Finite Group Databasebeta

← #34203412

Subgroups — #34203412

Order 20224.

548 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 123 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#23766065C211yesmaximal, characteristic
2C2×#19098352C212yesmaximal
3C4×#14579078C211yesmaximal, characteristic
4#23761517C212yesmaximal
5C4×#99×C79C211yesmaximal, characteristic
6C2×#14579142C2211yescharacteristic
7C22×Dic79⋊C4C411yescharacteristic, complemented
8C2×#14579078C2212yes—
9C4×#11929608—21no—
10#19098458—21no—
11#19098457—21no—
12#19098447—21no—
13#19098443C2211yescharacteristic
14#19098436—21no—
15#19098435C411yescharacteristic, complemented
16#19098424C411yescharacteristic, complemented
17C4×C22⋊Dic79C411yescharacteristic, complemented
18#19098352C418yescomplemented
19#19098332—21no—
20C2×C4×D4×C79C2211yescharacteristic
21C2×#99×C79C2212yes—
22C4×C22⋊C4×C79—21no—
23#342×C79—21no—
24C23×Dic79C2×C411yescharacteristic
25C23×Dic79C2×C411yescharacteristic
26C2×#11929608—22no—
27#14579142—22no—
28#14579142C2×C414yes—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—41no—
34C2×Dic79⋊C4—41no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79C2×C411yescharacteristic
38C2×C22⋊Dic79C2×C411yescharacteristic
39C2×C4×Dic79—21no—
40#14579078—22no—
41#14579078C2×C414yes—
42#14579072—41no—
43C4×C79⋊C8—41no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C22×D4×C79C2311yescharacteristic
50C23×C4×C79C2×C411yescharacteristic
51C23×C4×C79D411yescharacteristic
52C4×D4×C79—41no—
53C4×D4×C79—41no—
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79—22no—
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79—41no—
59C2×C42×C79D411yescharacteristic
60#99×C79—22no—
61#99×C79C2×C414yes—
62#95×C79—41no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—41no—
69C22×Dic79—41no—
70C22×Dic79—41no—
71C22×Dic79—81no—
72C22×Dic79C4212yescomplemented
73#11929608—24no—
74#11929608—42no—
75C2×C79⋊C8—42no—
76Dic79⋊C4—24no—
77Dic79⋊C4—42no—
78Dic79⋊C4—41no—
79Dic79⋊C4—41no—
80Dic79⋊C4—81no—
81C22⋊Dic79—21no—
82C22⋊Dic79—24no—
83C22⋊Dic79—41no—
84C22⋊Dic79—81no—
85C22⋊Dic79C4212yescomplemented
86C4×Dic79—81no—
87C24×C79C22×C411yescharacteristic
88C24×C79C2×D411yescharacteristic
89C2×D4×C79—41no—
90C2×D4×C79—41no—
91C2×D4×C79C22×C412yes—
92C2×D4×C79#4112yes—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—42no—
98C22×C4×C79—81no—
99C22×C4×C79—81no—
100C22×C4×C79C22⋊C411yescharacteristic
101C22×C4×C79C22⋊C411yescharacteristic
102C22×C4×C79C22⋊C411yescharacteristic
103C22×C4×C79C22⋊C411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C4⋊C4×C79—41no—
106C22⋊C4×C79—24no—
107C22⋊C4×C79—41no—
108C22⋊C4×C79—41no—
109C22⋊C4×C79—42no—
110C42×C79—21no—
111C42×C79—21no—
112C42×C79—82no—
113C23×C79—21no—
114C23×C79—21no—
115C23×C79—21no—
116C23×C79—21no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79C2×C4211yescharacteristic
122C23×C79C2×C22⋊C411yescharacteristic
123C23×C79#11711yescharacteristic
124C23×C79C4×D412yes—
125D4×C79—42no—
126D4×C79—42no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—41no—
133C2×C4×C79—42no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—44no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79—82no—
142C2×C4×C79—82no—
143C2×C4×C79#9911yescharacteristic
144C2×C4×C79#9911yescharacteristic
145C2×C4×C79C4×D412yes—
146C2×Dic79—22no—
147C2×Dic79—22no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—42no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—42no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—82no—
161C79⋊C8—44no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—22no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—41no—
168C22×C79—41no—
169C22×C79—42no—
170C22×C79—42no—
171C22×C79—41no—
172C22×C79—41no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79—81no—
176C22×C79C4×C22⋊C411yescharacteristic
177C22×C79C2×#9911yescharacteristic
178C22×C79#41011yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—22no—
182C4×C79—41no—
183C4×C79—41no—
184C4×C79—41no—
185C4×C79—84no—
186Dic79—42no—
