Finite Group Databasebeta

← C61×#10216 #29513649

Subgroups — C61×#10216 #29513649

Order 15616.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3255C211yesmaximal, characteristic, complemented
2C61×#3235C212yesmaximal
3C61×#1963C211yesmaximal, characteristic
4C61×#1506C212yesmaximal, complemented
5C61×#1307C211yesmaximal, characteristic, complemented
6C61×#533C2212yes—
7C61×#532C2211yescharacteristic
8C61×#501—22no—
9C61×#490—22no—
10C61×#482—24no—
11C61×#476—22no—
12C61×#475—22no—
13C61×#474—22no—
14C61×#470—22no—
15C61×#462—22no—
16C61×#432C2212yes—
17C61×#423C2211yescharacteristic
18C2×#104×C61—22no—
19C61×#376C2211yescharacteristic
20C61×#328—24no—
21C2×#41×C61—21no—
22C2×Q16×C61—22no—
23C2×M16×C61D412yes—
24C4⋊Q8×C61D411yescharacteristic
25C4⋊Q8×C61D412yes—
26C4⋊Q8×C61D412yes—
27#127×C61D411yescharacteristic
28#125×C61—22no—
29#124×C61—21no—
30C22⋊Q8×C61—22no—
31C22⋊Q8×C61—22no—
32#121×C61—22no—
33C4×Q8×C61—22no—
34C4×Q8×C61—22no—
35C4×D4×C61—22no—
36#117×C61—22no—
37C2×C42×C61—21no—
38C2×C42×C61C2311yescharacteristic
39#107×C61—24no—
40#106×C61—22no—
41#106×C61—24no—
42C4⋊C8×C61—22no—
43C4⋊C8×C61—22no—
44#104×C61—44no—
45#103×C61—24no—
46#102×C61—24no—
47#102×C61—24no—
48C22⋊C8×C61D414yes—
49#97×C61—22no—
50#95×C61—22no—
51#41×C61—42no—
52#41×C61—42no—
53C2×Q8×C61—22no—
54C2×Q8×C61—21no—
55C2×Q8×C61—22no—
56C2×D4×C61—22no—
57C2×D4×C61—21no—
58C2×D4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63Q16×C61—82no—
64M16×C61—42no—
65C2×C8×C61—24no—
66C2×C8×C61—22no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—22no—
72C4⋊C4×C61—42no—
73C4⋊C4×C61—42no—
74C22⋊C4×C61—42no—
75C22⋊C4×C61—42no—
76C42×C61—22no—
77C42×C61—41no—
78C42×C61—42no—
79C42×C61C2×D411yescharacteristic
80C42×C61C2×D412yes—
81C42×C61#4111yescharacteristic
82C23×C61—41no—
83C23×C61#12711yescharacteristic
84Q8×C61—42no—
85Q8×C61—42no—
86Q8×C61—42no—
87Q8×C61—82no—
88D4×C61—44no—
89D4×C61—42no—
90D4×C61—82no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—42no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—41no—
98C2×C4×C61—82no—
99C2×C4×C61—81no—
100C2×C4×C61—81no—
101C2×C4×C61#12012yes—
102C2×C4×C61#12011yescharacteristic
103C2×C4×C61#12112yes—
104C2×C4×C61#12111yescharacteristic
105C8×C61—44no—
106C8×C61—42no—
107#10216C6111yesmaximal, characteristic, direct product factor, Sylow
108C22×C61—21no—
109C22×C61—22no—
110C22×C61—81no—
111C22×C61—82no—
112C22×C61#39211yescharacteristic
113C4×C61—21no—
114C4×C61—22no—
115C4×C61—22no—
116C4×C61—82no—
117C4×C61—81no—
118C4×C61—82no—
119C4×C61—82no—
120C4×C61#45311yescharacteristic
121C4×C61#45411yescharacteristic
122#3255C2×C6111yescharacteristic, complemented
123#3235C2×C6112yes—
124#1963C2×C6111yescharacteristic
125#1506C2×C6112yescomplemented
126#1307C2×C6111yescharacteristic, complemented
127C2×C61—41no—
128C2×C61—161no—
129C2×C61#200611yescharacteristic
130C2×C61#200912yes—
131#533C22×C6112yes—
132#532C22×C6111yescharacteristic
133#501—22no—
134#490—22no—
135#482—24no—
136#476—22no—
137#475—22no—
138#474—22no—
139#470—22no—
140#462—22no—
141#432C22×C6112yes—
142#423C22×C6111yescharacteristic
143C2×#104—22no—
144#376C22×C6111yescharacteristic
145#328—24no—
146C61#1021611yescharacteristic, direct product factor, Sylow
147C2×#41—21no—
148C2×Q16—22no—
149C2×M16D4×C6112yes—
150C4⋊Q8D4×C6111yescharacteristic
151C4⋊Q8D4×C6112yes—
152C4⋊Q8D4×C6112yes—
153#127D4×C6111yescharacteristic
154#125—22no—
155#124—21no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158#121—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×D4—22no—
162#117—22no—
163C2×C42—21no—
164C2×C42C23×C6111yescharacteristic
165#107—24no—
166#106—22no—
167#106—24no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170#104—44no—
171#103—24no—
172#102—24no—
173#102—24no—
174C22⋊C8D4×C6114yes—
175#97—22no—
176#95—22no—
177#41—42no—
178#41—42no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×D4×C6111yescharacteristic
188C22×C4C2×D4×C6112yes—
189Q16—82no—
190M16—42no—
191C2×C8—24no—
192C2×C8—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—41no—
204C42—42no—
205C42C2×D4×C6111yescharacteristic
206C42C2×D4×C6112yes—
207C42#41×C6111yescharacteristic
208C23—41no—
209C23#127×C6111yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—82no—
214D4—44no—
215D4—42no—
216D4—82no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4#120×C6112yes—
228C2×C4#120×C6111yescharacteristic
229C2×C4#121×C6112yes—
230C2×C4#121×C6111yescharacteristic
231C8—44no—
232C8—42no—
233C22—22no—
234C22—21no—
235C22—82no—
236C22—81no—
237C22C61×#39211yescharacteristic
238C4—22no—
239C4—22no—
240C4—21no—
241C4—82no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4C61×#45311yescharacteristic
246C4C61×#45411yescharacteristic
247C2—41no—
248C2—161no—
249C2C61×#200611yescharacteristic
250C2C61×#200912yes—
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC2×C42C23×C61
prime radicalC61×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3255C2semidirect
4. C61×#1506C2semidirect
5. C61×#1307C2semidirect
107. #10216C61direct (N × H)
122. #3255C2×C61semidirect
125. #1506C2×C61semidirect
126. #1307C2×C61semidirect
146. C61#10216direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#32552
C61×#32352
C61×#19632
C61×#15062
C61×#13072
#1021661