Finite Group Databasebeta

← #34201242

Subgroups — #34201242

Order 20224.

464 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764753C211yesmaximal, characteristic, complemented
2#23762390C211yesmaximal, characteristic, complemented
3#23762322C211yesmaximal, characteristic
4#23761400C211yesmaximal, characteristic, complemented
5#23761397C211yesmaximal, characteristic
6#23761348C211yesmaximal, characteristic, complemented
7C79×#2693C211yesmaximal, characteristic
8#19099189C2211yescharacteristic
9#19098893—21no—
10#19098714—21no—
11#19098436—21no—
12#19098330C2211yescharacteristic
13#19098329—21no—
14#19098299—21no—
15#19098299—21no—
16#19098287C2211yescharacteristic
17#19098287C2211yescharacteristic
18C2×#123×C79—21no—
19#400×C79—21no—
20#397×C79C2211yescharacteristic
21#396×C79—21no—
22#392×C79C2211yescharacteristic
23#386×C79—21no—
24#386×C79—21no—
25#385×C79C2211yescharacteristic
26#380×C79—21no—
27C2×#11929612—41no—
28#14579140—21no—
29#14579140—21no—
30#14579138—41no—
31#14579138—41no—
32C2×Dic79⋊C4—41no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35#14579092—41no—
36#14579084—41no—
37#14579082—41no—
38C22⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—21no—
40C79⋊#95—41no—
41C22×D4×C79—21no—
42C22×D4×C79D411yescharacteristic
43C23×C4×C79—21no—
44C23×C4×C79D411yescharacteristic
45#123×C79—42no—
46#123×C79—42no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79D411yescharacteristic
57C2×C22⋊C4×C79C2311yescharacteristic
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—41no—
69#11929612—82no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76Dic79⋊C4—82no—
77C22⋊Dic79—22no—
78C22⋊Dic79—41no—
79C22⋊Dic79—41no—
80C22⋊Dic79—41no—
81C24×C79—21no—
82C24×C79C2×D411yescharacteristic
83C2×D4×C79—41no—
84C2×D4×C79—42no—
85C2×D4×C79—42no—
86C2×D4×C79—41no—
87C2×D4×C79—41no—
88C2×D4×C79—81no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—42no—
95C22×C4×C79—41no—
96C22×C4×C79—42no—
97C22×C4×C79—81no—
98C22×C4×C79—81no—
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101C4⋊C4×C79—42no—
102C4⋊C4×C79—42no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—81no—
105C22⋊C4×C79—21no—
106C22⋊C4×C79—41no—
107C22⋊C4×C79—42no—
108C22⋊C4×C79—41no—
109C22⋊C4×C79—41no—
110C22⋊C4×C79—81no—
111C22⋊C4×C79—81no—
112C22⋊C4×C79—81no—
113C22⋊C4×C79SD1612yescomplemented
114C23×C79—21no—
115C23×C79—21no—
116C23×C79—22no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79—41no—
125C23×C79#12011yescharacteristic
126C23×C79C2×SD1611yescharacteristic
127C23×C79AGL(1,ℤ/8)11yescharacteristic
128D4×C79—42no—
129D4×C79—81no—
130D4×C79—82no—
131D4×C79—82no—
132C2×C4×C79—22no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—84no—
153C2×C4×C79—81no—
154C2×Dic79—42no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—81no—
158C2×Dic79—81no—
159C2×Dic79—81no—
160C2×Dic79—81no—
161Dic158—81no—
162Dic158—81no—
163C79⋊C8—81no—
164C79⋊C8—81no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—41no—
170C22×C79—41no—
171C22×C79—42no—
172C22×C79—42no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79—42no—
176C22×C79—81no—
177C22×C79—81no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79#45011yescharacteristic
182C22×C79#45611yescharacteristic
183C22×C79B(4,𝔽2)11yescharacteristic
184C4×C79—42no—
185C4×C79—41no—
186C4×C79—41no—
187C4×C79—81no—
188C4×C79—81no—
189C4×C79—81no—
190C4×C79—81no—
191Dic79—82no—
192Dic79—81no—
