Finite Group Databasebeta

← #34201044

Subgroups — #34201044

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#23764561C211yesmaximal, characteristic, complemented
2#23762339C212yesmaximal, complemented
3#23761403C212yesmaximal, complemented
4#23761334C211yesmaximal, characteristic, complemented
5C79×#2733C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098714—22no—
8#19098572—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×#124×C79—21no—
14#394×C79C2212yes—
15#392×C79—22no—
16#391×C79C2211yescharacteristic
17#390×C79—21no—
18#388×C79—22no—
19C2×#11929612—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579085—41no—
26#14579085—41no—
27#14579082—42no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79D411yescharacteristic
30C22×D4×C79—21no—
31C22×D4×C79—21no—
32C22×D4×C79D412yes—
33#124×C79—42no—
34#124×C79—42no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79C2311yescharacteristic
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C42×C79—21no—
42C2×C42×C79D411yescharacteristic
43#95×C79—22no—
44#95×C79D412yes—
45C2×Dic158—22no—
46#11929612—84no—
47#11929608—41no—
48#11929608—41no—
49C2×C79⋊C8—42no—
50C2×C79⋊C8—41no—
51C4⋊Dic79—22no—
52C4⋊Dic79—21no—
53C4⋊Dic79—21no—
54Dic79⋊C4—42no—
55C22⋊Dic79—41no—
56C4×Dic79—41no—
57C24×C79—22no—
58C2×Q8×C79—21no—
59C2×Q8×C79—41no—
60C2×Q8×C79—41no—
61C2×Q8×C79SD1612yescomplemented
62C2×D4×C79—42no—
63C2×D4×C79—42no—
64C2×D4×C79—42no—
65C2×D4×C79—82no—
66C2×D4×C79—81no—
67C2×D4×C79—81no—
68C2×D4×C79—81no—
69C2×D4×C79—81no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D412yes—
75C4⋊C4×C79—41no—
76C4⋊C4×C79—81no—
77C22⋊C4×C79—44no—
78C22⋊C4×C79—44no—
79C22⋊C4×C79—42no—
80C22⋊C4×C79—42no—
81C22⋊C4×C79—82no—
82C42×C79—42no—
83C42×C79—41no—
84C23×C79—42no—
85C23×C79—42no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79—42no—
91C23×C79#12011yescharacteristic
92Q8×C79—42no—
93Q8×C79—81no—
94D4×C79—44no—
95D4×C79—82no—
96D4×C79—84no—
97D4×C79—84no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79C2×SD1611yescharacteristic
112C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
113C2×Dic79—41no—
114C2×Dic79—41no—
115Dic158—82no—
116C79⋊C8—81no—
117C79⋊C8—82no—
118C22×C79—21no—
119C22×C79—21no—
120C22×C79—82no—
121C22×C79—82no—
122C22×C79—82no—
123C22×C79—82no—
124C22×C79—82no—
125C22×C79—82no—
126C22×C79—82no—
127C22×C79#45011yescharacteristic
128C22×C79#45511yescharacteristic
129C22×C79B(4,𝔽2)11yescharacteristic
130C4×C79—21no—
131C4×C79—21no—
132C4×C79—42no—
133C4×C79—42no—
134C4×C79—42no—
135C4×C79—82no—
136Dic79—81no—
137Dic79—81no—
138#12212—791nomaximal, Sylow
139C2×C79—21no—
140C2×C79—21no—
141C2×C79—162no—
142C2×C79#160111yescharacteristic
143C2×C79#219911yescharacteristic
144C2×C79#220411yescharacteristic
145#2733D7911yescharacteristic
146#2028—791no—
147#1599—792no—
148#1345—792no—
149#1319—791no—
150C79#1221211yescharacteristic, complemented, Sylow
151#544—791no—
152C2×#124—21no—
153#450—1582no—
154#419—1581no—
155#394C2×D7912yes—
156#392—22no—
157#391C2×D7911yescharacteristic
158#390—21no—
159#388—22no—
160#343—791no—
161#339—1581no—
162#328—1582no—
163#324—792no—
164C22×Q8C22⋊D7911yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22⋊D7912yes—
168C2×SD16—3162no—
169C2×M16—1581no—
170C4⋊Q8—3161no—
171#125—3161no—
172#124—42no—
173#124—42no—
174C22⋊Q8—1582no—
175#117—1581no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×D7911yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—21no—
183C2×C42C22⋊D7911yescharacteristic
184#103—3161no—
185#103—3161no—
186#102—3162no—
187C22⋊C8—1582no—
188#95—22no—
189#95C22⋊D7912yes—
190C24—22no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—1582no—
195C2×Q8#1192961212yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×C22⋊D7911yescharacteristic
208C22×C4C2×C22⋊D7912yes—
209SD16—6324no—
210M16—3161no—
211M16—3161no—
212C2×C8—3162no—
213C2×C8—3161no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—3162no—
217C4⋊C4—1582no—
218C4⋊C4—1581no—
219C4⋊C4—1581no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C22⋊C4—3161no—
226C42—41no—
227C42—42no—
228C42—3161no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23#1457913011yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—6322no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3161no—
258C2×C4—3161no—
259C2×C4C2×#1192961211yescharacteristic
260C2×C4#1457918711yescharacteristic
261C8—6321no—
262C8—6322no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22#1909871411yescharacteristic
273C22#1909874111yescharacteristic
274C22#1909875011yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—6321no—
282C4—6321no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2#2376235211yescharacteristic
287C2#2376540111yescharacteristic
288C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
Fitting, prime radicalC79×#2733C2
socleC22×C79B(4,𝔽2)
hypercenter#2733D79
hypocenterC79#12212
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #23764561C2semidirect
2. #23762339C2semidirect
3. #23761403C2semidirect
4. #23761334C2semidirect
61. C2×Q8×C79SD16semidirect
150. C79#12212semidirect
195. C2×Q8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237645612
#237623392
#237614032
#237613342
C79×#27332
#1221279