Finite Group Databasebeta

← #34204274

Subgroups — #34204274

Order 20224.

480 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 101 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#23766113C211yesmaximal, characteristic
2C2×#19098360C211yesmaximal, characteristic
3#23763394C212yesmaximal, complemented
4#23761526C212yesmaximal, complemented
5C2×#355×C79C211yesmaximal, characteristic
6C2×Q8×Dic79C2211yescharacteristic
7C2×D4×Dic79C411yescharacteristic
8#19098586—22no—
9C2×#14579080C2212yes—
10#19098470C2211yescharacteristic
11#19098466C411yescharacteristic
12#19098456—22no—
13#19098438—22no—
14#19098433C411yescharacteristic
15#19098409C411yescharacteristic
16#19098360—22no—
17#19098331—22no—
18C2×#124×C79C2211yescharacteristic
19C2×#101×C79C2212yes—
20#355×C79—22no—
21C23×Dic79—21no—
22C2×#11929608—22no—
23Q8×Dic79—42no—
24D4×Dic79—42no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79D412yes—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—22no—
31C2×C4×Dic79C2×C411yescharacteristic
32C2×C4×Dic79C2×C411yescharacteristic
33C2×C4×Dic79D412yes—
34Dic79⋊C8—44no—
35#14579080—22no—
36#14579080—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C22×Q8×C79C2311yescharacteristic
41C22×D4×C79C2×C411yescharacteristic
42C2×M16×C79—22no—
43#124×C79—22no—
44#124×C79D412yes—
45#124×C79Q812yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79C2×C411yescharacteristic
48#101×C79—22no—
49#101×C79—22no—
50C22×Dic79—21no—
51C22×Dic79—21no—
52C22×Dic79—42no—
53C22×Dic79—82no—
54C22×Dic79C22⋊C411yescharacteristic
55C22×Dic79C22⋊C412yes—
56C22×Dic79C22⋊C411yescharacteristic
57#11929608—42no—
58#11929608—42no—
59C2×C79⋊C8—42no—
60C4⋊Dic79—41no—
61C4⋊Dic79—42no—
62C4⋊Dic79—41no—
63C4⋊Dic79—82no—
64C22⋊Dic79—42no—
65C22⋊Dic79—82no—
66C4×Dic79—24no—
67C4×Dic79—22no—
68C4×Dic79—41no—
69C24×C79—21no—
70C2×Q8×C79—41no—
71C2×Q8×C79—41no—
72C2×Q8×C79C2×D411yescharacteristic
73C2×Q8×C79C2×Q811yescharacteristic
74C2×Q8×C79#4112yes—
75C2×D4×C79—21no—
76C2×D4×C79—41no—
77C2×D4×C79—41no—
78C2×D4×C79C4⋊C412yes—
79C22×C4×C79—21no—
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D412yes—
82M16×C79—42no—
83M16×C79—42no—
84C2×C8×C79—42no—
85C22⋊C4×C79—24no—
86C22⋊C4×C79—24no—
87C42×C79—21no—
88C42×C79C4⋊C412yes—
89C23×C79—22no—
90C23×C79—42no—
91C23×C79—42no—
92C23×C79—41no—
93C23×C79—41no—
94C23×C79C2×C22⋊C411yescharacteristic
95Q8×C79—41no—
96Q8×C79—41no—
97Q8×C79—42no—
98D4×C79—42no—
99D4×C79—42no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79C2×C4⋊C411yescharacteristic
108C2×C4×C79#11711yescharacteristic
109C2×C4×C79#12112yes—
110C2×C4×C79C22⋊Q812yes—
111C2×Dic79—21no—
112C2×Dic79—22no—
113C2×Dic79—41no—
114C2×Dic79—42no—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—42no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—84no—
122C2×Dic79#9912yes—
123C8×C79—82no—
124C79⋊C8—82no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—44no—
128C22×C79—42no—
129C22×C79—82no—
130C22×C79—81no—
131C22×C79—81no—
132C22×C79#38011yescharacteristic
133C22×C79C2×#9911yescharacteristic
134C22×C79#41011yescharacteristic
135C4×C79—21no—
136C4×C79—21no—
137C4×C79—22no—
138C4×C79—22no—
139C4×C79—42no—
140Dic79—22no—
141Dic79—42no—
142Dic79—41no—
143Dic79—81no—
144Dic79—81no—
