Finite Group Databasebeta

← C73×#12366 #32921770

Subgroups — C73×#12366 #32921770

Order 18688.

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

normal subgroups only

This group has 364 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
1C73×#3292C211yesmaximal, characteristic
2C73×#2230C212yesmaximal, complemented
3C73×#1715C211yesmaximal, characteristic, complemented
4C73×#1357C211yesmaximal, characteristic, complemented
5C73×#1336C212yesmaximal, complemented
6#576×C73—22no—
7#498×C73C2211yescharacteristic, complemented
8#493×C73C2211yescharacteristic
9#492×C73—21no—
10#486×C73—21no—
11#472×C73—22no—
12#453×C73—21no—
13#453×C73—21no—
14#443×C73C2212yes—
15#365×C73—22no—
16#364×C73C2212yescomplemented
17#360×C73—22no—
18#322×C73C2211yescharacteristic, complemented
19H(5,𝔽2)×C73—21no—
20H(5,𝔽2)×C73—21no—
21AGL(1,ℤ/8)×C73—22no—
22AGL(1,ℤ/8)×C73—22no—
23AGL(1,ℤ/8)×C73—42no—
24AGL(1,ℤ/8)×C73—42no—
25#135×C73—42no—
26#135×C73D412yes—
27C2×SD16×C73—42no—
28C2×D8×C73—22no—
29C2×D8×C73—42no—
30#131×C73—22no—
31C4⋊Q8×C73—21no—
32#127×C73D411yescharacteristic
33#127×C73D411yescharacteristic
34M32×C73—22no—
35#108×C73D412yes—
36#106×C73—22no—
37#104×C73—22no—
38#102×C73—42no—
39#100×C73—21no—
40#100×C73—21no—
41C4×C8×C73—21no—
42C4×C8×C73C2311yescharacteristic
43#41×C73—22no—
44#41×C73—42no—
45#41×C73—42no—
46C2×Q8×C73—41no—
47C2×D4×C73—21no—
48C2×D4×C73—21no—
49C2×D4×C73—41no—
50C2×D4×C73—41no—
51C2×D4×C73—42no—
52C2×D4×C73—42no—
53C2×D4×C73—81no—
54C2×D4×C73—81no—
55Q16×C73SD1612yescomplemented
56SD16×C73—22no—
57SD16×C73—42no—
58SD16×C73—42no—
59SD16×C73—84no—
60D8×C73—42no—
61D8×C73—42no—
62D8×C73—42no—
63D8×C73—42no—
64D8×C73SD1612yescomplemented
65M16×C73—22no—
66M16×C73—42no—
67C2×C8×C73—22no—
68C2×C8×C73—42no—
69C2×C8×C73C2×D412yes—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C42×C73C2×D411yescharacteristic
73C16×C73—42no—
74C23×C73—41no—
75C23×C73—41no—
76C23×C73—81no—
77C23×C73—81no—
78Q8×C73—22no—
79Q8×C73—82no—
80D4×C73—22no—
81D4×C73—42no—
82D4×C73—42no—
83D4×C73—42no—
84D4×C73—82no—
85D4×C73—82no—
86D4×C73—82no—
87D4×C73—82no—
88C2×C4×C73—21no—
89C2×C4×C73—42no—
90C2×C4×C73—81no—
91C2×C4×C73—81no—
92C2×C4×C73—81no—
93C2×C4×C73#12011yescharacteristic
94C8×C73—42no—
95C8×C73—42no—
96C8×C73C2×SD1612yes—
97C8×C73AGL(1,ℤ/8)12yes—
98C22×C73—42no—
99C22×C73—81no—
100C22×C73—81no—
101C22×C73—161no—
102C22×C73—161no—
103C22×C73—321no—
104C22×C73—321no—
105C22×C73#45311yescharacteristic
106C4×C73—22no—
107C4×C73—42no—
108C4×C73—161no—
109C4×C73#45012yes—
110#12366C7311yesmaximal, characteristic, direct product factor, Sylow
111C2×C73—21no—
112C2×C73—82no—
113C2×C73—161no—
114C2×C73—161no—
115C2×C73#162911yescharacteristic
116#3292C2×C7311yescharacteristic
117#2230C2×C7312yescomplemented
118#1715C2×C7311yescharacteristic, complemented
