Finite Group Databasebeta

← #34207359

Subgroups — #34207359

Order 20224.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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#23773924C211yesmaximal, characteristic
2#23765121C212yesmaximal
3#23764271C211yesmaximal, characteristic
4#23761506C212yesmaximal
5C79×#1377C211yesmaximal, characteristic
6C22×Dic316C2211yescharacteristic
7#19099120—21no—
8#19098972—24no—
9#19098601—21no—
10#19098601C2212yes—
11C2×#14579093C2211yescharacteristic
12C2×#14579086—21no—
13#19098469—21no—
14#19098368—24no—
15C2×M32×C79C2212yes—
16C2×#97×C79C2211yescharacteristic
17C22×Dic158—21no—
18C2×Dic316—22no—
19C2×Dic316—22no—
20C2×Dic316—21no—
21C2×Dic316Q812yescomplemented
22#14579161—22no—
23#14579153—22no—
24#14579140—42no—
25C4×Dic158—42no—
26#14579120—22no—
27C2×C4⋊Dic79—21no—
28#14579095—22no—
29#14579095D414yes—
30#14579093—22no—
31#14579093—22no—
32#14579093—22no—
33#14579086—42no—
34C79⋊#95—42no—
35C22×C8×C79D411yescharacteristic
36C22×C8×C79C2311yescharacteristic
37C2×C42×C79D411yescharacteristic
38M32×C79—22no—
39M32×C79—22no—
40C2×C16×C79—22no—
41#97×C79—22no—
42C2×Dic158—22no—
43C2×Dic158—42no—
44C2×Dic158—41no—
45C2×Dic158—41no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C22×Dic79—41no—
50C22×Dic79—41no—
51Dic316—24no—
52Dic316—42no—
53C4⋊Dic79—24no—
54C4⋊Dic79—42no—
55C4⋊Dic79—41no—
56C4⋊Dic79—41no—
57Dic79⋊C4—44no—
58C22⋊Dic79—42no—
59C4×Dic79—42no—
60C22×C4×C79D812yes—
61C22×C4×C79C2×D411yescharacteristic
62C2×C8×C79—21no—
63C2×C8×C79—21no—
64C2×C8×C79SD1612yes—
65C2×C8×C79C2×D412yes—
66C2×C8×C79C2×Q811yescharacteristic
67C2×C8×C79#4111yescharacteristic
68C2×C8×C79#4112yes—
69C42×C79—21no—
70C42×C79—21no—
71C16×C79—42no—
72C23×C79C2×D811yescharacteristic
73C2×C4×C79—22no—
74C2×C4×C79—42no—
75C2×C4×C79#12011yescharacteristic
76C2×C4×C79C22⋊Q811yescharacteristic
77C2×C4×C79#12411yescharacteristic
78C2×C4×C79C2×SD1611yescharacteristic
79C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C79#13711yescharacteristic
81C2×Dic79—44no—
82C2×Dic79—42no—
83C2×Dic79—81no—
84C2×Dic79—81no—
85C2×Dic79—82no—
86C2×Dic79—81no—
87Dic158—42no—
88Dic158—42no—
89Dic158—84no—
90Dic158—81no—
91Dic158—81no—
92C8×C79—22no—
93C8×C79C22⋊Q812yes—
94C8×C79#12412yes—
95C22×C79—22no—
96C22×C79#44711yescharacteristic
97C22×C79#47411yescharacteristic
98C22×C79#48611yescharacteristic
99C4×C79—42no—
100C4×C79#39311yescharacteristic
101C4×C79#44811yescharacteristic
102C4×C79#47611yescharacteristic
103C4×C79#48911yescharacteristic
104Dic79—84no—
105Dic79—82no—
106Dic79—161no—
107#13368—791nomaximal, Sylow
108C2×C79—21no—
109C2×C79—21no—
110C2×C79#202311yescharacteristic
111C2×C79#219112yes—
112#3162—791no—
113#2141—792no—
114#1971—791no—
115#1386—792no—
116#1377D7911yescharacteristic
117C79#1336811yescharacteristic, complemented, Sylow
118C22×Q16—791no—
119#533—1581no—
120C2×M32C2×D7912yes—
121#501—1584no—
122#427—792no—
123#427—1581no—
124C2×#106—791no—
125C2×#103—1581no—
126C2×#97C2×D7911yescharacteristic
127#391—1581no—
128#358—1584no—
129C22×Q8—1581no—
130C2×Q16—792no—
131C2×Q16—1582no—
132C2×Q16—1581no—
133C2×Q16—1582no—
134C22×C8D31611yescharacteristic
135C22×C8C22×D7911yescharacteristic
136C4⋊Q8—1582no—
137#125—1582no—
138C22⋊Q8—3162no—
139C4×Q8—3162no—
140#117—1582no—
141C2×C4⋊C4—1581no—
142C2×C42D31611yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—794no—
147#107—1582no—
148#106—1582no—
149#106—1582no—
150#106—1582no—
151#103—3162no—
152#97—22no—
153#95—3162no—
154C2×Q8—3162no—
155C2×Q8—3161no—
156C2×Q8—3161no—
157C2×Q8—3161no—
158C2×Q8—3161no—
159C2×Q8—3161no—
160C2×Q8—1582no—
161C22×C4—3161no—
162C22×C4—3161no—
163C22×C4D63212yes—
164C22×C4C2×D31611yescharacteristic
165Q16—3162no—
166Q16—1584no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1192961312yes—
170C2×C8D4×D7912yes—
171C2×C8Q8×D7911yescharacteristic
172C2×C8#1192962311yescharacteristic
173C2×C8#1192962412yes—
174C4⋊C4—3164no—
175C4⋊C4—3162no—
176C4⋊C4—3161no—
177C4⋊C4—3161no—
178C4⋊C4—1584no—
179C22⋊C4—3162no—
180C42—21no—
181C42—21no—
182C42—3162no—
183C16—42no—
184C23C2×D63211yescharacteristic
185Q8—3162no—
186Q8—3162no—
187Q8—6324no—
188Q8—6321no—
189Q8—6321no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—3164no—
193C2×C4—3162no—
194C2×C4—6321no—
195C2×C4—6321no—
196C2×C4—6322no—
197C2×C4—6321no—
198C2×C4#1457913111yescharacteristic
199C2×C4#1457914111yescharacteristic
200C2×C4#1457914911yescharacteristic
201C2×C4C2×#1192961311yescharacteristic
202C2×C4#1457918911yescharacteristic
203C2×C4#1457919411yescharacteristic
204C8—22no—
205C8#1457913912yes—
206C8#1457914712yes—
207C22—22no—
208C22#1909869911yescharacteristic
209C22#1909884411yescharacteristic
210C22#1909891011yescharacteristic
211C4—42no—
212C4—6322no—
213C4—6324no—
214C4—12641no—
215C4#1909847611yescharacteristic
216C4#1909870411yescharacteristic
217C4#1909885411yescharacteristic
218C4#1909892311yescharacteristic
219C2—21no—
220C2—21no—
221C2#2376453711yescharacteristic
222C2#2376535312yes—
subgroupG/H
centerC22#19098699
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#1377C2
socleC22×C79#447
prime radicalC23×C79C2×D8
hypercenter#1377D79
hypocenterC79#13368
LCSG ▷ C22×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
21. C2×Dic316Q8semidirect
117. C79#13368semidirect

Maximal subgroups

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

subgroupindex
#237739242
#237651212
#237642712
#237615062
C79×#13772
#1336879