Finite Group Databasebeta

← C41×#12167 #23788368

Subgroups — C41×#12167 #23788368

Order 10496.

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3259C211yesmaximal, characteristic
2C41×#2185C212yesmaximal, complemented
3C41×#2047C211yesmaximal, characteristic, complemented
4C41×#1349C212yesmaximal, complemented
5C41×#1323C211yesmaximal, characteristic, complemented
6C22×Q16×C41C2211yescharacteristic
7C41×#540—21no—
8C2×SD32×C41—22no—
9C41×#478—22no—
10C41×#469—21no—
11C41×#466C2212yes—
12C41×#459—22no—
13C41×#424C2211yescharacteristic
14C41×#419—21no—
15C2×#101×C41—21no—
16C41×#358—22no—
17C41×C22⋊C16C2212yes—
18C41×#341C2211yescharacteristic
19C22×Q8×C41—21no—
20C2×Q16×C41—22no—
21C2×Q16×C41—42no—
22C2×D8×C41—21no—
23C2×D8×C41D412yes—
24C2×M16×C41—21no—
25C2×M16×C41D411yescharacteristic
26C22×C8×C41C2311yescharacteristic
27#127×C41—21no—
28#125×C41—21no—
29#123×C41—42no—
30#121×C41—22no—
31#121×C41—42no—
32C4×D4×C41—22no—
33#117×C41D411yescharacteristic
34SD32×C41—44no—
35C2×C16×C41—22no—
36#107×C41D412yes—
37#106×C41—21no—
38C4⋊C8×C41—22no—
39#103×C41—42no—
40#102×C41—22no—
41#102×C41—22no—
42#101×C41—42no—
43C2×Q8×C41—22no—
44C2×Q8×C41—41no—
45C2×Q8×C41—41no—
46C2×Q8×C41—41no—
47C2×Q8×C41—41no—
48C2×Q8×C41—41no—
49C2×D4×C41—22no—
50C2×D4×C41—42no—
51C2×D4×C41—42no—
52C22×C4×C41—42no—
53C22×C4×C41—41no—
54C22×C4×C41C2×D411yescharacteristic
55Q16×C41—24no—
56Q16×C41—42no—
57D8×C41—42no—
58D8×C41—81no—
59M16×C41—41no—
60M16×C41—41no—
61M16×C41—41no—
62C2×C8×C41—21no—
63C2×C8×C41—41no—
64C2×C8×C41—41no—
65C2×C8×C41C2×D412yes—
66C4⋊C4×C41—22no—
67C4⋊C4×C41—42no—
68C4⋊C4×C41D812yes—
69C22⋊C4×C41—21no—
70C22⋊C4×C41—42no—
71C22⋊C4×C41—42no—
72C42×C41—21no—
73C16×C41—42no—
74C23×C41—42no—
75C23×C41AGL(1,ℤ/8)11yescharacteristic
76Q8×C41—42no—
77Q8×C41—42no—
78Q8×C41—81no—
79Q8×C41—81no—
80D4×C41—42no—
81D4×C41—42no—
82D4×C41—162no—
83D4×C41—162no—
84C2×C4×C41—21no—
85C2×C4×C41—21no—
86C2×C4×C41—22no—
87C2×C4×C41—42no—
88C2×C4×C41—42no—
89C2×C4×C41—81no—
90C2×C4×C41—81no—
91C2×C4×C41—162no—
92C2×C4×C41#12011yescharacteristic
93C2×C4×C41C2×D811yescharacteristic
94C8×C41—22no—
95C8×C41—41no—
96C8×C41—81no—
97#12167C4111yesmaximal, characteristic, direct product factor, Sylow
98C22×C41—21no—
99C22×C41—22no—
100C22×C41—82no—
101C22×C41—82no—
102C22×C41—82no—
103C22×C41#44711yescharacteristic
104C4×C41—21no—
105C4×C41—42no—
106C4×C41—82no—
107C4×C41—82no—
108C4×C41#45411yescharacteristic
109C4×C41B(4,𝔽2)11yescharacteristic
110#3259C2×C4111yescharacteristic
111#2185C2×C4112yescomplemented
112#2047C2×C4111yescharacteristic, complemented
113#1349C2×C4112yescomplemented
114#1323C2×C4111yescharacteristic, complemented
115C2×C41—41no—
116C2×C41—162no—
117C2×C41#159811yescharacteristic
118C2×C41#219412yes—
119C22×Q16C22×C4111yescharacteristic
120#540—21no—
121C2×SD32—22no—
122#478—22no—
123#469—21no—
124#466C22×C4112yes—
125#459—22no—
126#424C22×C4111yescharacteristic
127#419—21no—
128C2×#101—21no—
129#358—22no—
130C22⋊C16C22×C4112yes—
131#341C22×C4111yescharacteristic
132C41#1216711yescharacteristic, direct product factor, Sylow
133C22×Q8—21no—
134C2×Q16—22no—
135C2×Q16—42no—
136C2×D8—21no—
137C2×D8D4×C4112yes—
138C2×M16—21no—
139C2×M16D4×C4111yescharacteristic
140C22×C8C23×C4111yescharacteristic
141#127—21no—
142#125—21no—
143#123—42no—
144#121—22no—
145#121—42no—
146C4×D4—22no—
147#117D4×C4111yescharacteristic
148SD32—44no—
149C2×C16—22no—
150#107D4×C4112yes—
151#106—21no—
152C4⋊C8—22no—
153#103—42no—
154#102—22no—
155#102—22no—
156#101—42no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C2×D4×C4111yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—42no—
172D8—81no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×D4×C4112yes—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4D8×C4112yes—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—21no—
187C16—42no—
188C23—42no—
189C23AGL(1,ℤ/8)×C4111yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—42no—
195D4—42no—
196D4—162no—
197D4—162no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—162no—
206C2×C4#120×C4111yescharacteristic
207C2×C4C2×D8×C4111yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—21no—
212C22—22no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22C41×#44711yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4C41×#45411yescharacteristic
222C4C41×B(4,𝔽2)11yescharacteristic
223C2—41no—
224C2—162no—
225C2C41×#159811yescharacteristic
226C2C41×#219412yes—
subgroupG/H
center, socleC22×C41#447
commutator, Frattini, cosocleC22×C8C23×C41
prime radicalC41×#3259C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C22×C4×C41 ◁ C41×#424 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2185C2semidirect
3. C41×#2047C2semidirect
4. C41×#1349C2semidirect
5. C41×#1323C2semidirect
97. #12167C41direct (N × H)
111. #2185C2×C41semidirect
112. #2047C2×C41semidirect
113. #1349C2×C41semidirect
114. #1323C2×C41semidirect
132. C41#12167direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#32592
C41×#21852
C41×#20472
C41×#13492
C41×#13232
#1216741