Finite Group Databasebeta

← #34185998

Subgroups — #34185998

Order 20224.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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#23767874C211yesmaximal, characteristic
2#23763480C211yesmaximal, characteristic
3#23761932C212yesmaximal
4#23761252C212yesmaximal
5C79×#2313C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7#19098527C2212yes—
8#19098494—22no—
9#19098468C412yescomplemented
10#19098436—21no—
11#19098436—21no—
12#19098432C412yescomplemented
13C4×C22⋊Dic79C2211yescharacteristic
14#19098332—22no—
15#19098299—24no—
16C22×C4⋊C4×C79C2211yescharacteristic
17#387×C79—24no—
18#381×C79C2212yes—
19#380×C79—21no—
20C4×C22⋊C4×C79—21no—
21C22×Dic158C2×C411yescharacteristic
22C2×#11929608—22no—
23#14579140—22no—
24#14579140—22no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×Dic79⋊C4—21no—
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—21no—
31C2×C4×Dic79C2×C412yes—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79C2×C412yes—
44C2×C4⋊C4×C79D412yes—
45C2×C22⋊C4×C79—22no—
46C2×C42×C79—22no—
47#95×C79—22no—
48#95×C79—22no—
49#95×C79—22no—
50#95×C79D414yes—
51C2×Dic158—22no—
52C2×Dic158—42no—
53C22×Dic79—22no—
54C22×Dic79—21no—
55C22×Dic79—21no—
56#11929608—44no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59Dic79⋊C4—42no—
60Dic79⋊C4—82no—
61Dic79⋊C4—82no—
62C22⋊Dic79—24no—
63C22⋊Dic79—41no—
64C22⋊Dic79—41no—
65C4×Dic79—42no—
66C24×C79C2×D411yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—24no—
72C22×C4×C79—42no—
73C22×C4×C79—81no—
74C22×C4×C79—81no—
75C22×C4×C79C22⋊C412yes—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D412yes—
80C22×C4×C79#4112yes—
81C4⋊C4×C79—24no—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—82no—
84C22⋊C4×C79—44no—
85C42×C79—82no—
86C23×C79—22no—
87C23×C79—22no—
88C23×C79—41no—
89C23×C79C2×C22⋊C411yescharacteristic
90C23×C79#12011yescharacteristic
91C23×C79#12311yescharacteristic
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—44no—
103C2×C4×C79—44no—
104C2×C4×C79—44no—
105C2×C4×C79—81no—
106C2×C4×C79—81no—
107C2×C4×C79C4×D412yes—
108C2×C4×C79#12112yes—
109C2×Dic79—24no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—41no—
114C2×Dic79—41no—
115C2×Dic79—42no—
116Dic158—82no—
117C79⋊C8—82no—
118C22×C79—21no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—44no—
126C22×C79#38611yescharacteristic
127C22×C79#41012yes—
128C22×C79#45612yes—
129C4×C79—22no—
130C4×C79—22no—
131C4×C79—42no—
132C4×C79—42no—
133C4×C79—84no—
134Dic79—44no—
135Dic79—81no—
136Dic79—81no—
137#9461—791nomaximal, Sylow
138C2×C79—22no—
139C2×C79—22no—
140C2×C79—41no—
141C2×C79#188611yescharacteristic
142C2×C79#189911yescharacteristic
143C2×C79#191111yescharacteristic
144#2502—791no—
145#2313D7911yescharacteristic
146#1821—791no—
147#1511—792no—
148#1284—792no—
149C79#946111yescharacteristic, complemented, Sylow
150C2×C22⋊Q8—791no—
151C22×C4⋊C4C2×D7911yescharacteristic
152#407—792no—
153#398—1582no—
154#391—792no—
155#387—24no—
156#385—1581no—
157#385—1581no—
158#384—792no—
159#381C2×D7912yes—
160#380—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—791no—
163#343—1582no—
164#328—1584no—
165C22×Q8—791no—
166C23×C4C22⋊D7911yescharacteristic
167C23×C4C22⋊D7911yescharacteristic
168C23×C4C22×D7911yescharacteristic
169C2×M16—1582no—
170C22⋊Q8—1582no—
171C22⋊Q8—1582no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—791no—
176C2×C4⋊C4—1581no—
177C2×C4⋊C4—1582no—
178C2×C4⋊C4—1582no—
179C2×C4⋊C4C4×D7912yes—
180C2×C4⋊C4D31612yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1581no—
183C2×C22⋊C4—1581no—
184C2×C42—22no—
185C2×C42—792no—
186C22⋊C8—1582no—
187C22⋊C8—1582no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—1582no—
192#95—1582no—
193#95—1581no—
194#95C22⋊D7914yes—
195C24C2×C22⋊D7911yescharacteristic
196C2×Q8—3162no—
197C2×Q8—1582no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—1581no—
207C22×C4—1581no—
208C22×C4—1582no—
209C22×C4(C2×D79)⋊C412yes—
210C22×C4(C2×D79)⋊C412yes—
211C22×C4C2×C4×D7911yescharacteristic
212C22×C4C2×C22⋊D7912yes—
213C22×C4C2×D31611yescharacteristic
214C22×C4#1192962412yes—
215M16—3164no—
216C2×C8—3162no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—82no—
220C4⋊C4—3162no—
221C4⋊C4—6322no—
222C4⋊C4—6322no—
223C4⋊C4—1582no—
224C22⋊C4—44no—
225C22⋊C4—3161no—
226C22⋊C4—3161no—
227C22⋊C4—1584no—
228C42—82no—
229C42—3162no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×(C2×D79)⋊C411yescharacteristic
234C23#1457913011yescharacteristic
235C23#1457914411yescharacteristic
236Q8—6322no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—3162no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4—3161no—
256C2×C4—3161no—
257C2×C4—3162no—
258C2×C4—1584no—
259C2×C4C4×C22⋊D7912yes—
260C2×C4#1457913512yes—
261C8—6322no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1909844211yescharacteristic
271C22#1909853612yes—
272C22#1909874612yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—84no—
278C4—3164no—
279C4—6321no—
280C4—6321no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2376379311yescharacteristic
285C2#2376385811yescharacteristic
286C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2313C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2313D79
hypocenterC79#9461
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
9. #19098468C4semidirect
12. #19098432C4semidirect
149. C79#9461semidirect

Maximal subgroups

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

subgroupindex
#237678742
#237634802
#237619322
#237612522
C79×#23132
#946179