Finite Group Databasebeta

← #34198666

Subgroups — #34198666

Order 20224.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#23761724C211yesmaximal, characteristic
3#23761426C212yesmaximal
4#23761405C212yesmaximal
5C4×#101×C79C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic
7#19098616—21no—
8#19098502C411yescharacteristic
9#19098502C2211yescharacteristic
10#19098501—21no—
11#19098493C411yescharacteristic
12#19098493C2211yescharacteristic
13#19098482—21no—
14#19098447—21no—
15#19098447—21no—
16#19098431C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20C2×#101×C79—21no—
21C4×M16×C79—21no—
22#343×C79C2212yes—
23#14579153—41no—
24#14579153—41no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C4⋊(C79⋊C8)—41no—
32C22⋊(C79⋊C8)—22no—
33C4×C79⋊C8—41no—
34C79⋊#95—21no—
35C79⋊#95—21no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×Q8×C79D411yescharacteristic
45C2×M16×C79—22no—
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
54#101×C79—24no—
55#97×C79—41no—
56C4×C8×C79—41no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C2×C79⋊C8—42no—
62C4⋊Dic79—41no—
63C4⋊Dic79—41no—
64Dic79⋊C4—42no—
65Dic79⋊C4—41no—
66Dic79⋊C4—41no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69C2×Q8×C79—22no—
70C2×Q8×C79—22no—
71C2×Q8×C79—41no—
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79M16×C79—24no—
80M16×C79—42no—
81C2×C8×C79—42no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—41no—
85C42×C79—21no—
86C42×C79—21no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C23×C79C2×C22⋊C411yescharacteristic
92Q8×C79—44no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
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—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79#9911yescharacteristic
108C2×C4×C79#9911yescharacteristic
109C2×C4×C79#10412yes—
110C2×C4×C79#13512yes—
111C2×Dic79—41no—
112C2×Dic79—41no—
113C2×Dic79—41no—
114C2×Dic79—41no—
115C2×Dic79—42no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C8×C79—44no—
123C79⋊C8—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79C2×#9911yescharacteristic
127C22×C79#41611yescharacteristic
128C22×C79C2×#10411yescharacteristic
129C4×C79—21no—
130C4×C79—21no—
131C4×C79—22no—
132C4×C79—22no—
133C4×C79—41no—
134C4×C79—41no—
135C4×C79—41no—
136C4×C79—41no—
137Dic79—81no—
138Dic79—81no—
139Dic79—81no—
140Dic79—81no—
141#11740—791nomaximal, Sylow
142C2×C79—21no—
143C2×C79—21no—
144C2×C79#150111yescharacteristic
145C2×C79#212311yescharacteristic
146C2×C79#213311yescharacteristic
147#2634—791no—
148C4×#101D7911yescharacteristic
149#1461—791no—
150#1353—792no—
151#1346—792no—
152C79#1174011yescharacteristic, complemented, Sylow
153C2×#125—791no—
154C2×C4×Q8C2×D7911yescharacteristic
155#431—1581no—
156C2×#101—21no—
157C4×M16—21no—
158#400—791no—
159#400—791no—
160#400—1581no—
161#398—791no—
162#398—791no—
163#395—1581no—
164#387—1581no—
165#387—1581no—
166#383—791no—
167#343C2×D7912yes—
168#324—792no—
169#324—1581no—
170C22×Q8C22⋊D7911yescharacteristic
171C2×M16—22no—
172#125—3161no—
173#125—3161no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—791no—
177C2×C4⋊C4—791no—
178C2×C4⋊C4—1581no—
179C2×C4⋊C4—1581no—
180C2×C4⋊C4—1581no—
181C2×C4⋊C4—1581no—
182C2×C4⋊C4C4×D7911yescharacteristic
183C2×C4⋊C4D31611yescharacteristic
184C2×C4⋊C4C22×D7911yescharacteristic
185C2×C42C4×D7911yescharacteristic
186C2×C42D31611yescharacteristic
187C2×C42C22⋊D7911yescharacteristic
188C4⋊C8—3161no—
189#101—24no—
190C22⋊C8—1582no—
191#97—41no—
192C4×C8—41no—
193C4×C8—3161no—
194#95—791no—
195#95—791no—
196#95—1581no—
197#95—1581no—
198#95—1581no—
199#95—1581no—
200#95—1581no—
201#95—1581no—
202#95—1581no—
203#95—1581no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C22×C4—1581no—
208C22×C4—1581no—
209C22×C4—1581no—
210C22×C4—1581no—
211C22×C4(C2×D79)⋊C411yescharacteristic
212C22×C4(C2×D79)⋊C411yescharacteristic
213C22×C4(C2×D79)⋊C411yescharacteristic
214C22×C4(C2×D79)⋊C411yescharacteristic
215C22×C4C2×C4×D7911yescharacteristic
216C22×C4C2×C22⋊D7911yescharacteristic
217C22×C4C2×D31611yescharacteristic
218M16—24no—
219M16—42no—
220C2×C8—42no—
221C2×C8—3162no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—3161no—
226C4⋊C4—3161no—
227C4⋊C4—3162no—
228C4⋊C4—3161no—
229C4⋊C4—3161no—
230C4⋊C4—3162no—
231C4⋊C4—3162no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23C2×(C2×D79)⋊C411yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—3161no—
255C2×C4—3161no—
256C2×C4—3161no—
257C2×C4—3161no—
258C2×C4—3162no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4#1457907511yescharacteristic
266C2×C4#1457907511yescharacteristic
267C2×C4#1457908912yes—
268C2×C4#1457918112yes—
269C8—44no—
270C8—6322no—
271C22—21no—
272C22—21no—
273C22C2×#1457907511yescharacteristic
274C22#1909855711yescharacteristic
275C22C2×#1457908911yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—6321no—
285C4—6321no—
286C4—6321no—
287C4—6321no—
288C2—21no—
289C2—21no—
290C2#2376188911yescharacteristic
291C2#2376505511yescharacteristic
292C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#101×C79C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#101D79
hypocenterC79#11740
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
152. C79#11740semidirect

Maximal subgroups

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

subgroupindex
#237692762
#237617242
#237614262
#237614052
C4×#101×C792
#1174079