Finite Group Databasebeta

← #34201112

Subgroups — #34201112

Order 20224.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764561C211yesmaximal, characteristic
2#23762373C212yesmaximal
3#23761430C212yesmaximal
4#23761334C211yesmaximal, characteristic
5C79×#2811C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098719—22no—
8#19098572—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#395×C79—22no—
14#395×C79C2212yes—
15#391×C79—21no—
16#391×C79—21no—
17#391×C79—22no—
18#391×C79C2211yescharacteristic
19C2×#11929614—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579085—42no—
26#14579085—41no—
27#14579085—41no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79—21no—
30C22×Q8×C79D411yescharacteristic
31C22×Q8×C79D412yes—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79C2311yescharacteristic
36C2×C42×C79—21no—
37C2×C42×C79D411yescharacteristic
38#95×C79—22no—
39#95×C79—22no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44#11929614—84no—
45#11929608—41no—
46#11929608—41no—
47C2×C79⋊C8—42no—
48C2×C79⋊C8—41no—
49C4⋊Dic79—21no—
50C4⋊Dic79—21no—
51C4⋊Dic79—22no—
52Dic79⋊C4—42no—
53C22⋊Dic79—41no—
54C4×Dic79—41no—
55C2×Q8×C79—24no—
56C2×Q8×C79—21no—
57C2×Q8×C79—41no—
58C2×Q8×C79—41no—
59C2×Q8×C79—42no—
60C2×Q8×C79—82no—
61C2×Q8×C79—81no—
62C2×Q8×C79—81no—
63C2×Q8×C79Q1612yescomplemented
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—22no—
67C22×C4×C79—22no—
68C22×C4×C79C2×D411yescharacteristic
69C22×C4×C79C2×D412yes—
70C4⋊C4×C79—41no—
71C4⋊C4×C79—82no—
72C4⋊C4×C79—81no—
73C4⋊C4×C79—81no—
74C42×C79—41no—
75C42×C79—81no—
76C23×C79#12011yescharacteristic
77Q8×C79—42no—
78Q8×C79—44no—
79Q8×C79—81no—
80Q8×C79—84no—
81Q8×C79—82no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—22no—
85C2×C4×C79—24no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—41no—
89C2×C4×C79—42no—
90C2×C4×C79—41no—
91C2×C4×C79—41no—
92C2×C4×C79—42no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79C2×Q1611yescharacteristic
99C2×C4×C79#13711yescharacteristic
100C2×Dic79—41no—
101C2×Dic79—41no—
102Dic158—82no—
103C79⋊C8—81no—
104C79⋊C8—82no—
105C22×C79—21no—
106C22×C79—21no—
107C22×C79#45111yescharacteristic
108C22×C79#45511yescharacteristic
109C22×C79B(4,𝔽2)11yescharacteristic
110C4×C79—21no—
111C4×C79—21no—
112C4×C79—42no—
113C4×C79—44no—
114C4×C79—42no—
115C4×C79—42no—
116C4×C79—82no—
117Dic79—81no—
118Dic79—81no—
119#12224—791nomaximal, Sylow
120C2×C79—21no—
121C2×C79—21no—
122C2×C79#160211yescharacteristic
123C2×C79#219911yescharacteristic
124C2×C79#220511yescharacteristic
125#2811D7911yescharacteristic
126#2028—791no—
127#1605—792no—
128#1354—792no—
129#1319—791no—
130C79#1222411yescharacteristic, complemented, Sylow
131#544—791no—
132#451—1582no—
133#419—1581no—
134#395—22no—
135#395C2×D7912yes—
136#391—21no—
137#391—21no—
138#391—22no—
139#391C2×D7911yescharacteristic
140#343—791no—
141#339—1581no—
142#328—1582no—
143#324—792no—
144C22×Q8—21no—
145C22×Q8C22⋊D7911yescharacteristic
146C22×Q8C22⋊D7912yes—
147C2×Q16—3162no—
148C2×M16—1581no—
149C4⋊Q8—3161no—
150#125—3161no—
151C22⋊Q8—1582no—
152#117—1581no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22×D7911yescharacteristic
157C2×C42—21no—
158C2×C42C22⋊D7911yescharacteristic
159#103—3161no—
160#103—3161no—
161#103—3162no—
162C22⋊C8—1582no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95C22⋊D7912yes—
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—1582no—
177C2×Q8#1192961412yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D7911yescharacteristic
183C22×C4C2×C22⋊D7912yes—
184Q16—6324no—
185M16—3161no—
186M16—3161no—
187C2×C8—3162no—
188C2×C8—3161no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—3162no—
194C4⋊C4—1581no—
195C4⋊C4—1581no—
196C4⋊C4—1582no—
197C22⋊C4—3161no—
198C42—41no—
199C42—81no—
200C42—3161no—
201C23#1457913011yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—6322no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—3161no—
225C2×C4—3161no—
226C2×C4C2×#1192961411yescharacteristic
227C2×C4#1457919211yescharacteristic
228C8—6321no—
229C8—6322no—
230C22—21no—
231C22—21no—
232C22#1909871911yescharacteristic
233C22#1909874111yescharacteristic
234C22#1909875011yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—6321no—
243C4—6321no—
244C2—21no—
245C2—21no—
246C2#2376235711yescharacteristic
247C2#2376540111yescharacteristic
248C2#2376543011yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2811C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2811D79
hypocenterC79#12224
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
63. C2×Q8×C79Q16semidirect
130. C79#12224semidirect
177. C2×Q8#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645612
#237623732
#237614302
#237613342
C79×#28112
#1222479