Finite Group Databasebeta

← #34202038

Subgroups — #34202038

Order 20224.

364 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#23765571C212yesmaximal, complemented
2#23763026C211yesmaximal, characteristic, complemented
3#23761437C211yesmaximal, characteristic, complemented
4#23761376C212yesmaximal, complemented
5C79×#3292C211yesmaximal, characteristic
6#19098961C2211yescharacteristic, complemented
7#19098935—21no—
8#19098909—21no—
9#19098387—22no—
10#19098384C2212yescomplemented
11#19098373—22no—
12#19098284C2211yescharacteristic, complemented
13#576×C79—22no—
14#493×C79C2211yescharacteristic
15#472×C79—22no—
16#453×C79—21no—
17#453×C79—21no—
18#443×C79C2212yes—
19C2×#11929612—42no—
20#14579161—21no—
21#14579101—22no—
22#14579093—22no—
23#14579082—42no—
24C4×C79⋊C8—21no—
25H(5,𝔽2)×C79—21no—
26H(5,𝔽2)×C79—21no—
27AGL(1,ℤ/8)×C79—22no—
28AGL(1,ℤ/8)×C79—22no—
29AGL(1,ℤ/8)×C79—42no—
30AGL(1,ℤ/8)×C79—42no—
31#135×C79—42no—
32#135×C79D412yes—
33C2×D8×C79—22no—
34C2×D8×C79—42no—
35#131×C79—22no—
36#127×C79D411yescharacteristic
37#127×C79D411yescharacteristic
38#108×C79D412yes—
39#104×C79—22no—
40#100×C79—21no—
41#100×C79—21no—
42C4×C8×C79C2311yescharacteristic
43C2×Dic158—41no—
44#11929612—84no—
45C2×C79⋊C8—22no—
46C4⋊Dic79—41no—
47C4⋊Dic79—41no—
48C79⋊C16—42no—
49#41×C79—22no—
50#41×C79—42no—
51#41×C79—42no—
52C2×D4×C79—21no—
53C2×D4×C79—21no—
54C2×D4×C79—41no—
55C2×D4×C79—41no—
56C2×D4×C79—42no—
57C2×D4×C79—42no—
58C2×D4×C79—81no—
59C2×D4×C79—81no—
60Q16×C79SD1612yescomplemented
61SD16×C79—22no—
62SD16×C79—42no—
63SD16×C79—42no—
64D8×C79—42no—
65D8×C79—42no—
66D8×C79—42no—
67D8×C79—42no—
68D8×C79SD1612yescomplemented
69M16×C79—22no—
70M16×C79—42no—
71C2×C8×C79—42no—
72C2×C8×C79C2×D412yes—
73C42×C79C2×D411yescharacteristic
74C23×C79—41no—
75C23×C79—41no—
76C23×C79—81no—
77C23×C79—81no—
78Q8×C79—22no—
79D4×C79—22no—
80D4×C79—42no—
81D4×C79—42no—
82D4×C79—42no—
83D4×C79—82no—
84D4×C79—82no—
85D4×C79—82no—
86D4×C79—82no—
87C2×C4×C79—21no—
88C2×C4×C79—42no—
89C2×C4×C79—81no—
90C2×C4×C79—81no—
91C2×C4×C79#12011yescharacteristic
92C2×Dic79—81no—
93Dic158—82no—
94C8×C79—42no—
95C8×C79C2×SD1612yes—
96C8×C79AGL(1,ℤ/8)12yes—
97C79⋊C8—42no—
98C22×C79—42no—
99C22×C79—81no—
100C22×C79—81no—
101C22×C79—161no—
102C22×C79—161no—
103C22×C79—321no—
104C22×C79—321no—
105C22×C79#45311yescharacteristic
106C4×C79—22no—
107C4×C79—42no—
108C4×C79#45012yes—
109Dic79—161no—
110#12366—791nomaximal, Sylow
111C2×C79—21no—
112C2×C79—82no—
113C2×C79—161no—
114C2×C79—161no—
115C2×C79#162911yescharacteristic
116#3292D7911yescharacteristic
117#2230—792no—
118#1715—791no—
119#1357—791no—
120#1336—792no—
121C79#1236611yescharacteristic, complemented, Sylow
122#576—22no—
123#498—791no—
124#493C2×D7911yescharacteristic
125#492—1581no—
126#486—1581no—
127#472—22no—
128#453—21no—
129#453—21no—
130#443C2×D7912yes—
131#365—1582no—
132#364—792no—
133#360—1582no—
134#322—791no—
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#135C22⋊D7912yes—
143C2×SD16—3162no—
144C2×D8—22no—
145C2×D8—42no—
146#131—22no—
147C4⋊Q8—1581no—
148#127C22⋊D7911yescharacteristic
149#127C22⋊D7911yescharacteristic
150M32—1582no—
151#108C22⋊D7912yes—
152#106—1582no—
153#104—22no—
154#102—3162no—
155#100—21no—
156#100—21no—
157C4×C8—1581no—
158C4×C8C22×D7911yescharacteristic
159#41—22no—
160#41—42no—
161#41—42no—
162C2×Q8—3161no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
171Q16#1192961212yescomplemented
172SD16—22no—
173SD16—42no—
174SD16—42no—
175SD16—6324no—
176D8—42no—
177D8—42no—
178D8—42no—
179D8—42no—
180D8#1192961212yescomplemented
181M16—22no—
182M16—42no—
183C2×C8—42no—
184C2×C8—1582no—
185C2×C8C2×C22⋊D7912yes—
186C4⋊C4—3161no—
187C4⋊C4—3161no—
188C42C2×C22⋊D7911yescharacteristic
189C16—3162no—
190C23—41no—
191C23—41no—
192C23—81no—
193C23—81no—
194Q8—22no—
195Q8—6322no—
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—6321no—
209C2×C4#1457913011yescharacteristic
210C8—42no—
211C8—3162no—
212C8C2×#1192961212yes—
213C8#1457918712yes—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—161no—
218C22—161no—
219C22—321no—
220C22—321no—
221C22#1909873111yescharacteristic
222C4—22no—
223C4—42no—
224C4—12641no—
225C4#1909871412yes—
226C2—21no—
227C2—82no—
228C2—161no—
229C2—161no—
230C2#2376251611yescharacteristic
subgroupG/H
centerC2#23762516
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
Fitting, prime radicalC79×#3292C2
socleC2×C79#1629
hypercenter#3292D79
hypocenterC79#12366
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3292

Complements

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

NH ≅ G/Ntype
1. #23765571C2semidirect
2. #23763026C2semidirect
3. #23761437C2semidirect
4. #23761376C2semidirect
6. #19098961C22semidirect
10. #19098384C22semidirect
12. #19098284C22semidirect
60. Q16×C79SD16semidirect
68. D8×C79SD16semidirect
121. C79#12366semidirect
171. Q16#11929612semidirect
180. D8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237655712
#237630262
#237614372
#237613762
C79×#32922
#1236679