Finite Group Databasebeta

← #34205093

Subgroups — #34205093

Order 20224.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#19098360C211yesmaximal, characteristic
2#23763804C211yesmaximal, characteristic
3#23763614C212yesmaximal, complemented
4#23761527C212yesmaximal, complemented
5C79×#2379C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098603—22no—
8#19098568—21no—
9C2×#14579080C2211yescharacteristic
10C2×#14579080C2212yes—
11#19098360C414yescomplemented
12#19098323—21no—
13#19098323—22no—
14C2×#124×C79C2211yescharacteristic
15#391×C79C2212yes—
16#388×C79—22no—
17#386×C79—21no—
18#385×C79—22no—
19#382×C79—21no—
20C2×#11929608—22no—
21C2×#11929608—21no—
22Q8×Dic79—41no—
23Q8×Dic79—41no—
24#14579120—21no—
25#14579120—22no—
26#14579094—42no—
27#14579092—42no—
28#14579082—41no—
29#14579082—41no—
30#14579080—22no—
31#14579080—22no—
32#14579080C2×C414yescomplemented
33C22×Q8×C79C2311yescharacteristic
34C22×D4×C79D411yescharacteristic
35C23×C4×C79—21no—
36#124×C79—22no—
37#124×C79C2×C412yes—
38#124×C79Q812yes—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79D412yes—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79—21no—
45C2×C42×C79D411yescharacteristic
46C2×C42×C79D412yes—
47#95×C79—22no—
48#95×C79—22no—
49#95×C79—22no—
50#11929608—24no—
51#11929608—42no—
52#11929608—41no—
53#11929608—41no—
54C2×C79⋊C8—42no—
55C2×C79⋊C8—41no—
56C4⋊Dic79—21no—
57C4⋊Dic79—21no—
58C4⋊Dic79—22no—
59C4⋊Dic79—22no—
60C22⋊Dic79—41no—
61C4×Dic79—42no—
62C4×Dic79—41no—
63C24×C79—21no—
64C2×Q8×C79—41no—
65C2×Q8×C79—41no—
66C2×Q8×C79C22×C411yescharacteristic
67C2×Q8×C79C2×Q811yescharacteristic
68C2×Q8×C79#4112yes—
69C2×D4×C79—21no—
70C2×D4×C79—41no—
71C2×D4×C79—41no—
72C2×D4×C79C4⋊C412yes—
73C22×C4×C79—21no—
74C22×C4×C79—22no—
75C22×C4×C79—22no—
76C22×C4×C79—82no—
77C22×C4×C79—81no—
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D412yes—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—81no—
82C4⋊C4×C79—81no—
83C22⋊C4×C79—24no—
84C22⋊C4×C79—24no—
85C22⋊C4×C79—82no—
86C22⋊C4×C79—81no—
87C42×C79—21no—
88C42×C79—42no—
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×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—84no—
115C2×C4×C79C2×C4⋊C411yescharacteristic
116C2×C4×C79C4×D412yes—
117C2×C4×C79C22⋊Q812yes—
118C2×C4×C79#12311yescharacteristic
119C2×Dic79—41no—
120C2×Dic79—41no—
121C79⋊C8—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—
138C4×C79—82no—
139C4×C79—82no—
140Dic79—81no—
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#2379D7911yescharacteristic
150C2×#355—791no—
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—792no—
160C2×#101—791no—
161#391C2×D7912yes—
162#388—22no—
163#386—21no—
164#385—22no—
165#382—21no—
166#355—794no—
167#339—1582no—
168#339—1581no—
169C22×Q8C22×D7911yescharacteristic
170C22×D4C22⋊D7911yescharacteristic
171C23×C4—21no—
172C2×M16—1581no—
173C2×M16—1582no—
174#124—22no—
175#124Dic15812yes—
176#124C4×D7912yes—
177C4×Q8—3161no—
178C4×Q8—3161no—
179#117—1581no—
180#117—1582no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊D7912yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42C22⋊D7911yescharacteristic
188C2×C42C22⋊D7912yes—
189#107—3162no—
190#106—3162no—
191#102—3161no—
192#102—3161no—
193#101—794no—
194#101—1582no—
195#101—1582no—
196#95—22no—
197#95—22no—
198#95—22no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×C4×D7911yescharacteristic
203C2×Q8C2×Dic15811yescharacteristic
204C2×Q8#1192962412yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4Dic79⋊C412yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—82no—
213C22×C4—81no—
214C22×C4C2×C22⋊D7911yescharacteristic
215C22×C4C2×C22⋊D7912yes—
216M16—3162no—
217M16—3161no—
218M16—3161no—
219M16—1584no—
220C2×C8—3162no—
221C2×C8—3161no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—1581no—
226C4⋊C4—1581no—
227C4⋊C4—1582no—
228C4⋊C4—1582no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—82no—
232C22⋊C4—81no—
233C22⋊C4—3161no—
234C42—21no—
235C42—42no—
236C42—3162no—
237C42—3161no—
238C42Dic79⋊C412yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1457913011yescharacteristic
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—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4C2×Dic79⋊C411yescharacteristic
267C2×C4C4×C22⋊D7912yes—
268C2×C4#1457914012yes—
269C2×C4#1457914411yescharacteristic
270C8—3164no—
271C8—6321no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1909843611yescharacteristic
280C22#1909875011yescharacteristic
281C22#1909875311yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—82no—
289C4—6321no—
290C4—6321no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2376368111yescharacteristic
295C2#2376541711yescharacteristic
296C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#2379C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
cosocleC22×Q8×C79C23
hypercenter#2379D79
hypocenterC79#12970
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2379

Complements

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

NH ≅ G/Ntype
3. #23763614C2semidirect
4. #23761527C2semidirect
11. #19098360C4semidirect
32. #14579080C2×C4semidirect
154. C79#12970semidirect

Maximal subgroups

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

subgroupindex
C2×#190983602
#237638042
#237636142
#237615272
C79×#23792
#1297079