Finite Group Databasebeta

← #34192124

Subgroups — #34192124

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098854C211yesmaximal, characteristic
2#23764361C211yesmaximal, characteristic, complemented
3#23764194C211yesmaximal, characteristic
4#23763401C211yesmaximal, characteristic
5C2×#19098311C211yesmaximal, characteristic, complemented
6#23761284C211yesmaximal, characteristic, complemented
7C2×#335×C79C211yesmaximal, characteristic, complemented
8C2×#14579161C2211yescharacteristic
9C2×C4×D316C2211yescharacteristic
10#19098854—22no—
11#19098854—22no—
12C2×#14579093—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579083—21no—
15C2×#14579083—21no—
16C2×(C2×D79)⋊C8—21no—
17C2×(C2×D79)⋊C8—21no—
18#19098433C2211yescharacteristic
19#19098326—21no—
20#19098326—21no—
21#19098326—21no—
22#19098311—22no—
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×C4×C8×C79C2211yescharacteristic
26#335×C79—22no—
27C22×Dic158—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C22×C79⋊C8—21no—
31#14579161—21no—
32#14579161—21no—
33#14579161—21no—
34#14579161D412yes—
35C4×D316—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79—21no—
44C2×C4⋊Dic79D411yescharacteristic
45C2×(C2×D79)⋊C4—21no—
46#14579093—24no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49#14579083—24no—
50#14579083—42no—
51(C2×D79)⋊C8—42no—
52(C2×D79)⋊C8—42no—
53C22×C8×C79—21no—
54C22×C8×C79—21no—
55C22×C8×C79D411yescharacteristic
56C22×C8×C79D411yescharacteristic
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79—21no—
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C4⋊C8×C79D412yes—
62C4×C8×C79—21no—
63C4×C8×C79—21no—
64C23×D79—21no—
65C2×Dic158—22no—
66C2×Dic158—41no—
67C2×Dic158—81no—
68C2×Dic158—81no—
69C2×D316—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316Q1612yes—
75C22×Dic79—21no—
76C22×Dic79—41no—
77C22×Dic79—41no—
78C2×C4×D79—41no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C4⋊Dic79—22no—
88C4⋊Dic79—21no—
89C4⋊Dic79—22no—
90C4⋊Dic79—21no—
91C4⋊Dic79—41no—
92C4⋊Dic79—41no—
93C4⋊Dic79—41no—
94C4⋊Dic79—41no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—81no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79#4111yescharacteristic
103C2×C8×C79—22no—
104C2×C8×C79—21no—
105C2×C8×C79—21no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—21no—
109C2×C8×C79—41no—
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C2×C8×C79—41no—
113C2×C8×C79—41no—
114C2×C8×C79SD1612yes—
115C42×C79C2×D411yescharacteristic
116C42×C79C2×D411yescharacteristic
117C42×C79#4111yescharacteristic
118C42×C79#4111yescharacteristic
119C23×C79#12111yescharacteristic
120C22×D79—22no—
121C22×D79—41no—
122C22×D79—41no—
123C22×D79—41no—
124C22×D79—41no—
125C22×D79—41no—
126C22×D79—41no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79#12011yescharacteristic
130C2×C4×C79#12111yescharacteristic
131C2×C4×C79#12111yescharacteristic
132C2×C4×C79#12311yescharacteristic
133C2×C4×C79#12311yescharacteristic
134C2×C4×C79#12411yescharacteristic
135C2×C4×C79C2×SD1611yescharacteristic
136C2×C4×C79C2×Q1611yescharacteristic
137C2×C4×C79#13511yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C79#13711yescharacteristic
142C2×C4×C79#13711yescharacteristic
143C2×Dic79—41no—
144C2×Dic79—42no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—81no—
148C2×Dic79—81no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151C2×Dic79—81no—
152D316—22no—
153D316—22no—
154D316—41no—
155D316—41no—
156C4×D79—81no—
157C4×D79—81no—
158C4×D79—81no—
159C4×D79—81no—
160Dic158—82no—
161Dic158—82no—
162C8×C79—22no—
163C8×C79—42no—
164C8×C79—41no—
165C8×C79—81no—
166C79⋊C8—81no—
167C22×C79#39411yescharacteristic
168C22×C79#46111yescharacteristic
169C22×C79#46311yescharacteristic
170C22×C79#46711yescharacteristic
171C22×C79#46811yescharacteristic
172C22×C79#47111yescharacteristic
173C22×C79#47211yescharacteristic
174C2×D79—42no—
175C2×D79—42no—
176C2×D79—42no—
177C2×D79—81no—
178C2×D79—81no—
179C2×D79—81no—
180C2×D79—81no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79#44911yescharacteristic
184C4×C79#45111yescharacteristic
185C4×C79#46011yescharacteristic
186C4×C79#46411yescharacteristic
187C4×C79#46511yescharacteristic
