Finite Group Databasebeta

← C61×#13596 #29517029

Subgroups — C61×#13596 #29517029

Order 15616.

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3142C211yesmaximal, characteristic
2C61×#2126C212yesmaximal
3C61×#1412C214yesmaximal, complemented
4C61×#532C2211yescharacteristic
5C2×#125×C61C411yescharacteristic, complemented
6C61×#491C412yescomplemented
7C61×#490C2212yes—
8C61×#488—22no—
9C61×#417—22no—
10C61×#410—22no—
11C4×M16×C61C411yescharacteristic, complemented
12C61×#354—22no—
13C61×#353C2214yes—
14C2×#41×C61D412yes—
15C2×M16×C61—21no—
16C4⋊Q8×C61C2×C411yescharacteristic
17#127×C61C2311yescharacteristic
18#125×C61—21no—
19#125×C61—41no—
20#125×C61C2×C412yes—
21#124×C61D412yes—
22#121×C61—41no—
23C4×D4×C61—41no—
24#117×C61—42no—
25C2×C4⋊C4×C61—21no—
26C2×C4⋊C4×C61—41no—
27C2×C42×C61C2×C411yescharacteristic
28#103×C61—24no—
29#102×C61—24no—
30#99×C61—24no—
31#99×C61—42no—
32#97×C61C2×C412yes—
33C4×C8×C61—21no—
34#41×C61—42no—
35#41×C61—42no—
36C2×Q8×C61C22⋊C412yes—
37C2×D4×C61—22no—
38C2×D4×C61—41no—
39C2×D4×C61C2×D412yes—
40C22×C4×C61—21no—
41C22×C4×C61—22no—
42C22×C4×C61—41no—
43C22×C4×C61C22⋊C411yescharacteristic
44M16×C61—42no—
45C2×C8×C61—22no—
46C4⋊C4×C61—22no—
47C4⋊C4×C61—41no—
48C4⋊C4×C61—42no—
49C4⋊C4×C61—42no—
50C4⋊C4×C61—41no—
51C4⋊C4×C61—82no—
52C22⋊C4×C61—42no—
53C22⋊C4×C61—44no—
54C42×C61—21no—
55C42×C61—82no—
56C42×C61C22⋊C411yescharacteristic
57C42×C61C22×C411yescharacteristic
58C23×C61—22no—
59C23×C61#9911yescharacteristic
60Q8×C61—42no—
61D4×C61—42no—
62D4×C61—82no—
63D4×C61—82no—
64C2×C4×C61—22no—
65C2×C4×C61—21no—
66C2×C4×C61—21no—
67C2×C4×C61—22no—
68C2×C4×C61—42no—
69C2×C4×C61—42no—
70C2×C4×C61—42no—
71C2×C4×C61—81no—
72C2×C4×C61—82no—
73C2×C4×C61—84no—
74C2×C4×C61#9911yescharacteristic
75C2×C4×C61C2×C22⋊C411yescharacteristic
76C8×C61—42no—
77#13596C6111yesmaximal, characteristic, direct product factor, Sylow
78C22×C61—21no—
79C22×C61—41no—
80C22×C61—42no—
81C22×C61—82no—
82C22×C61C2×#9911yescharacteristic
83C4×C61—21no—
84C4×C61—41no—
85C4×C61—41no—
86C4×C61—42no—
87C4×C61—82no—
88C4×C61—164no—
89C4×C61#35412yes—
90#3142C2×C6111yescharacteristic
91#2126C2×C6112yes—
92#1412C2×C6114yescomplemented
93C2×C61—21no—
94C2×C61—41no—
95C2×C61—82no—
96C2×C61C2×#35411yescharacteristic
97#532C22×C6111yescharacteristic
98C2×#125C4×C6111yescharacteristic, complemented
99#491C4×C6112yescomplemented
100#490C22×C6112yes—
101#488—22no—
102#417—22no—
103#410—22no—
104C4×M16C4×C6111yescharacteristic, complemented
105#354—22no—
106#353C22×C6114yes—
107C61#1359611yescharacteristic, direct product factor, Sylow
108C2×#41D4×C6112yes—
109C2×M16—21no—
110C4⋊Q8C2×C4×C6111yescharacteristic
111#127C23×C6111yescharacteristic
112#125—21no—
113#125—41no—
114#125C2×C4×C6112yes—
115#124D4×C6112yes—
116#121—41no—
117C4×D4—41no—
118#117—42no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C42C2×C4×C6111yescharacteristic
122#103—24no—
123#102—24no—
124#99—24no—
125#99—42no—
126#97C2×C4×C6112yes—
127C4×C8—21no—
128#41—42no—
129#41—42no—
130C2×Q8C22⋊C4×C6112yes—
131C2×D4—22no—
132C2×D4—41no—
133C2×D4C2×D4×C6112yes—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4C22⋊C4×C6111yescharacteristic
138M16—42no—
139C2×C8—22no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—82no—
146C22⋊C4—42no—
147C22⋊C4—44no—
148C42—21no—
149C42—82no—
150C42C22⋊C4×C6111yescharacteristic
151C42C22×C4×C6111yescharacteristic
152C23—22no—
153C23#99×C6111yescharacteristic
154Q8—42no—
155D4—42no—
156D4—82no—
157D4—82no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—81no—
166C2×C4—82no—
167C2×C4—84no—
168C2×C4#99×C6111yescharacteristic
169C2×C4C2×C22⋊C4×C6111yescharacteristic
170C8—42no—
171C22—21no—
172C22—41no—
173C22—42no—
174C22—82no—
175C22C2×#99×C6111yescharacteristic
176C4—21no—
177C4—41no—
178C4—41no—
179C4—42no—
180C4—82no—
181C4—164no—
182C4C61×#35412yes—
183C2—21no—
184C2—41no—
185C2—82no—
186C2C2×C61×#35411yescharacteristic
subgroupG/H
center, socleC2×C61C2×#354
commutatorC42C22×C4×C61
Frattini, cosocle#127C23×C61
prime radicalC61×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C61 ◁ C2×C4×C61 ◁ C22×C4×C61 ◁ C61×#532 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#1412C2semidirect
5. C2×#125×C61C4semidirect
6. C61×#491C4semidirect
11. C4×M16×C61C4semidirect
77. #13596C61direct (N × H)
92. #1412C2×C61semidirect
98. C2×#125C4×C61semidirect
99. #491C4×C61semidirect
104. C4×M16C4×C61semidirect
107. C61#13596direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31422
C61×#21262
C61×#14122
#1359661