Finite Group Databasebeta

← #34198844

Subgroups — #34198844

Order 20224.

480 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 87 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#23768392C211yesmaximal, characteristic
2C4×#14579076C211yesmaximal, characteristic
3#23761401C214yesmaximal, complemented
4C79×#1465C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic, complemented
6C2×#14579148—21no—
7C2×#14579076—21no—
8#19098488C412yescomplemented
9#19098488C2212yes—
10#19098468—21no—
11#19098423—21no—
12#19098423—21no—
13C4×C4⋊Dic79C411yescharacteristic, complemented
14C4×C22⋊Dic79—21no—
15#19098330C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17#407×C79—21no—
18#324×C79—21no—
19#323×C79C2212yes—
20C22×Dic158—21no—
21#14579161—41no—
22#14579161—41no—
23#14579148—44no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C412yes—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C4×Dic79—21no—
31C2×C4×Dic79—41no—
32#14579076—24no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—41no—
36C79⋊#95C2×C412yes—
37C22×D4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—41no—
41C4×D4×C79—41no—
42C4×D4×C79—41no—
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79C2×C411yescharacteristic
45C2×C22⋊C4×C79C2311yescharacteristic
46C2×C42×C79C2×C411yescharacteristic
47C22⋊C8×C79—22no—
48C22⋊C8×C79—41no—
49C2×Dic158—42no—
50C2×Dic158—42no—
51C2×Dic158—42no—
52C22×Dic79—22no—
53C22×Dic79—22no—
54C22×Dic79—42no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—41no—
58C4⋊Dic79—41no—
59C22⋊Dic79—24no—
60C22⋊Dic79—44no—
61C22⋊Dic79—42no—
62C22⋊Dic79—44no—
63C4×Dic79—42no—
64C4×Dic79—82no—
65C24×C79C22⋊C411yescharacteristic
66C24×C79C2×D411yescharacteristic
67C2×D4×C79—22no—
68C2×D4×C79—22no—
69C2×D4×C79—41no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81M16×C79—82no—
82C2×C8×C79—42no—
83C4⋊C4×C79—41no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C42×C79—21no—
87C42×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—41no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79C2×C22⋊C411yescharacteristic
98C23×C79AGL(1,ℤ/8)12yes—
99D4×C79—44no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—81no—
112C2×C4×C79—81no—
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79#10412yes—
116C2×Dic79—44no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—82no—
124C2×Dic79—82no—
125Dic158—84no—
126C8×C79—82no—
127C22×C79—21no—
128C22×C79—21no—
129C22×C79—22no—
130C22×C79—42no—
131C22×C79—42no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—41no—
137C22×C79—81no—
138C22×C79—81no—
139C22×C79C2×#9911yescharacteristic
140C22×C79#41811yescharacteristic
141C22×C79C2×#10411yescharacteristic
142C4×C79—21no—
143C4×C79—22no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79#35412yes—
148Dic79—82no—
149Dic79—84no—
150Dic79—82no—
151#11777—791nomaximal, Sylow
152C2×C79—21no—
153C2×C79—21no—
154C2×C79—42no—
155C2×C79—81no—
156C2×C79#150711yescharacteristic
157C2×C79#212311yescharacteristic
158C2×C79C2×#35411yescharacteristic
159#2547—791no—
160C4×#99—791no—
161#1465D7911yescharacteristic
162#1344—794no—
163C79#1177711yescharacteristic, complemented, Sylow
164C2×C4⋊Q8—791no—
165C2×#124—1581no—
166C2×C4×D4C2×D7911yescharacteristic
167C2×#99—1581no—
168#407—21no—
169#397—792no—
170#397—792no—
171#391—1581no—
172#380—1581no—
173#380—1581no—
174C4×C4⋊C4—791no—
175C4×C22⋊C4—1581no—
176#342—792no—
177#324—21no—
178#323C2×D7912yes—
179C22×Q8—1581no—
180C22×D4D31611yescharacteristic
181C23×C4D31611yescharacteristic
182C23×C4C22⋊D7911yescharacteristic
183C2×M16—41no—
184C4⋊Q8—3161no—
185C4⋊Q8—3161no—
186#124—3164no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—792no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4C22⋊D7911yescharacteristic
193C2×C22⋊C4—1582no—
194C2×C22⋊C4—1582no—
195C2×C22⋊C4—1582no—
196C2×C22⋊C4C4×D7911yescharacteristic
197C2×C22⋊C4C22×D7911yescharacteristic
198C2×C42—3161no—
199C2×C42—1581no—
200C2×C42C4×D7911yescharacteristic
201#99—1584no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204#95—3161no—
205#95—792no—
206#95—1582no—
207#95—1582no—
208C24(C2×D79)⋊C411yescharacteristic
209C24C2×D31611yescharacteristic
210C2×Q8—3162no—
211C2×Q8—3162no—
212C2×Q8—3162no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—3162no—
223C22×C4—1582no—
224C22×C4—1582no—
225C22×C4(C2×D79)⋊C411yescharacteristic
226C22×C4(C2×D79)⋊C411yescharacteristic
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4C2×C4×D7911yescharacteristic
229C22×C4C2×C22⋊D7911yescharacteristic
230M16—82no—
231C2×C8—42no—
232C4⋊C4—41no—
233C4⋊C4—3162no—
234C4⋊C4—3162no—
235C4⋊C4—3161no—
236C4⋊C4—3161no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—3164no—
240C22⋊C4—3162no—
241C22⋊C4—3164no—
242C22⋊C4—1584no—
243C42—21no—
244C42—21no—
245C42—3162no—
246C42—6322no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D79)⋊C411yescharacteristic
257C23#1457918912yes—
258Q8—6324no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—3164no—
274C2×C4—3162no—
275C2×C4—3162no—
276C2×C4—3162no—
277C2×C4—3162no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—6322no—
281C2×C4—6322no—
282C2×C4#1457907511yescharacteristic
283C2×C4#1457907511yescharacteristic
284C2×C4#1457908712yes—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#1457907511yescharacteristic
299C22#1909856911yescharacteristic
300C22C2×#1457908711yescharacteristic
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—6322no—
307C4—6324no—
308C4—6322no—
309C4#1909835712yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2376191711yescharacteristic
315C2#2376505511yescharacteristic
316C2C2×#1909835711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1465C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1465D79
hypocenterC79#11777
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #23761401C2semidirect
5. C2×#14579161C4semidirect
8. #19098488C4semidirect
13. C4×C4⋊Dic79C4semidirect
163. C79#11777semidirect

Maximal subgroups

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

subgroupindex
#237683922
C4×#145790762
#237614012
C79×#14652
#1177779