187Dic79—42no—
188Dic79—81no—
189Dic79—81no—
190#12621—791nomaximal, Sylow
191C2×C79—21no—
192C2×C79—21no—
193C2×C79—42no—
194C2×C79—41no—
195C2×C79—41no—
196C2×C79C4×#9911yescharacteristic
197C2×C79#212111yescharacteristic
198C2×C79#212311yescharacteristic
199#2322—791no—
200C2×#352—792no—
201C4×#100—791no—
202C4×#99D7911yescharacteristic
203#1390—792no—
204C79#1262111yescharacteristic, complemented, Sylow
205C2×#123—791no—
206C2×C4×D4C2×D7911yescharacteristic
207C22×C4⋊C4—791no—
208C2×#100—792no—
209C2×#99C2×D7912yes—
210C4×M16—1581no—
211#389—1581no—
212#388—1581no—
213#387—1581no—
214#386—791no—
215#385—791no—
216#385—1581no—
217#381—791no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—791no—
220#352—798no—
221#343—1581no—
222#342—21no—
223C22×D4C22×D7911yescharacteristic
224C23×C4—791no—
225C23×C4—791no—
226C23×C4C4×D7911yescharacteristic
227C23×C4C22⋊D7911yescharacteristic
228C2×M16—1582no—
229#123—794no—
230#123—1582no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—3161no—
234C2×C4⋊C4—3161no—
235C2×C4⋊C4—1581no—
236C2×C4⋊C4—1581no—
237C2×C4⋊C4—1581no—
238C2×C4⋊C4—1581no—
239C2×C4⋊C4D31611yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—791no—
242C2×C22⋊C4—791no—
243C2×C22⋊C4—1581no—
244C2×C22⋊C4—1581no—
245C2×C22⋊C4C4×D7911yescharacteristic
246C2×C22⋊C4C22⋊D7911yescharacteristic
247C2×C42—41no—
248C2×C42—1581no—
249C2×C42D31611yescharacteristic
250#100—794no—
251#100—1582no—
252#99—22no—
253#99C4×D7914yes—
254#97—3161no—
255C4×C8—3161no—
256#95—41no—
257#95—1581no—
258#95—1581no—
259#95—1581no—
260#95—1581no—
261#95—1581no—
262C24C2×C4×D7911yescharacteristic
263C24C2×C22⋊D7911yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C2×C4×D7912yes—
267C2×D4#1192962412yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—3161no—
276C22×C4—3161no—
277C22×C4—3161no—
278C22×C4—792no—
279C22×C4—6321no—
280C22×C4—1581no—
281C22×C4—1581no—
282C22×C4—1581no—
283C22×C4—1581no—
284C22×C4—1581no—
285C22×C4(C2×D79)⋊C411yescharacteristic
286C22×C4(C2×D79)⋊C411yescharacteristic
287C22×C4(C2×D79)⋊C411yescharacteristic
288C22×C4(C2×D79)⋊C411yescharacteristic
289C22×C4C2×D31611yescharacteristic
290M16—3162no—
291M16—1584no—
292C2×C8—3162no—
293C4⋊C4—41no—
294C4⋊C4—3162no—
295C4⋊C4—3161no—
296C4⋊C4—3161no—
297C4⋊C4—6321no—
298C4⋊C4—1584no—
299C22⋊C4—24no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—3161no—
304C22⋊C4—792no—
305C22⋊C4—6321no—
306C22⋊C4—1581no—
307C22⋊C4—1584no—
308C42—21no—
309C42—21no—
310C42—82no—
311C42—6321no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C42×D7911yescharacteristic
321C23C2×(C2×D79)⋊C411yescharacteristic
322C23#1457911811yescharacteristic
323C23C4×C22⋊D7912yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—3161no—
343C2×C4—3162no—
344C2×C4—3161no—
345C2×C4—3161no—
346C2×C4—3161no—
347C2×C4—3162no—
348C2×C4—3161no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—6322no—
355C2×C4—1582no—
356C2×C4—1582no—
357C2×C4#1457907511yescharacteristic
358C2×C4#1457907511yescharacteristic
359C2×C4C4×D31612yes—
360C8—3164no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C4×(C2×D79)⋊C411yescharacteristic
376C22C2×#1457907511yescharacteristic
377C22#1909853611yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—3162no—
386C4—3162no—
387C4—6321no—
388C4—6321no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#1457907511yescharacteristic
395C2#2376504411yescharacteristic
396C2#2376505311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC23×C79C2×C42
FrattiniC22×D4C22×D79
FittingC4×#99×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#99D79
hypocenterC79#12621
LCSG ▷ C23×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
7. C22×Dic79⋊C4C4semidirect
15. #19098435C4semidirect
16. #19098424C4semidirect
17. C4×C22⋊Dic79C4semidirect
18. #19098352C4semidirect
72. C22×Dic79C42semidirect
85. C22⋊Dic79C42semidirect
204. C79#12621semidirect

Maximal subgroups

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

subgroupindex
#237660652
C2×#190983522
C4×#145790782
#237615172
C4×#99×C792
#1262179