Finite Group Databasebeta

← C41×#12310 #23788511

Subgroups — C41×#12310 #23788511

Order 10496.

406 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C41×#3292C211yesmaximal, characteristic, complemented
2C41×#2213C212yesmaximal, complemented
3C41×#1712C211yesmaximal, characteristic, complemented
4C41×#1357C211yesmaximal, characteristic, complemented
5C41×#1335C212yesmaximal, complemented
6C41×#576—22no—
7C41×#509—22no—
8C41×#493—22no—
9C41×#493C2211yescharacteristic
10C41×#493C2211yescharacteristic, complemented
11C41×#472—22no—
12C41×#453—22no—
13C41×#443C2212yes—
14C41×#364C2212yescomplemented
15C41×#361—22no—
16C82×C41C2211yescharacteristic, complemented
17H(5,𝔽2)×C41—22no—
18AGL(1,ℤ/8)×C41—24no—
19AGL(1,ℤ/8)×C41—44no—
20#135×C41—42no—
21#135×C41D412yes—
22C2×D8×C41—22no—
23C2×D8×C41—22no—
24C2×D8×C41—44no—
25C2×D8×C41—42no—
26#131×C41—22no—
27#127×C41—21no—
28#127×C41D412yes—
29SD32×C41—42no—
30D16×C41—42no—
31M32×C41—22no—
32#108×C41D412yes—
33#104×C41—22no—
34#100×C41—22no—
35C4×C8×C41—21no—
36C4×C8×C41C2311yescharacteristic
37#41×C41—22no—
38#41×C41—44no—
39C2×D4×C41—22no—
40C2×D4×C41—42no—
41C2×D4×C41—42no—
42C2×D4×C41—44no—
43C2×D4×C41—41no—
44C2×D4×C41—82no—
45Q16×C41D812yescomplemented
46SD16×C41—22no—
47SD16×C41—44no—
48D8×C41—44no—
49D8×C41—42no—
50D8×C41—42no—
51D8×C41—42no—
52D8×C41—42no—
53D8×C41—88no—
54D8×C41D812yescomplemented
55M16×C41—22no—
56M16×C41—42no—
57C2×C8×C41—22no—
58C2×C8×C41—42no—
59C2×C8×C41C2×D412yes—
60C42×C41C2×D411yescharacteristic
61C16×C41—42no—
62C23×C41—42no—
63C23×C41—82no—
64C23×C41—81no—
65Q8×C41—22no—
66D4×C41—22no—
67D4×C41—42no—
68D4×C41—44no—
69D4×C41—84no—
70D4×C41—84no—
71D4×C41—84no—
72D4×C41—82no—
73C2×C4×C41—21no—
74C2×C4×C41—42no—
75C2×C4×C41—82no—
76C2×C4×C41#12011yescharacteristic
77C8×C41—24no—
78C8×C41—42no—
79C8×C41C2×D812yes—
80C8×C41AGL(1,ℤ/8)12yes—
81#12310C4111yesmaximal, characteristic, direct product factor, Sylow
82C22×C41—42no—
83C22×C41—82no—
84C22×C41—162no—
85C22×C41—162no—
86C22×C41—161no—
87C22×C41—322no—
88C22×C41#45311yescharacteristic
89C4×C41—22no—
90C4×C41—42no—
91C4×C41#44712yes—
92#3292C2×C4111yescharacteristic, complemented
93#2213C2×C4112yescomplemented
94#1712C2×C4111yescharacteristic, complemented
95#1357C2×C4111yescharacteristic, complemented
96#1335C2×C4112yescomplemented
97C2×C41—21no—
98C2×C41—82no—
99C2×C41—162no—
100C2×C41—321no—
101C2×C41#161911yescharacteristic
102#576—22no—
103#509—22no—
104#493—22no—
105#493C22×C4111yescharacteristic
106#493C22×C4111yescharacteristic, complemented
107#472—22no—
108#453—22no—
109#443C22×C4112yes—
110#364C22×C4112yescomplemented
111#361—22no—
112C82C22×C4111yescharacteristic, complemented
113C41#1231011yescharacteristic, direct product factor, Sylow
114H(5,𝔽2)—22no—
115AGL(1,ℤ/8)—24no—
116AGL(1,ℤ/8)—44no—
117#135—42no—
118#135D4×C4112yes—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—44no—
122C2×D8—42no—
123#131—22no—
124#127—21no—
125#127D4×C4112yes—
126SD32—42no—
127D16—42no—
128M32—22no—
129#108D4×C4112yes—
130#104—22no—
131#100—22no—
132C4×C8—21no—
133C4×C8C23×C4111yescharacteristic
134#41—22no—
135#41—44no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—44no—
140C2×D4—41no—
141C2×D4—82no—
142Q16D8×C4112yescomplemented
143SD16—22no—
144SD16—44no—
145D8—44no—
146D8—42no—
147D8—42no—
148D8—42no—
149D8—42no—
150D8—88no—
151D8D8×C4112yescomplemented
152M16—22no—
153M16—42no—
154C2×C8—22no—
155C2×C8—42no—
156C2×C8C2×D4×C4112yes—
157C42C2×D4×C4111yescharacteristic
158C16—42no—
159C23—42no—
160C23—82no—
161C23—81no—
162Q8—22no—
163D4—22no—
164D4—42no—
165D4—44no—
166D4—84no—
167D4—84no—
168D4—84no—
169D4—82no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4#120×C4111yescharacteristic
174C8—24no—
175C8—42no—
176C8C2×D8×C4112yes—
177C8AGL(1,ℤ/8)×C4112yes—
178C22—42no—
179C22—82no—
180C22—162no—
181C22—162no—
182C22—161no—
183C22—322no—
184C22C41×#45311yescharacteristic
185C4—22no—
186C4—42no—
187C4C41×#44712yes—
188C2—21no—
189C2—82no—
190C2—162no—
191C2—321no—
192C2C41×#161911yescharacteristic
subgroupG/H
center, socleC2×C41#1619
commutator, Frattini, cosocleC4×C8C23×C41
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C41 ◁ C2×C4×C41 ◁ C4×C8×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C41×#3292C2semidirect
2. C41×#2213C2semidirect
3. C41×#1712C2semidirect
4. C41×#1357C2semidirect
5. C41×#1335C2semidirect
10. C41×#493C22semidirect
14. C41×#364C22semidirect
16. C82×C41C22semidirect
45. Q16×C41D8semidirect
54. D8×C41D8semidirect
81. #12310C41direct (N × H)
92. #3292C2×C41semidirect
93. #2213C2×C41semidirect
94. #1712C2×C41semidirect
95. #1357C2×C41semidirect
96. #1335C2×C41semidirect
106. #493C22×C41semidirect
110. #364C22×C41semidirect
112. C82C22×C41semidirect
113. C41#12310direct (N × H)
142. Q16D8×C41semidirect
151. D8D8×C41semidirect

Maximal subgroups

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

subgroupindex
C41×#32922
C41×#22132
C41×#17122
C41×#13572
C41×#13352
#1231041