Finite Group Databasebeta

← #34181048

Subgroups — #34181048

Order 20224.

540 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 121 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#124×Dic79C211yesmaximal, characteristic
2#23763254C212yesmaximal, complemented
3C2×#19098303C211yesmaximal, characteristic
4#23761282C212yesmaximal, complemented
5C2×#329×C79C211yesmaximal, characteristic
6C2×Q8×Dic79C411yescharacteristic
7C2×D4×Dic79C411yescharacteristic
8C2×Dic79⋊C8—22no—
9C2×#14579072C2212yes—
10#19098470C411yescharacteristic
11#19098466C411yescharacteristic
12#19098456—22no—
13#19098429C2211yescharacteristic
14C22⋊C4×Dic79—22no—
15C42×Dic79C2211yescharacteristic
16#19098317—22no—
17#19098303—22no—
18C2×#124×C79C2211yescharacteristic
19C2×#97×C79C2212yes—
20#329×C79—22no—
21C23×Dic79—21no—
22C22×C79⋊C8—22no—
23Q8×Dic79—42no—
24D4×Dic79—42no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—21no—
30C2×C4×Dic79—21no—
31C2×C4×Dic79C2×C411yescharacteristic
32C2×C4×Dic79C2×C411yescharacteristic
33C2×C4×Dic79D412yes—
34C2×C4×Dic79D412yes—
35Dic79⋊C8—28no—
36#14579072—22no—
37#14579072—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×Q8×C79C2×C411yescharacteristic
41C22×D4×C79C2×C411yescharacteristic
42C22×C8×C79—22no—
43#124×C79—22no—
44#124×C79D412yes—
45#124×C79Q812yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79C2311yescharacteristic
48#97×C79—22no—
49#97×C79—22no—
50C22×Dic79—21no—
51C22×Dic79—21no—
52C22×Dic79—44no—
53C22×Dic79—42no—
54C22×Dic79C22⋊C411yescharacteristic
55C22×Dic79C22⋊C411yescharacteristic
56C22×Dic79C22⋊C412yes—
57C2×C79⋊C8—24no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—42no—
60C4⋊Dic79—41no—
61C4⋊Dic79—41no—
62C4⋊Dic79—42no—
63C22⋊Dic79—42no—
64C22⋊Dic79—44no—
65C4×Dic79—22no—
66C4×Dic79—24no—
67C4×Dic79—41no—
68C4×Dic79—42no—
69C4×Dic79—42no—
70C24×C79—21no—
71C2×Q8×C79—21no—
72C2×Q8×C79—41no—
73C2×Q8×C79—41no—
74C2×Q8×C79C4⋊C412yes—
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—
82C2×C8×C79—24no—
83C2×C8×C79—42no—
84C2×C8×C79—42no—
85C22⋊C4×C79—24no—
86C22⋊C4×C79—24no—
87C42×C79C2×D411yescharacteristic
88C42×C79C2×Q811yescharacteristic
89C42×C79#4112yes—
90C23×C79—22no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79C2×C22⋊C411yescharacteristic
96Q8×C79—42no—
97Q8×C79—42no—
98D4×C79—42no—
99D4×C79—42no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
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—22no—
112C2×Dic79—22no—
113C2×Dic79—22no—
114C2×Dic79—21no—
115C2×Dic79—22no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—84no—
121C2×Dic79#10012yes—
122C2×Dic79#10414yes—
123C8×C79—44no—
124C79⋊C8—44no—
125C22×C79—42no—
126C22×C79—44no—
127C22×C79—81no—
128C22×C79—82no—
129C22×C79—81no—
130C22×C79#38011yescharacteristic
131C22×C79C2×#10011yescharacteristic
132C22×C79#41311yescharacteristic
133C22×C79C2×#10412yes—
134C22×C79#42112yes—
135C4×C79—21no—
136C4×C79—21no—
137C4×C79—22no—
138C4×C79—22no—
139C4×C79—42no—
140Dic79—24no—
141Dic79—22no—
142Dic79—41no—
143Dic79—81no—
144Dic79—81no—
145#8583—791nomaximal, Sylow
146C2×C79—82no—
147C2×C79#152511yescharacteristic
148C2×C79#153212yes—
149C2×C79#153311yescharacteristic
