211yescharacteristic
19C61×#339—22no—
20C61×Q8⋊C8—24no—
21C22×Q8×C61D412yes—
22C2×Q16×C61—24no—
23C2×M16×C61—21no—
24C4⋊Q8×C61D412yes—
25#125×C61—21no—
26C22⋊Q8×C61—22no—
27C4×Q8×C61—22no—
28C4×Q8×C61—22no—
29C4×Q8×C61—22no—
30C4×Q8×C61—22no—
31C4×Q8×C61—22no—
32#117×C61—21no—
33C2×C4⋊C4×C61—24no—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61D412yes—
36C2×C42×C61—22no—
37C2×C42×C61D412yes—
38C2×C42×C61C2311yescharacteristic
39#107×C61—24no—
40C4⋊C8×C61—22no—
41C4⋊C8×C61—22no—
42#103×C61—24no—
43#103×C61—24no—
44C22⋊C8×C61—22no—
45C4×C8×C61D414yes—
46#95×C61—22no—
47#95×C61—22no—
48#95×C61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61—22no—
52C2×Q8×C61—44no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61C2×D412yes—
57C22×C4×C61C2×D411yescharacteristic
58Q16×C61—48no—
59M16×C61—42no—
60C2×C8×C61—24no—
61C2×C8×C61—41no—
62C4⋊C4×C61—22no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—22no—
66C4⋊C4×C61—48no—
67C4⋊C4×C61—42no—
68C4⋊C4×C61—42no—
69C4⋊C4×C61—42no—
70C22⋊C4×C61—41no—
71C42×C61—22no—
72C42×C61—42no—
73C42×C61—42no—
74C42×C61—41no—
75C42×C61Q1614yescomplemented
76C42×C61C2×D411yescharacteristic
77C42×C61C2×D412yes—
78C42×C61#4111yescharacteristic
79C23×C61#12011yescharacteristic
80Q8×C61—24no—
81Q8×C61—24no—
82Q8×C61—42no—
83Q8×C61—42no—
84Q8×C61—44no—
85C2×C4×C61—24no—
86C2×C4×C61—24no—
87C2×C4×C61—42no—
88C2×C4×C61—44no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—44no—
92C2×C4×C61—42no—
93C2×C4×C61—42no—
94C2×C4×C61#12012yes—
95C2×C4×C61#12112yes—
96C2×C4×C61#12111yescharacteristic
97C2×C4×C61#12711yescharacteristic
98C2×C4×C61C2×Q1612yes—
99C2×C4×C61#13712yes—
100C8×C61—44no—
101C8×C61—81no—
102#10017C6111yesmaximal, characteristic, direct product factor, Sylow
103C22×C61—22no—
104C22×C61#39211yescharacteristic
105C22×C61#45112yes—
106C4×C61—22no—
107C4×C61—22no—
108C4×C61—44no—
109C4×C61—44no—
110C4×C61—82no—
111C4×C61—82no—
112C4×C61#46212yes—
113C4×C61#49411yescharacteristic
114C4×C61#49711yescharacteristic
115#3104C2×C6112yes—
116#2399C2×C6111yescharacteristic
117#1995C2×C6111yescharacteristic
118#1487C2×C6112yes—
119#1274C2×C6111yescharacteristic
120C2×C61—22no—
121C2×C61#201012yes—
122C2×C61#201611yescharacteristic
123#533C22×C6111yescharacteristic
124C2×C4×Q8C22×C6112yes—
125#484—22no—
126#477—22no—
127#462—22no—
128#451—22no—
129C4×Q16—24no—
130#431C22×C6112yes—
131#423C22×C6111yescharacteristic
132#398—22no—
133#395—22no—
134C4×C4⋊C4—22no—
135#376C22×C6111yescharacteristic
136#339—22no—
137Q8⋊C8—24no—
138C61#1001711yescharacteristic, direct product factor, Sylow
139C22×Q8D4×C6112yes—
140C2×Q16—24no—
141C2×M16—21no—
142C4⋊Q8D4×C6112yes—
143#125—21no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—21no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4D4×C6112yes—
154C2×C42—22no—
155C2×C42D4×C6112yes—
156C2×C42C23×C6111yescharacteristic
157#107—24no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160#103—24no—
161#103—24no—
162C22⋊C8—22no—
163C4×C8D4×C6114yes—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—44no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×D4×C6112yes—
175C22×C4C2×D4×C6111yescharacteristic
176Q16—48no—
177M16—42no—
178C2×C8—24no—
179C2×C8—41no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—48no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C22⋊C4—41no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—41no—
193C42Q16×C6114yescomplemented
194C42C2×D4×C6111yescharacteristic
195C42C2×D4×C6112yes—
196C42#41×C6111yescharacteristic
197C23#120×C6111yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—44no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4#120×C6112yes—
213C2×C4#121×C6112yes—
214C2×C4#121×C6111yescharacteristic
215C2×C4#127×C6111yescharacteristic
216C2×C4C2×Q16×C6112yes—
217C2×C4#137×C6112yes—
218C8—44no—
219C8—81no—
220C22—22no—
221C22C61×#39211yescharacteristic
222C22C61×#45112yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—82no—
229C4C61×#46212yes—
230C4C61×#49411yescharacteristic
231C4C61×#49711yescharacteristic
232C2—22no—
233C2C61×#201012yes—
234C2C61×#201611yescharacteristic
subgroupG/H
centerC2×C4×C61#127
commutator, Frattini, cosocleC2×C42C23×C61
socleC22×C61#392
prime radicalC23×C61#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
75. C42×C61Q16semidirect
102. #10017C61direct (N × H)
138. C61#10017direct (N × H)
193. C42Q16×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#31042
C61×#23992
C61×#19952
C61×#14872
C61×#12742
#1001761