Finite Group Databasebeta

← #34269539

Subgroups — #34269539

Order 20224.

850 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 171 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#23774606C212yesmaximal, complemented
2#23773786C211yesmaximal, characteristic
3#23762734C218yesmaximal, complemented
4#23762024C211yesmaximal, characteristic
5#23762002C212yesmaximal, complemented
6C2×#563×C79C211yesmaximal, characteristic
7#19099120C2211yescharacteristic
8#19098973C2214yescomplemented
9#19098933C2214yescomplemented
10#19098922C2214yes—
11#19098917—22no—
12#19098898—22no—
13#19098893—22no—
14#19098867—24no—
15#19098620C2212yes—
16#19098572—21no—
17#19098568—21no—
18C4×#11929608C2211yescharacteristic
19#19098305C2218yescomplemented
20#19098303C2214yes—
21#563×C79—24no—
22#563×C79—22no—
23#563×C79C2214yes—
24C2×C4⋊Q8×C79—21no—
25C2×#126×C79—22no—
26C2×#125×C79C2212yes—
27C2×#124×C79C2211yescharacteristic
28C2×#123×C79—21no—
29C2×C22⋊Q8×C79—22no—
30C2×#11929608—21no—
31#14579161C2312yes—
32#14579153—21no—
33#14579140—22no—
34C4×Dic158—22no—
35#14579120—21no—
36#14579094—24no—
37#14579092—24no—
38C4⋊(C79⋊C8)—22no—
39#14579085—24no—
40#14579084—24no—
41#14579084—24no—
42#14579082—24no—
43C22⋊(C79⋊C8)—22no—
44#14579072C2312yes—
45#14579072C2314yes—
46C4×C79⋊C8—21no—
47C22×Q8×C79—21no—
48C22×Q8×C79D411yescharacteristic
49C22×D4×C79D411yescharacteristic
50C23×C4×C79—21no—
51C4⋊Q8×C79—24no—
52C4⋊Q8×C79—24no—
53#126×C79—28no—
54#126×C79—28no—
55#125×C79—22no—
56#125×C79—22no—
57#125×C79D414yes—
58#125×C79C2314yes—
59#124×C79—22no—
60#124×C79D412yes—
61#124×C79C2312yes—
62#123×C79—24no—
63#123×C79—24no—
64C22⋊Q8×C79—28no—
65C22⋊Q8×C79—28no—
66C2×C4⋊C4×C79—22no—
67C2×C4⋊C4×C79—22no—
68C2×C4⋊C4×C79—22no—
69C2×C4⋊C4×C79D412yes—
70C2×C4⋊C4×C79D412yes—
71C2×C22⋊C4×C79—22no—
72C2×C22⋊C4×C79—21no—
73C2×C22⋊C4×C79—22no—
74C2×C42×C79—21no—
75C2×C42×C79C2311yescharacteristic
76C2×Dic158—22no—
77#11929608—42no—
78C2×C79⋊C8—22no—
79C2×C79⋊C8—24no—
80C4⋊Dic79—22no—
81C4⋊Dic79—22no—
82Dic79⋊C4—42no—
83C22⋊Dic79—41no—
84C4×Dic79—41no—
85C24×C79—21no—
86C2×Q8×C79—24no—
87C2×Q8×C79—21no—
88C2×Q8×C79—42no—
89C2×Q8×C79—82no—
90C2×Q8×C79C2×D412yes—
91C2×D4×C79—21no—
92C2×D4×C79—42no—
93C2×D4×C79C2×D412yes—
94C22×C4×C79—22no—
95C22×C4×C79—21no—
96C22×C4×C79—22no—
97C22×C4×C79—24no—
98C22×C4×C79—82no—
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D412yes—
101C4⋊C4×C79—28no—
102C4⋊C4×C79—24no—
103C4⋊C4×C79—24no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—22no—
106C4⋊C4×C79—24no—
107C4⋊C4×C79—24no—
108C4⋊C4×C79C2×D414yes—
109C4⋊C4×C79C2×D414yes—
110C22⋊C4×C79—28no—
111C22⋊C4×C79—24no—
112C22⋊C4×C79—24no—
113C22⋊C4×C79—24no—
114C42×C79—24no—
115C42×C79C2×D412yes—
116C42×C79C2×D411yescharacteristic
117C42×C79C2411yescharacteristic
118C23×C79—22no—
119C23×C79—44no—
120C23×C79—42no—
121C23×C79#12011yescharacteristic
122Q8×C79—42no—
123Q8×C79—81no—
124Q8×C79—84no—
125D4×C79—42no—
126D4×C79—81no—
127C2×C4×C79—22no—
128C2×C4×C79—24no—
129C2×C4×C79—22no—
130C2×C4×C79—24no—
131C2×C4×C79—42no—
132C2×C4×C79—44no—
133C2×C4×C79—44no—
134C2×C4×C79—42no—
135C2×C4×C79—84no—
136C2×C4×C79#12011yescharacteristic
137C2×C4×C79#12012yes—
138C2×C4×C79C22×D411yescharacteristic
139C2×C4×C79C22×D412yes—
140C2×Dic79—42no—
141Dic158—44no—
142C79⋊C8—44no—
143C79⋊C8—42no—
144C22×C79—22no—
145C22×C79—42no—
146C22×C79—44no—
147C22×C79—84no—
148C22×C79#45512yes—
149C22×C79C2×#12011yescharacteristic
150C4×C79—21no—