119#1357C2×C7311yescharacteristic, complemented
120#1336C2×C7312yescomplemented
121C73#1236611yescharacteristic, direct product factor, Sylow
122#576—22no—
123#498C22×C7311yescharacteristic, complemented
124#493C22×C7311yescharacteristic
125#492—21no—
126#486—21no—
127#472—22no—
128#453—21no—
129#453—21no—
130#443C22×C7312yes—
131#365—22no—
132#364C22×C7312yescomplemented
133#360—22no—
134#322C22×C7311yescharacteristic, complemented
135H(5,𝔽2)—21no—
136H(5,𝔽2)—21no—
137AGL(1,ℤ/8)—22no—
138AGL(1,ℤ/8)—22no—
139AGL(1,ℤ/8)—42no—
140AGL(1,ℤ/8)—42no—
141#135—42no—
142#135D4×C7312yes—
143C2×SD16—42no—
144C2×D8—22no—
145C2×D8—42no—
146#131—22no—
147C4⋊Q8—21no—
148#127D4×C7311yescharacteristic
149#127D4×C7311yescharacteristic
150M32—22no—
151#108D4×C7312yes—
152#106—22no—
153#104—22no—
154#102—42no—
155#100—21no—
156#100—21no—
157C4×C8—21no—
158C4×C8C23×C7311yescharacteristic
159#41—22no—
160#41—42no—
161#41—42no—
162C2×Q8—41no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
171Q16SD16×C7312yescomplemented
172SD16—22no—
173SD16—42no—
174SD16—42no—
175SD16—84no—
176D8—42no—
177D8—42no—
178D8—42no—
179D8—42no—
180D8SD16×C7312yescomplemented
181M16—22no—
182M16—42no—
183C2×C8—22no—
184C2×C8—42no—
185C2×C8C2×D4×C7312yes—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C42C2×D4×C7311yescharacteristic
189C16—42no—
190C23—41no—
191C23—41no—
192C23—81no—
193C23—81no—
194Q8—22no—
195Q8—82no—
196D4—22no—
197D4—42no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—82no—
203D4—82no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#120×C7311yescharacteristic
210C8—42no—
211C8—42no—
212C8C2×SD16×C7312yes—
213C8AGL(1,ℤ/8)×C7312yes—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—161no—
218C22—161no—
219C22—321no—
220C22—321no—
221C22#453×C7311yescharacteristic
222C4—22no—
223C4—42no—
224C4—161no—
225C4#450×C7312yes—
226C2—21no—
227C2—82no—
228C2—161no—
229C2—161no—
230C2C73×#162911yescharacteristic
subgroupG/H
center, socleC2×C73#1629
commutator, Frattini, cosocleC4×C8C23×C73
prime radicalC73×#3292C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C73 ◁ C2×C4×C73 ◁ C4×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2230C2semidirect
3. C73×#1715C2semidirect
4. C73×#1357C2semidirect
5. C73×#1336C2semidirect
7. #498×C73C22semidirect
16. #364×C73C22semidirect
18. #322×C73C22semidirect
55. Q16×C73SD16semidirect
64. D8×C73SD16semidirect
110. #12366C73direct (N × H)
117. #2230C2×C73semidirect
118. #1715C2×C73semidirect
119. #1357C2×C73semidirect
120. #1336C2×C73semidirect
121. C73#12366direct (N × H)
123. #498C22×C73semidirect
132. #364C22×C73semidirect
134. #322C22×C73semidirect
171. Q16SD16×C73semidirect
180. D8SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32922
C73×#22302
C73×#17152
C73×#13572
C73×#13362
#1236673