Finite Group Databasebeta

← #34202028

Subgroups — #34202028

Order 20224.

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

normal subgroups only

This group has 358 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#23762986C211yesmaximal, characteristic, complemented
3#23761441C211yesmaximal, characteristic, complemented
4#23761373C212yesmaximal, complemented
5C79×#3291C211yesmaximal, characteristic
6#19098961C2211yescharacteristic, complemented
7#19098959—21no—
8#19098935—21no—
9#19098387—22no—
10#19098384C2212yescomplemented
11#19098373—22no—
12C8×C79⋊C8C2211yescharacteristic, complemented
13#577×C79—22no—
14#492×C79C2211yescharacteristic
15#471×C79—22no—
16#456×C79—21no—
17#453×C79—21no—
18#443×C79C2212yes—
19C2×#11929612—42no—
20#14579161—21no—
21#14579101—22no—
22#14579093—22no—
23#14579092—42no—
24C4×C79⋊C8—21no—
252−1+4×C79—21no—
26H(5,𝔽2)×C79—21no—
27#137×C79—22no—
28#137×C79—42no—
29AGL(1,ℤ/8)×C79—22no—
30AGL(1,ℤ/8)×C79—42no—
31#135×C79—42no—
32#135×C79D412yes—
33C2×SD16×C79—22no—
34C2×SD16×C79—42no—
35#131×C79—22no—
36C4⋊Q8×C79D411yescharacteristic
37#127×C79D411yescharacteristic
38#108×C79D412yes—
39#104×C79—22no—
40#101×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—
52#41×C79—81no—
53C2×Q8×C79—21no—
54C2×Q8×C79—42no—
55C2×D4×C79—21no—
56C2×D4×C79—41no—
57C2×D4×C79—42no—
58C2×D4×C79—81no—
59Q16×C79—42no—
60Q16×C79SD1612yescomplemented
61SD16×C79—22no—
62SD16×C79—42no—
63SD16×C79—42no—
64SD16×C79—42no—
65SD16×C79—42no—
66D8×C79—42no—
67D8×C79SD1612yescomplemented
68M16×C79—22no—
69M16×C79—42no—
70C2×C8×C79—42no—
71C2×C8×C79C2×D412yes—
72C4⋊C4×C79—41no—
73C42×C79C2×D411yescharacteristic
74C23×C79—41no—
75C23×C79—81no—
76Q8×C79—22no—
77Q8×C79—42no—
78Q8×C79—81no—
79D4×C79—22no—
80D4×C79—42no—
81D4×C79—42no—
82D4×C79—82no—
83D4×C79—82no—
84D4×C79—81no—
85C2×C4×C79—21no—
86C2×C4×C79—42no—
87C2×C4×C79—41no—
88C2×C4×C79—81no—
89C2×C4×C79—82no—
90C2×C4×C79#12011yescharacteristic
91C2×Dic79—81no—
92Dic158—82no—
93C8×C79—42no—
94C8×C79C2×SD1612yes—
95C8×C79AGL(1,ℤ/8)12yes—
96C79⋊C8—24no—
97C22×C79—42no—
98C22×C79—81no—
99C22×C79—161no—
100C22×C79—321no—
101C22×C79#45311yescharacteristic
102C4×C79—22no—
103C4×C79—42no—
104C4×C79—81no—
105C4×C79#45012yes—
106Dic79—161no—
107#12364—791nomaximal, Sylow
108C2×C79—21no—
109C2×C79—82no—
110C2×C79—161no—
111C2×C79#162911yescharacteristic
112#3291D7911yescharacteristic
113#2230—792no—
114#1705—791no—
115#1359—791no—
116#1335—792no—
117C79#1236411yescharacteristic, complemented, Sylow
118#577—22no—
119#498—791no—
120#498—1581no—
121#492—1581no—
122#492C2×D7911yescharacteristic
123#471—22no—
124#456—21no—
125#453—21no—
126#443C2×D7912yes—
127#365—1582no—
128#364—792no—
129#360—1582no—
130C82—791no—
1312−1+4—21no—
132H(5,𝔽2)—21no—
133#137—22no—
134#137—42no—
135AGL(1,ℤ/8)—22no—
136AGL(1,ℤ/8)—42no—
137#135—42no—
138#135C22⋊D7912yes—
139C2×SD16—22no—
140C2×SD16—42no—
141C2×SD16—3162no—
142#131—22no—
143C4⋊Q8—1581no—
144C4⋊Q8C22⋊D7911yescharacteristic
145#127C22⋊D7911yescharacteristic
146M32—1582no—
147#108C22⋊D7912yes—
148#106—3162no—
149#106—1582no—
150#104—22no—
151#101—21no—
152#100—21no—
153C4×C8—1581no—
154C4×C8C22×D7911yescharacteristic
155#41—22no—
156#41—42no—
157#41—42no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—3161no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—81no—
166Q16—42no—
167Q16#1192961212yescomplemented
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—6324no—
174D8—42no—
175D8#1192961212yescomplemented
176M16—22no—
177M16—42no—
178C2×C8—42no—
179C2×C8—1582no—
180C2×C8C2×C22⋊D7912yes—
181C4⋊C4—41no—
182C4⋊C4—3161no—
183C4⋊C4—3161no—
184C42C2×C22⋊D7911yescharacteristic
185C16—3162no—
186C23—41no—
187C23—81no—
188Q8—22no—
189Q8—42no—
190Q8—81no—
191Q8—6322no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—6321no—
204C2×C4#1457913011yescharacteristic
205C8—42no—
206C8—1584no—
207C8C2×#1192961212yes—
208C8#1457918712yes—
209C22—42no—
210C22—81no—
211C22—161no—
212C22—321no—
213C22#1909873111yescharacteristic
214C4—22no—
215C4—42no—
216C4—81no—
217C4—12641no—
218C4#1909871412yes—
219C2—21no—
220C2—82no—
221C2—161no—
222C2#2376251611yescharacteristic
subgroupG/H
centerC2#23762516
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
Fitting, prime radicalC79×#3291C2
socleC2×C79#1629
hypercenter#3291D79
hypocenterC79#12364
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #23765571C2semidirect
2. #23762986C2semidirect
3. #23761441C2semidirect
4. #23761373C2semidirect
6. #19098961C22semidirect
10. #19098384C22semidirect
12. C8×C79⋊C8C22semidirect
60. Q16×C79SD16semidirect
67. D8×C79SD16semidirect
117. C79#12364semidirect
167. Q16#11929612semidirect
175. D8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237655712
#237629862
#237614412
#237613732
C79×#32912
#1236479