Finite Group Databasebeta

← #34183994

Subgroups — #34183994

Order 20224.

694 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 139 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#23771739C211yesmaximal, characteristic
2#23763468C211yesmaximal, characteristic
3C2×#19098299C212yesmaximal
4#23761279C212yesmaximal
5C2×#376×C79C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098469C412yescomplemented
9#19098461C412yescomplemented
10#19098447—22no—
11#19098436—21no—
12#19098426—21no—
13#19098423C2211yescharacteristic
14#19098318—22no—
15#19098299—24no—
16C22×C42×C79—21no—
17C22×C42×C79C2211yescharacteristic
18#376×C79—24no—
19C2×#95×C79—21no—
20C2×#95×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C22×C79⋊C8—22no—
23#14579140—22no—
24#14579140—22no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—21no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C23×C4×C79—22no—
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C23×C4×C79C2311yescharacteristic
41C2×C42×C79—28no—
42C2×C42×C79—24no—
43C2×C42×C79—22no—
44C2×C42×C79—22no—
45C2×C42×C79C2×C412yes—
46C2×C42×C79D412yes—
47#95×C79—24no—
48#95×C79—22no—
49#95×C79D414yes—
50C2×Dic158—22no—
51C2×Dic158—42no—
52C22×Dic79—21no—
53C22×Dic79—22no—
54C22×Dic79—21no—
55C2×C79⋊C8—44no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—22no—
58C4⋊Dic79—42no—
59Dic79⋊C4—44no—
60Dic79⋊C4—42no—
61C22⋊Dic79—41no—
62C22⋊Dic79—41no—
63C22⋊Dic79—42no—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—24no—
68C22×C4×C79—28no—
69C22×C4×C79—21no—
70C22×C4×C79—24no—
71C22×C4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79—22no—
74C22×C4×C79—42no—
75C22×C4×C79—44no—
76C22×C4×C79—42no—
77C22×C4×C79—42no—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22⋊C412yes—
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D412yes—
83C22×C4×C79#4112yes—
84C42×C79—216no—
85C42×C79—24no—
86C42×C79—24no—
87C23×C79—22no—
88C23×C79—22no—
89C23×C79C2×C22⋊C411yescharacteristic
90C23×C79#12011yescharacteristic
91C23×C79#12311yescharacteristic
92C23×C79#13512yes—
93C23×C79#13712yes—
94C2×C4×C79—24no—
95C2×C4×C79—24no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—28no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—44no—
107C2×C4×C79—48no—
108C2×C4×C79—48no—
109C2×C4×C79—42no—
110C2×C4×C79—44no—
111C2×C4×C79—42no—
112C2×C4×C79#10414yes—
113C2×C4×C79C4×D412yes—
114C2×C4×C79#12112yes—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122Dic158—82no—
123C79⋊C8—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—21no—
128C22×C79—24no—
129C22×C79—24no—
130C22×C79#38611yescharacteristic
131C22×C79#41012yes—
132C22×C79#41611yescharacteristic
133C22×C79#41911yescharacteristic
134C22×C79C2×#10412yes—
135C22×C79#45212yes—
136C22×C79#48512yes—
137C4×C79—24no—
138C4×C79—22no—
139C4×C79—22no—
140C4×C79—48no—
141C4×C79—42no—
142Dic79—81no—
143Dic79—81no—
144Dic79—82no—
145#9145—791nomaximal, Sylow
146C2×C79—22no—
147C2×C79—22no—
148C2×C79#150012yes—
149C2×C79#150612yes—
150C2×C79#187311yescharacteristic
151C2×C79#189611yescharacteristic
152C2×C79#189911yescharacteristic
153#2855—791no—
154C2×#376D7911yescharacteristic
155#1817—791no—
156C2×#328—792no—
157#1294—792no—
158C79#914511yescharacteristic, complemented, Sylow
159C2×C22⋊Q8—791no—
160C22×C42—21no—
161C22×C42C2×D7911yescharacteristic
162C2×C22⋊C8—792no—
163#391—792no—
164#389—792no—
165#387—1582no—
166#385—1581no—
167#381—1581no—
168#380—791no—
169#376—24no—
170C2×#95—21no—
171C2×#95C2×D7912yes—
172#336—1582no—
173#328—1584no—
174C22×Q8—791no—
175C23×C4—22no—
176C23×C4C22⋊D7911yescharacteristic
177C23×C4C22⋊D7911yescharacteristic
178C23×C4C22×D7911yescharacteristic
179C22×C8—1582no—
180C22⋊Q8—1582no—
181C22⋊Q8—1582no—
182C2×C4⋊C4—792no—
183C2×C4⋊C4—791no—
184C2×C4⋊C4—1582no—
185C2×C4⋊C4—1581no—
186C2×C22⋊C4—1581no—
187C2×C22⋊C4—1581no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C4×D7912yes—
193C2×C42D31612yes—
194C22⋊C8—1582no—
195C22⋊C8—1582no—
196#95—24no—
197#95—22no—
198#95—1582no—
199#95—1582no—
200#95—1582no—
201#95—1581no—
202#95C22⋊D7914yes—
203C24C2×C22⋊D7911yescharacteristic
204C2×Q8—3162no—
205C2×Q8—1582no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1582no—
220C22×C4—1581no—
221C22×C4—1581no—
222C22×C4(C2×D79)⋊C412yes—
223C22×C4(C2×D79)⋊C412yes—
224C22×C4C2×C4×D7911yescharacteristic
225C22×C4C2×C22⋊D7912yes—
226C22×C4C2×D31611yescharacteristic
227C22×C4#1192962412yes—
228C2×C8—3164no—
229C2×C8—3162no—
230C4⋊C4—3162no—
231C4⋊C4—3164no—
232C4⋊C4—3162no—
233C4⋊C4—1582no—
234C22⋊C4—3161no—
235C22⋊C4—3161no—
236C22⋊C4—3162no—
237C42—216no—
238C42—24no—
239C42—24no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D79)⋊C411yescharacteristic
243C23#1457913011yescharacteristic
244C23#1457914411yescharacteristic
245C23#1457918112yes—
246C23#1457919112yes—
247Q8—6322no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—48no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—3161no—
267C2×C4—3161no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4—3162no—
273C2×C4#1457908914yes—
274C2×C4C4×C22⋊D7912yes—
275C2×C4#1457913512yes—
276C8—6322no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#1909844211yescharacteristic
284C22#1909853612yes—
285C22#1909855711yescharacteristic
286C22#1909857111yescharacteristic
287C22C2×#1457908912yes—
288C22#1909872512yes—
289C22#1909890412yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—48no—
294C4—42no—
295C4—6321no—
296C4—6321no—
297C4—6322no—
298C2—22no—
299C2—22no—
300C2#2376188412yes—
301C2#2376191112yes—
302C2#2376371211yescharacteristic
303C2#2376384311yescharacteristic
304C2#2376385811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#376×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#376D79
hypocenterC79#9145
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
8. #19098469C4semidirect
9. #19098461C4semidirect
158. C79#9145semidirect

Maximal subgroups

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

subgroupindex
#237717392
#237634682
C2×#190982992
#237612792
C2×#376×C792
#914579