193Dic79—81no—
194#12243—791nomaximal, Sylow
195C2×C79—21no—
196C2×C79—21no—
197C2×C79—41no—
198C2×C79—41no—
199C2×C79—82no—
200C2×C79#160411yescharacteristic
201C2×C79#219711yescharacteristic
202C2×C79#220011yescharacteristic
203#2693D7911yescharacteristic
204#2062—791no—
205#1608—791no—
206#1596—791no—
207#1344—791no—
208#1343—791no—
209#1326—791no—
210C79#1224311yescharacteristic, complemented, Sylow
211#543—791no—
212C2×#123—21no—
213#483—1581no—
214#450—1581no—
215#400—21no—
216#397C2×D7911yescharacteristic
217#396—21no—
218#392C2×D7911yescharacteristic
219#386—21no—
220#386—21no—
221#385—1581no—
222#385C2×D7911yescharacteristic
223#380—21no—
224#342—791no—
225#342—1581no—
226#328—1581no—
227#328—1581no—
228#323—791no—
229#323—791no—
230C22×D4—21no—
231C22×D4C22⋊D7911yescharacteristic
232C23×C4—21no—
233C23×C4C22⋊D7911yescharacteristic
234C2×SD16—3161no—
235#123—42no—
236#123—42no—
237C22⋊Q8—3161no—
238C22⋊Q8—3161no—
239C22⋊Q8—1581no—
240C22⋊Q8—1581no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—3161no—
245C2×C4⋊C4C22⋊D7911yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—1581no—
252C2×C22⋊C4—1581no—
253C2×C22⋊C4C22⋊D7911yescharacteristic
254C2×C22⋊C4C22×D7911yescharacteristic
255#106—3161no—
256#103—3161no—
257#102—3161no—
258C22⋊C8—1581no—
259C22⋊C8—1581no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—3161no—
265#95C22⋊D7911yescharacteristic
266#95C22⋊D7911yescharacteristic
267C24—21no—
268C24C2×C22⋊D7911yescharacteristic
269C2×Q8—1581no—
270C2×Q8—1581no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—3161no—
288C22×C4—1581no—
289C22×C4—1581no—
290C22×C4C2×C22⋊D7911yescharacteristic
291C22×C4C2×C22⋊D7911yescharacteristic
292SD16—6322no—
293C2×C8—3161no—
294C2×C8—3161no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C4⋊C4—6322no—
302C4⋊C4—1581no—
303C4⋊C4—1581no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—3161no—
313C22⋊C4—3161no—
314C22⋊C4—3161no—
315C22⋊C4—1582no—
316C22⋊C4#1192961212yescomplemented
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#1457913011yescharacteristic
329C23C2×#1192961211yescharacteristic
330C23#1457918711yescharacteristic
331Q8—6321no—
332Q8—6321no—
333D4—42no—
334D4—81no—
335D4—82no—
336D4—82no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—84no—
358C2×C4—81no—
359C2×C4—3162no—
360C2×C4—3161no—
361C2×C4—3161no—
362C2×C4—6321no—
363C2×C4—6321no—
364C2×C4—6321no—
365C2×C4—6321no—
366C8—6321no—
367C8—6321no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#1909871411yescharacteristic
385C22#1909874611yescharacteristic
386C22#1909875011yescharacteristic
387C4—42no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—6322no—
395C4—6321no—
396C4—6321no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—82no—
402C2#2376236911yescharacteristic
403C2#2376539011yescharacteristic
404C2#2376540711yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2693C2
socleC22×C79B(4,𝔽2)
prime radical#392×C79C22
hypercenter#2693D79
hypocenterC79#12243
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2693

Complements

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

NH ≅ G/Ntype
1. #23764753C2semidirect
2. #23762390C2semidirect
4. #23761400C2semidirect
6. #23761348C2semidirect
113. C22⋊C4×C79SD16semidirect
210. C79#12243semidirect
316. C22⋊C4#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647532
#237623902
#237623222
#237614002
#237613972
#237613482
C79×#26932
#1224379