150C2×C79#180011yescharacteristic
151C2×C79#180612yes—
152C4×#124—791no—
153#1767—792no—
154C2×#329—791no—
155C2×#329D7911yescharacteristic
156#1295—792no—
157C79#858311yescharacteristic, complemented, Sylow
158C2×#124C2×D7911yescharacteristic
159C2×C4×Q8—791no—
160C2×C4×D4—791no—
161C2×C4⋊C8—1582no—
162C2×#97—792no—
163C2×#97C2×D7912yes—
164#391—791no—
165#390—791no—
166#388—1582no—
167#382—791no—
168C4×C22⋊C4—1582no—
169C43—791no—
170#336—1582no—
171#329—22no—
172#329—1582no—
173C22×Q8C4×D7911yescharacteristic
174C22×D4C4×D7911yescharacteristic
175C23×C4—1581no—
176C22×C8—22no—
177C22×C8—1582no—
178#124—22no—
179#124Dic15812yes—
180#124C22⋊D7912yes—
181C4×Q8—3162no—
182C4×D4—3162no—
183C2×C4⋊C4—791no—
184C2×C4⋊C4—791no—
185C2×C4⋊C4—1581no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—1581no—
188C2×C22⋊C4—1582no—
189C2×C42—791no—
190C2×C42—792no—
191C2×C42—791no—
192C2×C42—792no—
193C2×C42—1581no—
194C2×C42C22×D7911yescharacteristic
195C4⋊C8—1588no—
196#97—22no—
197#97—22no—
198#97—1582no—
199#97—1582no—
200#95—1582no—
201#95—1582no—
202C24—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8Dic79⋊C412yes—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4Dic79⋊C412yes—
211C22×C4—21no—
212C22×C4—3162no—
213C22×C4—3164no—
214C22×C4—792no—
215C22×C4—791no—
216C22×C4—791no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4D4×D7912yes—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—3162no—
225C2×C8—3162no—
226C2×C8—1584no—
227C4⋊C4—3161no—
228C4⋊C4—3161no—
229C4⋊C4—3162no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—3164no—
233C22⋊C4—3162no—
234C42—3161no—
235C42—3162no—
236C42—3162no—
237C42—1584no—
238C42—1582no—
239C42C2×C22⋊D7911yescharacteristic
240C42C2×Dic15811yescharacteristic
241C42#1192962312yes—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22⋊C4×D7911yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3161no—
260C2×C4—3161no—
261C2×C4—3162no—
262C2×C4—3162no—
263C2×C4—792no—
264C2×C4—794no—
265C2×C4—6324no—
266C2×C4—1581no—
267C2×C4—1582no—
268C2×C4—1582no—
269C2×C4—1582no—
270C2×C4—1582no—
271C2×C4C2×Dic79⋊C411yescharacteristic
272C2×C4#1457911711yescharacteristic
273C2×C4#1457913312yes—
274C2×C4#1457913812yes—
275C8—44no—
276C8—3164no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22#1909842111yescharacteristic
283C22#100×D7911yescharacteristic
284C22#1909854411yescharacteristic
285C22#104×D7912yes—
286C22#1909857812yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—3161no—
293C4—6321no—
294C4—6321no—
295C4—1584no—
296C4—1582no—
297C2—82no—
298C2#2376199011yescharacteristic
299C2#2376201712yes—
300C2#2376202211yescharacteristic
301C2#2376339011yescharacteristic
302C2#2376341812yes—
subgroupG/H
centerC23C22⋊C4×D79
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#329×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenterC2×#329D79
hypocenterC79#8583
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #23763254C2semidirect
4. #23761282C2semidirect
157. C79#8583semidirect

Maximal subgroups

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

subgroupindex
#124×Dic792
#237632542
C2×#190983032
#237612822
C2×#329×C792
#858379