Finite Group Databasebeta

← #34201639

Subgroups — #34201639

Order 20224.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765617C211yesmaximal, characteristic, complemented
2#23765612C211yesmaximal, characteristic
3#23764980C211yesmaximal, characteristic, complemented
4#23761394C211yesmaximal, characteristic, complemented
5#23761391C211yesmaximal, characteristic
6#23761353C211yesmaximal, characteristic, complemented
7C79×#3263C211yesmaximal, characteristic
8#19098612C2211yescharacteristic
9#19098385—21no—
10#19098385—21no—
11#19098379—21no—
12#19098376—21no—
13#19098345C2211yescharacteristic
14#19098345C2211yescharacteristic
15#19098333—21no—
16#19098328—21no—
17C79⋊#338C2211yescharacteristic
18#542×C79—21no—
19#476×C79—21no—
20#475×C79—21no—
21#470×C79C2211yescharacteristic
22#469×C79C2211yescharacteristic
23#465×C79—21no—
24#464×C79—21no—
25#463×C79—21no—
26#424×C79C2211yescharacteristic
27C2×#11929608—21no—
28C2×#11929608—21no—
29#14579101—21no—
30#14579101—21no—
31#14579097—21no—
32#14579097—21no—
33C2×Q16×C79D411yescharacteristic
34C2×SD16×C79—21no—
35C2×D8×C79D411yescharacteristic
36C2×M16×C79C2311yescharacteristic
37C22×C8×C79D411yescharacteristic
38C4⋊Q8×C79—21no—
39#126×C79—41no—
40#124×C79—21no—
41#124×C79—41no—
42#123×C79—41no—
43C22⋊Q8×C79—21no—
44C22⋊Q8×C79—41no—
45#121×C79—21no—
46C4×Q8×C79—21no—
47C4×D4×C79—21no—
48#117×C79D411yescharacteristic
49#106×C79—21no—
50#106×C79D411yescharacteristic
51#106×C79D411yescharacteristic
52C4⋊C8×C79—21no—
53C4⋊C8×C79—21no—
54#103×C79—21no—
55#103×C79—21no—
56#102×C79—21no—
57#102×C79—21no—
58#11929608—41no—
59#11929608—41no—
60#11929608—41no—
61#11929608—41no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8—41no—
64C79⋊C16—41no—
65C79⋊C16—41no—
66C2×Q8×C79—21no—
67C2×Q8×C79—41no—
68C2×D4×C79—21no—
69C2×D4×C79—41no—
70C22×C4×C79—41no—
71C22×C4×C79C2×D411yescharacteristic
72Q16×C79—41no—
73SD16×C79—81no—
74D8×C79—41no—
75M16×C79—41no—
76C2×C8×C79—21no—
77C2×C8×C79—41no—
78C2×C8×C79C2×D411yescharacteristic
79C2×C8×C79C2×D411yescharacteristic
80C4⋊C4×C79—21no—
81C4⋊C4×C79—21no—
82C4⋊C4×C79—21no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—41no—
89C22⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C22⋊C4×C79SD1611yescharacteristic
92C22⋊C4×C79SD1611yescharacteristic
93C42×C79—21no—
94C42×C79—41no—
95C23×C79—41no—
96C23×C79C2×SD1611yescharacteristic
97Q8×C79—41no—
98Q8×C79—41no—
99Q8×C79—161no—
100D4×C79—41no—
101D4×C79—41no—
102D4×C79—161no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—161no—
111C2×C4×C79#12011yescharacteristic
112C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
113C8×C79—21no—
114C8×C79—21no—
115C8×C79—41no—
116C79⋊C8—81no—
117C79⋊C8—81no—
118C22×C79—21no—
119C22×C79—41no—
120C22×C79—81no—
121C22×C79—81no—
122C22×C79—81no—
123C22×C79#45011yescharacteristic
124C4×C79—21no—
125C4×C79—41no—
126C4×C79—41no—
127C4×C79—81no—
128C4×C79—81no—
129C4×C79—81no—
130C4×C79#45411yescharacteristic
131C4×C79#45811yescharacteristic
132#12306—791nomaximal, Sylow
133C2×C79—21no—
134C2×C79—41no—
135C2×C79—161no—
136C2×C79#161611yescharacteristic
137#3263D7911yescharacteristic
138#2239—791no—
139#2238—791no—
140#2103—791no—
141#1342—791no—
142#1341—791no—
143#1328—791no—
144C79#1230611yescharacteristic, complemented, Sylow
145#542—21no—
146#476—21no—
147#475—21no—
148#470C2×D7911yescharacteristic
149#469C2×D7911yescharacteristic
150#465—21no—
151#464—21no—
152#463—21no—
153#430—791no—
154#424C2×D7911yescharacteristic
155#365—1581no—
156#365—1581no—
157#362—1581no—
158#361—1581no—
159#349—791no—
160#349—791no—
161#344—1581no—
162#341—1581no—
163#338—791no—
164C2×Q16C22⋊D7911yescharacteristic
165C2×SD16—21no—
166C2×D8C22⋊D7911yescharacteristic
167C2×M16—1581no—
168C2×M16—1581no—
169C2×M16C22×D7911yescharacteristic
170C22×C8C22⋊D7911yescharacteristic
171C4⋊Q8—21no—
172#126—41no—
173#124—21no—
174#124—41no—
175#123—41no—
176C22⋊Q8—21no—
177C22⋊Q8—41no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181#117C22⋊D7911yescharacteristic
182M32—1581no—
183M32—1581no—
184#108—1581no—
185#108—1581no—
186#106—21no—
187#106C22⋊D7911yescharacteristic
188#106C22⋊D7911yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191#103—21no—
192#103—21no—
193#102—21no—
194#102—21no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—41no—
200C22×C4C2×C22⋊D7911yescharacteristic
201Q16—41no—
202SD16—81no—
203D8—41no—
204M16—41no—
205M16—3161no—
206M16—3161no—
207M16—3161no—
208M16—3161no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—3161no—
212C2×C8—3161no—
213C2×C8C2×C22⋊D7911yescharacteristic
214C2×C8C2×C22⋊D7911yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4#1192961211yescharacteristic
227C22⋊C4#1192961211yescharacteristic
228C42—21no—
229C42—41no—
230C16—3161no—
231C16—3161no—
232C23—41no—
233C23C2×#1192961211yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—161no—
237D4—41no—
238D4—41no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—161no—
248C2×C4#1457913011yescharacteristic
249C2×C4#1457918711yescharacteristic
250C8—21no—
251C8—21no—
252C8—41no—
253C8—6321no—
254C8—6321no—
255C22—21no—
256C22—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#1909871411yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4#1909873611yescharacteristic
268C4#1909875311yescharacteristic
269C2—21no—
270C2—41no—
271C2—161no—
272C2#2376243711yescharacteristic
subgroupG/H
centerC2#23762437
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#3263C2
socleC2×C79#1616
prime radical#469×C79C22
hypercenter#3263D79
hypocenterC79#12306
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3263

Complements

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

NH ≅ G/Ntype
1. #23765617C2semidirect
3. #23764980C2semidirect
4. #23761394C2semidirect
6. #23761353C2semidirect
144. C79#12306semidirect

Maximal subgroups

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

subgroupindex
#237656172
#237656122
#237649802
#237613942
#237613912
#237613532
C79×#32632
#1230679