151C4×C79—21no—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—44no—
155C4×C79—42no—
156C4×C79—44no—
157Dic79—82no—
158#18999—791nomaximal, Sylow
159C2×C79—22no—
160C2×C79—82no—
161C2×C79C2×#45511yescharacteristic
162C2×C79#302012yes—
163C2×#563D7911yescharacteristic
164#3236—792no—
165#3140—791no—
166#1664—798no—
167#1533—791no—
168#1528—792no—
169C79#1899911yescharacteristic, complemented, Sylow
170#563—24no—
171#563—22no—
172#563C2×D7914yes—
173#533—791no—
174C2×C4⋊Q8—21no—
175C2×#126—22no—
176C2×#125C2×D7912yes—
177C2×#124C2×D7911yescharacteristic
178C2×#123—21no—
179C2×C22⋊Q8—22no—
180#501—794no—
181#491—794no—
182#489—794no—
183#488—1582no—
184#484—1582no—
185#483—1582no—
186#479—1584no—
187#432—792no—
188#419—1581no—
189#418—1581no—
190C4×M16—791no—
191#330—798no—
192#329—794no—
193C22×Q8—21no—
194C22×Q8C22⋊D7911yescharacteristic
195C22×D4C22⋊D7911yescharacteristic
196C23×C4—21no—
197C2×M16—1581no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—792no—
201#126—28no—
202#126—28no—
203#125—22no—
204#125—22no—
205#125—1581no—
206#125C22⋊D7914yes—
207#125C22×D7914yes—
208#124—22no—
209#124C22⋊D7912yes—
210#124C22×D7912yes—
211#123—24no—
212#123—24no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C22⋊Q8—1582no—
216C4×Q8—1582no—
217#117—1581no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊D7912yes—
222C2×C4⋊C4C22⋊D7912yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227C2×C42C22×D7911yescharacteristic
228#107—1584no—
229#106—1584no—
230C4⋊C8—1582no—
231#103—1584no—
232#103—1584no—
233#103—1584no—
234#102—1584no—
235C22⋊C8—1582no—
236#97—794no—
237#97—792no—
238C4×C8—1581no—
239C24—21no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—1582no—
245C2×Q8C2×C22⋊D7912yes—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C2×C22⋊D7912yes—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—82no—
254C22×C4C2×C22⋊D7911yescharacteristic
255C22×C4C2×C22⋊D7912yes—
256M16—3162no—
257C2×C8—1582no—
258C2×C8—1584no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—3162no—
267C4⋊C4—1582no—
268C4⋊C4—1582no—
269C4⋊C4C2×C22⋊D7914yes—
270C4⋊C4C2×C22⋊D7914yes—
271C22⋊C4—28no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—3161no—
276C42—24no—
277C42—3161no—
278C42C2×C22⋊D7911yescharacteristic
279C42C2×C22⋊D7912yes—
280C42C23×D7911yescharacteristic
281C23—22no—
282C23—44no—
283C23—42no—
284C23#1457913011yescharacteristic
285Q8—42no—
286Q8—81no—
287Q8—84no—
288Q8—3164no—
289D4—42no—
290D4—81no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—3162no—
301C2×C4#1457913011yescharacteristic
302C2×C4#1457913012yes—
303C2×C4C22×C22⋊D7912yes—
304C2×C4C22×C22⋊D7911yescharacteristic
305C8—3162no—
306C8—3164no—
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#1909874112yes—
312C22C2×#1457913011yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—6322no—
321C2—22no—
322C2—82no—
323C2C2×#1909874111yescharacteristic
324C2#2377276412yes—
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#563×C79C2
socleC22×C79C2×#120
prime radicalC22×D4×C79D4
hypercenterC2×#563D79
hypocenterC79#18999
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #23774606C2semidirect
3. #23762734C2semidirect
5. #23762002C2semidirect
8. #19098973C22semidirect
9. #19098933C22semidirect
19. #19098305C22semidirect
169. C79#18999semidirect

Maximal subgroups

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

subgroupindex
#237746062
#237737862
#237627342
#237620242
#237620022
C2×#563×C792
#1899979