Finite Group Databasebeta

← #34199106

Subgroups — #34199106

Order 20224.

340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 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#23769276C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic
3#23761428C212yesmaximal
4#23761407C212yesmaximal
5C79×#1467C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic
7#19098587—21no—
8C2×#14579080—21no—
9#19098502C411yescharacteristic
10#19098502C2211yescharacteristic
11#19098501—21no—
12#19098493C411yescharacteristic
13#19098493C2211yescharacteristic
14#19098482—21no—
15#19098447—21no—
16#19098447—21no—
17#19098431C411yescharacteristic
18#19098332C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20#432×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C2×#11929608—22no—
24#14579153—41no—
25#14579153—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C4⋊(C79⋊C8)—41no—
33C4⋊(C79⋊C8)—41no—
34#14579080—42no—
35C79⋊#95—21no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95C2×C411yescharacteristic
44C79⋊#95C2×C411yescharacteristic
45C22×Q8×C79D411yescharacteristic
46C4×Q8×C79—41no—
47C4×Q8×C79—41no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C4⋊C8×C79—41no—
55C22⋊C8×C79—22no—
56#97×C79—41no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61#11929608—42no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—41no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68Dic79⋊C4—41no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71Dic79⋊C4—41no—
72C2×Q8×C79—21no—
73C2×Q8×C79—21no—
74C2×Q8×C79—41no—
75C2×Q8×C79—41no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C2×C8×C79—42no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79C2×C22⋊C411yescharacteristic
94Q8×C79—42no—
95Q8×C79—42no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
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—41no—
111C2×C4×C79#9911yescharacteristic
112C2×C4×C79#9911yescharacteristic
113C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C79#13711yescharacteristic
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—42no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C8×C79—82no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79C2×#9911yescharacteristic
131C22×C79#41711yescharacteristic
132C22×C79#42111yescharacteristic
133C4×C79—21no—
134C4×C79—21no—
135C4×C79—21no—
136C4×C79—21no—
137C4×C79—41no—
138C4×C79—41no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—41no—
142Dic79—81no—
143Dic79—81no—
144Dic79—81no—
145Dic79—81no—
146#11835—791nomaximal, Sylow
147C2×C79—21no—
148C2×C79—21no—
149C2×C79#151311yescharacteristic
150C2×C79#212311yescharacteristic
151C2×C79#213311yescharacteristic
152#2634—791no—
153#1801—791no—
154#1467D7911yescharacteristic
155#1353—792no—
156#1346—792no—
157C79#1183511yescharacteristic, complemented, Sylow
158C2×#125—791no—
159C2×C4×Q8C2×D7911yescharacteristic
160#432—21no—
161#423—1581no—
162C2×#101—1581no—
163#400—791no—
164#400—791no—
165#400—1581no—
166#398—791no—
167#398—791no—
168#395—1581no—
169#387—1581no—
170#387—1581no—
171#383—791no—
172#343—792no—
173#324—21no—
174#324C2×D7912yes—
175C22×Q8D31611yescharacteristic
176C2×M16—1582no—
177#125—3161no—
178#125—3161no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—791no—
182C2×C4⋊C4—791no—
183C2×C4⋊C4—1581no—
184C2×C4⋊C4—1581no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4—1581no—
187C2×C4⋊C4C4×D7911yescharacteristic
188C2×C4⋊C4C22⋊D7911yescharacteristic
189C2×C4⋊C4C22×D7911yescharacteristic
190C2×C42C4×D7911yescharacteristic
191C2×C42D31611yescharacteristic
192C2×C42C22⋊D7911yescharacteristic
193C4⋊C8—41no—
194C4⋊C8—3161no—
195C4⋊C8—3161no—
196#101—3162no—
197C22⋊C8—22no—
198#97—41no—
199#95—791no—
200#95—791no—
201#95—1581no—
202#95—1581no—
203#95—1581no—
204#95—1581no—
205#95—1581no—
206#95—1581no—
207#95—1581no—
208#95—1581no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C22×C4—1581no—
214C22×C4—1581no—
215C22×C4—1581no—
216C22×C4—1581no—
217C22×C4(C2×D79)⋊C411yescharacteristic
218C22×C4(C2×D79)⋊C411yescharacteristic
219C22×C4(C2×D79)⋊C411yescharacteristic
220C22×C4(C2×D79)⋊C411yescharacteristic
221C22×C4C2×C4×D7911yescharacteristic
222C22×C4C2×C22⋊D7911yescharacteristic
223C22×C4C2×D31611yescharacteristic
224M16—3162no—
225M16—3162no—
226C2×C8—42no—
227C2×C8—3162no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—3161no—
232C4⋊C4—3161no—
233C4⋊C4—3162no—
234C4⋊C4—3162no—
235C4⋊C4—3161no—
236C4⋊C4—3161no—
237C4⋊C4—3161no—
238C4⋊C4—3161no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C23C2×(C2×D79)⋊C411yescharacteristic
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—3161no—
267C2×C4—3162no—
268C2×C4—3161no—
269C2×C4—3161no—
270C2×C4—3161no—
271C2×C4—3161no—
272C2×C4—3161no—
273C2×C4—3161no—
274C2×C4#1457907511yescharacteristic
275C2×C4#1457907511yescharacteristic
276C2×C4#1457918911yescharacteristic
277C2×C4#1457919411yescharacteristic
278C8—82no—
279C8—6322no—
280C22—21no—
281C22—21no—
282C22C2×#1457907511yescharacteristic
283C22#1909856511yescharacteristic
284C22#1909857911yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—6321no—
295C4—6321no—
296C4—6321no—
297C4—6321no—
298C2—21no—
299C2—21no—
300C2#2376194411yescharacteristic
301C2#2376505311yescharacteristic
302C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1467C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1467D79
hypocenterC79#11835
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
157. C79#11835semidirect

Maximal subgroups

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

subgroupindex
#237692762
#237633962
#237614282
#237614072
C79×#14672
#1183579