Finite Group Databasebeta

← #34206565

Subgroups — #34206565

Order 20224.

540 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 107 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#23767708C211yesmaximal, characteristic
2C2×#19098352C214yesmaximal
3#23763392C211yesmaximal, characteristic
4C4×#99×C79C211yesmaximal, characteristic
5C2×#14579142C2212yes—
6C22×C4⋊Dic79C411yescharacteristic, complemented
7#19098587—21no—
8C2×#14579078C2212yes—
9#19098493—21no—
10#19098488—21no—
11#19098461—21no—
12#19098424C412yescomplemented
13#19098423—21no—
14#19098423C411yescharacteristic, complemented
15#19098352—28no—
16#19098332—21no—
17C2×C4×D4×C79C2211yescharacteristic
18C2×#99×C79C2212yes—
19C4×C22⋊C4×C79—21no—
20#342×C79—21no—
21C23×Dic79C2×C412yes—
22C2×#11929608—22no—
23#14579142—24no—
24#14579142—24no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—41no—
29C2×C4⋊Dic79—41no—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79C2×C412yes—
33C4⋊(C79⋊C8)—41no—
34C4⋊(C79⋊C8)—41no—
35#14579078—24no—
36C79⋊#95—22no—
37C79⋊#95—21no—
38C79⋊#95—22no—
39C22×D4×C79C2311yescharacteristic
40C23×C4×C79C2×C411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C4×D4×C79—41no—
43C4×D4×C79—41no—
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79C2×C411yescharacteristic
47C2×C22⋊C4×C79D411yescharacteristic
48C2×C42×C79—41no—
49C2×C42×C79D411yescharacteristic
50#99×C79—22no—
51#99×C79D414yes—
52#95×C79—41no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C22×Dic79—24no—
56C22×Dic79—42no—
57C22×Dic79—42no—
58#11929608—42no—
59#11929608—42no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—42no—
62C4⋊Dic79—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67C22⋊Dic79—24no—
68C22⋊Dic79—44no—
69C24×C79C22×C411yescharacteristic
70C24×C79C2×D411yescharacteristic
71C2×D4×C79—41no—
72C2×D4×C79—41no—
73C2×D4×C79C2×D412yes—
74C2×D4×C79#4112yes—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C4⋊C4×C79—41no—
88C22⋊C4×C79—24no—
89C22⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C22⋊C4×C79—42no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—82no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79C4×D412yes—
105C23×C79#12112yes—
106D4×C79—42no—
107D4×C79—42no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—44no—
121C2×C4×C79—81no—
122C2×C4×C79—82no—
123C2×C4×C79—82no—
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#9911yescharacteristic
126C2×C4×C79#12411yescharacteristic
127C2×C4×C79#12711yescharacteristic
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—42no—
131C2×Dic79—42no—
132C2×Dic79—42no—
133C2×Dic79—42no—
134C2×Dic79—84no—
135C79⋊C8—82no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—22no—
139C22×C79—22no—
140C22×C79—42no—
141C22×C79—42no—
142C22×C79—42no—
143C22×C79—41no—
144C22×C79—41no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—81no—
148C22×C79#39011yescharacteristic
149C22×C79C2×#9911yescharacteristic
150C22×C79#41011yescharacteristic
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—84no—
158C4×C79#35212yescomplemented
159Dic79—82no—
160Dic79—82no—
161#13220—791nomaximal, Sylow
162C2×C79—21no—
163C2×C79—21no—
164C2×C79—42no—
165C2×C79—42no—
166C2×C79#196411yescharacteristic
167C2×C79C2×#35211yescharacteristic
168C2×C79#212311yescharacteristic
169#2486—791no—
170C2×#352—794no—
171#1800—791no—
172C4×#99D7911yescharacteristic
173C79#1322011yescharacteristic, complemented, Sylow
174C2×#123—792no—
175C2×C4×D4C2×D7911yescharacteristic
176C22×C4⋊C4—791no—
177#423—1581no—
178C2×#100—792no—
179C2×#99C2×D7912yes—
180#398—1581no—
181#397—1581no—
182#389—1581no—
183#381—792no—
184#380—791no—
185#380—1581no—
186C4×C22⋊C4—21no—
187#352—1588no—
188#343—1581no—
189#342—21no—
190C22×D4C22×D7911yescharacteristic
191C23×C4—792no—
192C23×C4C4×D7911yescharacteristic
193C23×C4C22⋊D7911yescharacteristic
194C2×M16—1582no—
195#123—1584no—
196#123—1584no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—3161no—
200C2×C4⋊C4—3161no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1582no—
205C2×C4⋊C4D31611yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—792no—
208C2×C22⋊C4—1582no—
209C2×C22⋊C4C4×D7911yescharacteristic
210C2×C22⋊C4C22⋊D7911yescharacteristic
211C2×C42—41no—
212C2×C42D31611yescharacteristic
213C4⋊C8—3161no—
214C4⋊C8—3161no—
215#100—1584no—
216#99—22no—
217#99D31614yes—
218#95—41no—
219#95—1581no—
220#95—1582no—
221#95—1582no—
222C24C2×C4×D7911yescharacteristic
223C24C2×C22⋊D7911yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×D31612yes—
227C2×D4#1192962412yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—3162no—
236C22×C4—3162no—
237C22×C4—1582no—
238C22×C4—1582no—
239C22×C4—1584no—
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4C2×D31611yescharacteristic
245M16—3162no—
246M16—3162no—
247C2×C8—3162no—
248C4⋊C4—41no—
249C4⋊C4—3162no—
250C4⋊C4—3162no—
251C4⋊C4—3161no—
252C4⋊C4—3161no—
253C4⋊C4—3162no—
254C4⋊C4—3162no—
255C22⋊C4—24no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—3164no—
260C22⋊C4—1584no—
261C42—21no—
262C42—21no—
263C42—82no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×(C2×D79)⋊C411yescharacteristic
273C23C4×D31612yes—
274C23#1457913512yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—6324no—
300C2×C4#1457907511yescharacteristic
301C2×C4#1457907511yescharacteristic
302C2×C4#1457914911yescharacteristic
303C2×C4#1457915811yescharacteristic
304C8—6322no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1909846511yescharacteristic
318C22C2×#1457907511yescharacteristic
319C22#1909853611yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—84no—
327C4—6322no—
328C4—6322no—
329C4#1909835212yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2376423611yescharacteristic
335C2C2×#1909835211yescharacteristic
336C2#2376505311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC4×#99×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#99D79
hypocenterC79#13220
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic79C4semidirect
12. #19098424C4semidirect
14. #19098423C4semidirect
158. C4×C79#352semidirect
173. C79#13220semidirect
329. C4#19098352semidirect

Maximal subgroups

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

subgroupindex
#237677082
C2×#190983522
#237633922
C4×#99×C792
#1322079