Finite Group Databasebeta

← #34205095

Subgroups — #34205095

Order 20224.

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

normal subgroups only

This group has 472 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#23766574C211yesmaximal, characteristic
2#23763804C211yesmaximal, characteristic
3#23763617C212yesmaximal, complemented
4#23761526C212yesmaximal, complemented
5C2×#355×C79C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098605—22no—
8#19098568—21no—
9C2×#14579080C2211yescharacteristic
10#19098470C2212yes—
11#19098456—22no—
12#19098443—21no—
13#19098438—22no—
14#19098429—21no—
15#19098324—22no—
16#19098323—21no—
17C2×#124×C79C2211yescharacteristic
18C2×#101×C79C2212yes—
19#355×C79C414yescomplemented
20C23×Dic79—21no—
21C2×#11929608—21no—
22Q8×Dic79—41no—
23Q8×Dic79—41no—
24#14579120—21no—
25#14579120—22no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79D412yes—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—21no—
31C2×C4×Dic79D412yes—
32#14579095—42no—
33#14579093—42no—
34#14579082—41no—
35#14579082—41no—
36#14579080—22no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×Q8×C79C2311yescharacteristic
41C22×D4×C79D411yescharacteristic
42C2×M16×C79—22no—
43#124×C79—22no—
44#124×C79C2×C412yes—
45#124×C79Q812yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79D411yescharacteristic
48#101×C79—22no—
49#101×C79C2×C414yescomplemented
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—82no—
53C22×Dic79—81no—
54#11929608—41no—
55#11929608—41no—
56C2×C79⋊C8—41no—
57C4⋊Dic79—21no—
58C4⋊Dic79—21no—
59C4⋊Dic79—22no—
60C4⋊Dic79—22no—
61C4⋊Dic79—42no—
62C4⋊Dic79—81no—
63C4⋊Dic79—81no—
64C22⋊Dic79—41no—
65C22⋊Dic79—82no—
66C22⋊Dic79—81no—
67C4×Dic79—42no—
68C4×Dic79—42no—
69C4×Dic79—41no—
70C24×C79—21no—
71C2×Q8×C79—41no—
72C2×Q8×C79—41no—
73C2×Q8×C79C22×C411yescharacteristic
74C2×Q8×C79C2×Q811yescharacteristic
75C2×Q8×C79#4112yes—
76C2×D4×C79—21no—
77C2×D4×C79—41no—
78C2×D4×C79—41no—
79C2×D4×C79C4⋊C412yes—
80C22×C4×C79—21no—
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D412yes—
83M16×C79—24no—
84M16×C79—42no—
85C2×C8×C79—42no—
86C22⋊C4×C79—24no—
87C22⋊C4×C79—24no—
88C42×C79—21no—
89C42×C79C4⋊C412yes—
90C23×C79—22no—
91C23×C79—42no—
92C23×C79—42no—
93C23×C79—41no—
94C23×C79—41no—
95C23×C79#12011yescharacteristic
96Q8×C79—41no—
97Q8×C79—41no—
98Q8×C79—42no—
99D4×C79—42no—
100D4×C79—81no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—22no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79C2×C4⋊C411yescharacteristic
109C2×C4×C79C4×D412yes—
110C2×C4×C79C22⋊Q812yes—
111C2×C4×C79#12311yescharacteristic
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—84no—
121C8×C79—44no—
122C79⋊C8—81no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—44no—
126C22×C79—42no—
127C22×C79—82no—
128C22×C79—81no—
129C22×C79—81no—
130C22×C79#38511yescharacteristic
131C22×C79B(4,𝔽2)11yescharacteristic
132C22×C79#45811yescharacteristic
133C4×C79—21no—
134C4×C79—22no—
135C4×C79—22no—
136C4×C79—21no—
137C4×C79—42no—
138Dic79—82no—
139Dic79—81no—
140Dic79—82no—
141Dic79—81no—
142#12970—791nomaximal, Sylow
143C2×C79—21no—
144C2×C79—21no—
145C2×C79—82no—
146C2×C79#186711yescharacteristic