145#12803—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79—82no—
149C2×C79#179911yescharacteristic
150C2×C79#213311yescharacteristic
151C2×C79#213411yescharacteristic
152#2326—791no—
153C2×#355—791no—
154C2×#355D7911yescharacteristic
155#1801—792no—
156#1394—792no—
157C79#1280311yescharacteristic, complemented, Sylow
158C2×#124C2×D7911yescharacteristic
159C2×C4×Q8—791no—
160C2×C4×D4—791no—
161#423—1582no—
162C2×#101—792no—
163C2×#101C2×D7912yes—
164#391—791no—
165#390—791no—
166#388—1582no—
167#385—1582no—
168#384—791no—
169#376—791no—
170#355—22no—
171#355—1582no—
172#343—1582no—
173C22×Q8C22×D7911yescharacteristic
174C22×D4C4×D7911yescharacteristic
175C23×C4—1581no—
176C2×M16—22no—
177C2×M16—1582no—
178#124—22no—
179#124Dic15812yes—
180#124C22⋊D7912yes—
181C4×Q8—3162no—
182C4×D4—3162no—
183C2×C4⋊C4—791no—
184C2×C4⋊C4—792no—
185C2×C4⋊C4—791no—
186C2×C4⋊C4—1582no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1581no—
189C2×C22⋊C4—1582no—
190C2×C42—791no—
191C2×C42—791no—
192C2×C42—792no—
193C2×C42C4×D7911yescharacteristic
194C4⋊C8—3164no—
195#101—22no—
196#101—22no—
197#101—1582no—
198#101—1582no—
199#95—1581no—
200#95—1581no—
201#95—1582no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C22⋊D7911yescharacteristic
206C2×Q8C2×Dic15811yescharacteristic
207C2×Q8#1192962312yes—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4Dic79⋊C412yes—
212C22×C4—21no—
213C22×C4—3162no—
214C22×C4—791no—
215C22×C4—791no—
216C22×C4—792no—
217C22×C4—6322no—
218C22×C4—1581no—
219C22×C4—1581no—
220C22×C4C2×C4×D7911yescharacteristic
221C22×C4D4×D7912yes—
222M16—42no—
223M16—42no—
224M16—3162no—
225M16—3162no—
226C2×C8—42no—
227C2×C8—3162no—
228C4⋊C4—3161no—
229C4⋊C4—3161no—
230C4⋊C4—3162no—
231C4⋊C4—6322no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—3162no—
235C22⋊C4—6322no—
236C42—21no—
237C42—3161no—
238C42—1584no—
239C42—1582no—
240C42Dic79⋊C412yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C22⋊C4×D7911yescharacteristic
247Q8—41no—
248Q8—41no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3162no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3162no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—3161no—
267C2×C4—792no—
268C2×C4—6324no—
269C2×C4—1581no—
270C2×C4—1582no—
271C2×C4C2×Dic79⋊C411yescharacteristic
272C2×C4#1457911711yescharacteristic
273C2×C4#1457913312yes—
274C2×C4#1457913812yes—
275C8—82no—
276C8—6322no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1909842111yescharacteristic
285C22#99×D7911yescharacteristic
286C22#1909853511yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—3161no—
293C4—3162no—
294C4—6321no—
295C4—6321no—
296C4—1582no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2376338611yescharacteristic
301C2#2376508611yescharacteristic
302C2#2376509011yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#355×C79C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC22×Q8×C79C23
hypercenterC2×#355D79
hypocenterC79#12803
LCSG ▷ C22×C4×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. #23763394C2semidirect
4. #23761526C2semidirect
157. C79#12803semidirect

Maximal subgroups

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

subgroupindex
#237661132
C2×#190983602
#237633942
#237615262
C2×#355×C792
#1280379