Finite Group Databasebeta

← C61×#2816 #21442857

Subgroups — C61×#2816 #21442857

Order 7808.

528 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C2×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
2C2×#121×C61C211yesmaximal, characteristic
3C61×#399C212yesmaximal, complemented
4C61×#398C212yesmaximal, complemented
5C61×#394C212yesmaximal, complemented
6C61×#394C214yesmaximal, complemented
7C61×#393C212yesmaximal, complemented
8C61×#381C211yesmaximal, characteristic, complemented
9C22×Q8×C61C2211yescharacteristic
10C22×D4×C61C2212yes—
11C22×D4×C61C2211yescharacteristic
12C23×C4×C61C2211yescharacteristic
13C22⋊Q8×C61—24no—
14#121×C61—24no—
15C2×C4⋊C4×C61C2212yes—
16C2×C4⋊C4×C61C2211yescharacteristic
17C2×C4⋊C4×C61C2211yescharacteristic
18C2×C4⋊C4×C61C2214yes—
19C2×C22⋊C4×C61C2212yes—
20C2×C22⋊C4×C61C2212yes—
21C2×C22⋊C4×C61C2212yes—
22C2×C22⋊C4×C61C2214yes—
23C2×C22⋊C4×C61C2214yes—
24#95×C61C2212yes—
25#95×C61C2212yes—
26#95×C61C2214yes—
27C24×C61C2312yes—
28C24×C61C2311yescharacteristic
29C2×Q8×C61—22no—
30C2×Q8×C61—42no—
31C2×D4×C61—24no—
32C2×D4×C61—22no—
33C2×D4×C61—42no—
34C2×D4×C61—44no—
35C22×C4×C61—42no—
36C22×C4×C61D414yes—
37C22×C4×C61C2314yes—
38C22×C4×C61C2312yes—
39C22×C4×C61C2312yes—
40C22×C4×C61C2312yes—
41C22×C4×C61C2311yescharacteristic
42C22×C4×C61C2311yescharacteristic
43C4⋊C4×C61—22no—
44C4⋊C4×C61—22no—
45C4⋊C4×C61—24no—
46C4⋊C4×C61—44no—
47C22⋊C4×C61—24no—
48C22⋊C4×C61—24no—
49C22⋊C4×C61—42no—
50C22⋊C4×C61—44no—
51C22⋊C4×C61—44no—
52C23×C61—22no—
53C23×C61—22no—
54C23×C61—22no—
55C23×C61—22no—
56C23×C61—24no—
57C23×C61—24no—
58C23×C61—24no—
59C23×C61C2×D412yes—
60C23×C61C2411yescharacteristic
61Q8×C61—44no—
62D4×C61—44no—
63D4×C61—48no—
64C2×C4×C61—22no—
65C2×C4×C61—22no—
66C2×C4×C61—22no—
67C2×C4×C61—22no—
68C2×C4×C61—22no—
69C2×C4×C61—22no—
70C2×C4×C61—24no—
71C2×C4×C61—24no—
72C2×C4×C61—24no—
73C2×C4×C61—22no—
74C2×C4×C61—24no—
75C2×C4×C61—24no—
76C2×C4×C61—44no—
77C2×C4×C61C2×D412yes—
78C2×C4×C61C2×D412yes—
79C22×C61—22no—
80C22×C61—22no—
81C22×C61—22no—
82C22×C61—44no—
83C22×C61—42no—
84C22×C61—42no—
85C22×C61—42no—
86C22×C61—48no—
87C22×C61C22×D411yescharacteristic
88C22×C61H(5,𝔽2)12yes—
89C22×C61H(5,𝔽2)12yes—
90C22×C612−1+412yes—
91C4×C61—22no—
92C4×C61—22no—
93C4×C61—42no—
94C4×C61—42no—
95C4×C61—42no—
96C4×C61—44no—
97#2816C6111yesmaximal, characteristic, direct product factor, Sylow
98C2×C61—42no—
99C2×C61—82no—
100C2×C61#53411yescharacteristic
101C2×C61#53511yescharacteristic
102C2×C61#53611yescharacteristic
103C2×C61#56214yes—
104C2×C22⋊Q8C2×C6111yescharacteristic, complemented
105C2×#121C2×C6111yescharacteristic
106#399C2×C6112yescomplemented
107#398C2×C6112yescomplemented
108#394C2×C6112yescomplemented
109#394C2×C6114yescomplemented
110#393C2×C6112yescomplemented
111#381C2×C6111yescharacteristic, complemented
112C61#281611yescharacteristic, direct product factor, Sylow
113C22×Q8C22×C6111yescharacteristic
114C22×D4C22×C6112yes—
115C22×D4C22×C6111yescharacteristic
116C23×C4C22×C6111yescharacteristic
117C22⋊Q8—24no—
118#121—24no—
119C2×C4⋊C4C22×C6111yescharacteristic
120C2×C4⋊C4C22×C6112yes—
121C2×C4⋊C4C22×C6111yescharacteristic
122C2×C4⋊C4C22×C6114yes—
123C2×C22⋊C4C22×C6112yes—
124C2×C22⋊C4C22×C6112yes—
125C2×C22⋊C4C22×C6112yes—
126C2×C22⋊C4C22×C6114yes—
127C2×C22⋊C4C22×C6114yes—
128#95C22×C6112yes—
129#95C22×C6112yes—
130#95C22×C6114yes—
131C24C23×C6112yes—
132C24C23×C6111yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—44no—
139C22×C4—42no—
140C22×C4D4×C6114yes—
141C22×C4C23×C6114yes—
142C22×C4C23×C6112yes—
143C22×C4C23×C6112yes—
144C22×C4C23×C6112yes—
145C22×C4C23×C6111yescharacteristic
146C22×C4C23×C6111yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—44no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23—24no—
163C23C2×D4×C6112yes—
164C23C24×C6111yescharacteristic
165Q8—44no—
166D4—44no—
167D4—48no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4C2×D4×C6112yes—
182C2×C4C2×D4×C6112yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—48no—
191C22C22×D4×C6111yescharacteristic
192C22H(5,𝔽2)×C6112yes—
193C22H(5,𝔽2)×C6112yes—
194C222−1+4×C6112yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—44no—
201C2—42no—
202C2—82no—
203C2C61×#53411yescharacteristic
204C2C61×#53511yescharacteristic
205C2C61×#53611yescharacteristic
206C2C61×#56214yes—
subgroupG/H
center, socleC23×C61C24
commutator, Frattini, cosocleC23C24×C61
prime radicalC2×#121×C61C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊Q8×C61C2semidirect
3. C61×#399C2semidirect
4. C61×#398C2semidirect
5. C61×#394C2semidirect
6. C61×#394C2semidirect
7. C61×#393C2semidirect
8. C61×#381C2semidirect
97. #2816C61direct (N × H)
104. C2×C22⋊Q8C2×C61semidirect
106. #399C2×C61semidirect
107. #398C2×C61semidirect
108. #394C2×C61semidirect
109. #394C2×C61semidirect
110. #393C2×C61semidirect
111. #381C2×C61semidirect
112. C61#2816direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×C612
C2×#121×C612
C61×#3992
C61×#3982
C61×#3942
C61×#3932
C61×#3812
#281661