147C2×C79#220211yescharacteristic
148C2×C79#220311yescharacteristic
149#2379—791no—
150C2×#355D7911yescharacteristic
151#1888—791no—
152#1857—792no—
153#1394—792no—
154C79#1297011yescharacteristic, complemented, Sylow
155C2×#124C2×D7911yescharacteristic
156#519—791no—
157#428—1582no—
158#418—1581no—
159C2×#101—791no—
160C2×#101C2×D7912yes—
161#391—792no—
162#388—1582no—
163#386—1581no—
164#385—1582no—
165#382—1581no—
166#355Dic7914yescomplemented
167#339—1582no—
168#339—1581no—
169C22×Q8C22×D7911yescharacteristic
170C22×D4D31611yescharacteristic
171C23×C4—1581no—
172C2×M16—22no—
173C2×M16—1581no—
174#124—22no—
175#124Dic15812yes—
176#124C2×Dic7912yes—
177C4×Q8—3161no—
178C4×Q8—3161no—
179#117—1582no—
180#117—1581no—
181C2×C4⋊C4—792no—
182C2×C4⋊C4—1581no—
183C2×C4⋊C4—1581no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—1582no—
186C2×C22⋊C4—1581no—
187C2×C42—792no—
188C2×C42D31611yescharacteristic
189#107—3162no—
190#106—3162no—
191#102—3161no—
192#102—3161no—
193#101—22no—
194#101—1582no—
195#101C2×Dic7914yescomplemented
196#95—1582no—
197#95—1582no—
198#95—1582no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C22×Dic7911yescharacteristic
203C2×Q8C2×Dic15811yescharacteristic
204C2×Q8#1192962312yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊Dic7912yes—
209C22×C4—21no—
210C22×C4—6322no—
211C22×C4—6321no—
212C22×C4—1582no—
213C22×C4—1582no—
214C22×C4D4×D7912yes—
215C22×C4C2×D31611yescharacteristic
216M16—24no—
217M16—42no—
218M16—3161no—
219M16—3161no—
220C2×C8—42no—
221C2×C8—3161no—
222C4⋊C4—3162no—
223C4⋊C4—6321no—
224C4⋊C4—6321no—
225C4⋊C4—1581no—
226C4⋊C4—1581no—
227C4⋊C4—1582no—
228C4⋊C4—1582no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—3161no—
232C22⋊C4—6322no—
233C22⋊C4—6321no—
234C42—21no—
235C42—3162no—
236C42—3162no—
237C42—3161no—
238C42C4⋊Dic7912yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1457913111yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—3162no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4—3162no—
261C2×C4—3161no—
262C2×C4—3161no—
263C2×C4—3162no—
264C2×C4—3162no—
265C2×C4—6324no—
266C2×C4C2×C4⋊Dic7911yescharacteristic
267C2×C4D4×Dic7912yes—
268C2×C4#1457913812yes—
269C2×C4#1457914511yescharacteristic
270C8—44no—
271C8—6321no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1909843811yescharacteristic
280C22#1909875111yescharacteristic
281C22#1909875411yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—6322no—
288C4—6322no—
289C4—6321no—
290C4—6321no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2376368311yescharacteristic
295C2#2376541811yescharacteristic
296C2#2376542311yescharacteristic
subgroupG/H
centerC22#19098751
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#355×C79C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
cosocleC22×Q8×C79C23
hypercenterC2×#355D79
hypocenterC79#12970
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #23763617C2semidirect
4. #23761526C2semidirect
19. #355×C79C4semidirect
49. #101×C79C2×C4semidirect
154. C79#12970semidirect
166. #355Dic79semidirect
195. #101C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237665742
#237638042
#237636172
#237615262
C2×#355×C792
#1297079