Finite Group Databasebeta

← #34201088

Subgroups — #34201088

Order 20224.

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

normal subgroups only

This group has 460 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#23764556C211yesmaximal, characteristic
2#23762373C212yesmaximal, complemented
3#23761430C212yesmaximal, complemented
4#23761332C211yesmaximal, characteristic
5C79×#2683C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098719—22no—
8#19098568—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×C22⋊Q8×C79—22no—
14#396×C79—22no—
15#395×C79C2212yes—
16#390×C79C2211yescharacteristic
17#389×C79—21no—
18#386×C79—21no—
19C2×#11929614—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579085—42no—
26#14579082—41no—
27#14579082—41no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79D412yes—
30C22×D4×C79D411yescharacteristic
31C23×C4×C79—21no—
32C22⋊Q8×C79—48no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79C2311yescharacteristic
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—21no—
41C2×C42×C79D411yescharacteristic
42#95×C79—22no—
43#95×C79D412yes—
44C2×Dic158—22no—
45#11929614—84no—
46#11929608—41no—
47#11929608—41no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—41no—
50C4⋊Dic79—21no—
51C4⋊Dic79—21no—
52C4⋊Dic79—22no—
53Dic79⋊C4—42no—
54C22⋊Dic79—41no—
55C4×Dic79—41no—
56C24×C79—21no—
57C2×Q8×C79—44no—
58C2×Q8×C79—42no—
59C2×D4×C79—21no—
60C2×D4×C79—41no—
61C2×D4×C79—41no—
62C2×D4×C79Q1612yescomplemented
63C22×C4×C79—22no—
64C22×C4×C79—21no—
65C22×C4×C79—22no—
66C22×C4×C79—44no—
67C22×C4×C79—81no—
68C22×C4×C79C2×D411yescharacteristic
69C22×C4×C79C2×D412yes—
70C4⋊C4×C79—24no—
71C4⋊C4×C79—41no—
72C4⋊C4×C79—44no—
73C4⋊C4×C79—44no—
74C4⋊C4×C79—81no—
75C22⋊C4×C79—44no—
76C22⋊C4×C79—44no—
77C22⋊C4×C79—81no—
78C42×C79—41no—
79C23×C79—22no—
80C23×C79—42no—
81C23×C79—42no—
82C23×C79—41no—
83C23×C79—41no—
84C23×C79#12011yescharacteristic
85Q8×C79—44no—
86Q8×C79—82no—
87D4×C79—42no—
88D4×C79—81no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—22no—
92C2×C4×C79—24no—
93C2×C4×C79—22no—
94C2×C4×C79—42no—
95C2×C4×C79—41no—
96C2×C4×C79—41no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—84no—
103C2×C4×C79C2×Q1611yescharacteristic
104C2×C4×C79#13711yescharacteristic
105C2×Dic79—41no—
106C2×Dic79—41no—
107Dic158—82no—
108C79⋊C8—81no—
109C79⋊C8—82no—
110C22×C79—21no—
111C22×C79—21no—
112C22×C79—44no—
113C22×C79—42no—
114C22×C79—82no—
115C22×C79—81no—
116C22×C79—81no—
117C22×C79#45111yescharacteristic
118C22×C79#45511yescharacteristic
119C22×C79B(4,𝔽2)11yescharacteristic
120C4×C79—21no—
121C4×C79—21no—
122C4×C79—42no—
123C4×C79—44no—
124C4×C79—42no—
125C4×C79—82no—
126Dic79—81no—
127Dic79—81no—
128#12220—791nomaximal, Sylow
129C2×C79—21no—
130C2×C79—21no—
131C2×C79—82no—
132C2×C79#160211yescharacteristic
133C2×C79#219611yescharacteristic
134C2×C79#220311yescharacteristic
135#2683D7911yescharacteristic
136#2027—791no—
137#1605—792no—
138#1354—792no—
139#1318—791no—
140C79#1222011yescharacteristic, complemented, Sylow
141#544—791no—
142C2×C22⋊Q8—22no—
143#451—1582no—
144#418—1581no—
145#396—22no—
146#395C2×D7912yes—
147#390C2×D7911yescharacteristic
148#389—21no—
149#386—21no—
150#343—791no—
151#339—1581no—
152#328—1582no—
153#324—792no—
154C22×Q8C22⋊D7912yes—
155C22×D4C22⋊D7911yescharacteristic
156C23×C4—21no—
157C2×Q16—3162no—
158C2×M16—1581no—
159C4⋊Q8—3161no—
160#125—3161no—
161C22⋊Q8—48no—
162C22⋊Q8—1582no—
163#117—1581no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×D7911yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42C22⋊D7911yescharacteristic
173#103—3162no—
174#102—3161no—
175#102—3161no—
176C22⋊C8—1582no—
177#95—22no—
178#95C22⋊D7912yes—
179C24—21no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—1582no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1192961412yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×C22⋊D7911yescharacteristic
193C22×C4C2×C22⋊D7912yes—
194Q16—6324no—
195M16—3161no—
196M16—3161no—
197C2×C8—3162no—
198C2×C8—3161no—
199C4⋊C4—24no—
200C4⋊C4—41no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—3162no—
205C4⋊C4—1581no—
206C4⋊C4—1581no—
207C4⋊C4—1582no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—81no—
211C22⋊C4—3161no—
212C42—41no—
213C42—3161no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1457913011yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—6322no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—3161no—
240C2×C4—3161no—
241C2×C4C2×#1192961411yescharacteristic
242C2×C4#1457919211yescharacteristic
243C8—6321no—
244C8—6322no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1909871911yescharacteristic
253C22#1909874111yescharacteristic
254C22#1909875011yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—6321no—
262C4—6321no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2376235711yescharacteristic
267C2#2376538411yescharacteristic
268C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2683C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2683D79
hypocenterC79#12220
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #23762373C2semidirect
3. #23761430C2semidirect
62. C2×D4×C79Q16semidirect
140. C79#12220semidirect
186. C2×D4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645562
#237623732
#237614302
#237613322
C79×#26832
#1222079