188C4×C79#46811yescharacteristic
189C4×C79#48411yescharacteristic
190C4×C79#48511yescharacteristic
191Dic79—82no—
192Dic79—81no—
193Dic79—161no—
194#10481—791nomaximal, Sylow
195C2×C79#167511yescharacteristic
196C2×C79#168311yescharacteristic
197C2×C79#169111yescharacteristic
198C2×C79#169711yescharacteristic
199C2×C79#203611yescharacteristic
200C2×C79#205111yescharacteristic
201C2×C79#205511yescharacteristic
202D79—82no—
203C2×#476—791no—
204#1990—791no—
205#1954—791no—
206#1802—791no—
207C2×#335D7911yescharacteristic, complemented
208C2×#333—791no—
209#1296—791no—
210C79#1048111yescharacteristic, complemented, Sylow
211C2×C4⋊Q8—791no—
212C2×C4×D4—791no—
213#476—1582no—
214#476—1582no—
215C2×#106—1581no—
216C2×C4⋊C8—791no—
217C2×C4⋊C8C2×D7911yescharacteristic
218C2×C4⋊C8C2×D7911yescharacteristic
219C2×#102—1581no—
220C2×#102—1581no—
221C2×C22⋊C8—1581no—
222C2×C22⋊C8—1581no—
223C2×C4×C8C2×D7911yescharacteristic
224#384—791no—
225#340—1581no—
226#340—1581no—
227#340—1581no—
228#335—22no—
229#333—1582no—
230C22×Q8—1581no—
231C22×D4—791no—
232C23×C4—1581no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—1581no—
236C22×C8D31611yescharacteristic
237C22×C8D31611yescharacteristic
238C4⋊Q8—792no—
239C4⋊Q8—1581no—
240C4⋊Q8—1581no—
241C4⋊Q8—1581no—
242C4×D4—1581no—
243C4×D4—1581no—
244C4×D4—1581no—
245C4×D4—1581no—
246C2×C4⋊C4—791no—
247C2×C4⋊C4—1581no—
248C2×C4⋊C4—1581no—
249C2×C4⋊C4—1581no—
250C2×C4⋊C4—1581no—
251C2×C4⋊C4—1581no—
252C2×C22⋊C4—1581no—
253C2×C42C22×D7911yescharacteristic
254#106—1584no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—1581no—
259C4⋊C8—1581no—
260C4⋊C8C22⋊D7912yes—
261#102—3162no—
262#102—1584no—
263C22⋊C8—3162no—
264C22⋊C8—3162no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1581no—
268C2×Q8—3161no—
269C2×Q8—6321no—
270C2×Q8—6321no—
271C2×Q8—1582no—
272C2×D4—792no—
273C2×D4—1581no—
274C2×D4—1581no—
275C2×D4—1581no—
276C2×D4—1581no—
277C2×D4—1581no—
278C22×C4—3161no—
279C22×C4—3161no—
280C22×C4—3161no—
281C22×C4—3161no—
282C22×C4—3161no—
283C22×C4—3161no—
284C22×C4—3161no—
285C22×C4—3161no—
286C22×C4—1581no—
287C22×C4D4×D7911yescharacteristic
288C22×C4C2×D31611yescharacteristic
289C22×C4#1192962511yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—3161no—
302C2×C8—3161no—
303C2×C8—3161no—
304C2×C8#1192961312yes—
305C4⋊C4—3161no—
306C4⋊C4—3161no—
307C4⋊C4—3161no—
308C4⋊C4—3161no—
309C4⋊C4—3161no—
310C4⋊C4—3161no—
311C4⋊C4—6321no—
312C4⋊C4—1582no—
313C4⋊C4—1581no—
314C4⋊C4—1582no—
315C4⋊C4—1581no—
316C22⋊C4—3161no—
317C22⋊C4—3161no—
318C42D4×D7911yescharacteristic
319C42C2×C22⋊D7911yescharacteristic
320C42#1192962411yescharacteristic
321C42#1192962511yescharacteristic
322C23—3161no—
323C23—3161no—
324C23—3161no—
325C23—3161no—
326C23—3161no—
327C23—3161no—
328C23—1582no—
329C23#1457913611yescharacteristic
330Q8—6322no—
331Q8—6322no—
332D4—3161no—
333D4—3161no—
334D4—1582no—
335D4—1582no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3161no—
339C2×C4—3162no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—6321no—
343C2×C4—6321no—
344C2×C4—6321no—
345C2×C4—6321no—
346C2×C4—6321no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4—6321no—
350C2×C4—6321no—
351C2×C4#1457912911yescharacteristic
352C2×C4#1457913511yescharacteristic
353C2×C4#1457913611yescharacteristic
354C2×C4#1457914611yescharacteristic
355C2×C4#1457914611yescharacteristic
356C2×C4#1457915011yescharacteristic
357C2×C4C2×#1192961311yescharacteristic
358C2×C4Q16×D7911yescharacteristic
359C2×C4#1457918011yescharacteristic
360C2×C4#1457918311yescharacteristic
361C2×C4#1457918511yescharacteristic
362C2×C4#1457918911yescharacteristic
363C2×C4#1457919311yescharacteristic
364C2×C4#1457919411yescharacteristic
365C8—22no—
366C8—42no—
367C8—41no—
368C8—81no—
369C8—6321no—
370C22—3162no—
371C22—3162no—
372C22—3162no—
373C22—6321no—
374C22—6321no—
375C22—6321no—
376C22—6321no—
377C22#1909848011yescharacteristic
378C22#1909876911yescharacteristic
379C22#1909877911yescharacteristic
380C22#1909880511yescharacteristic
381C22#1909881011yescharacteristic
382C22#1909882711yescharacteristic
383C22#1909883411yescharacteristic
384C4—21no—
385C4—21no—
386C4—6322no—
387C4—6321no—
388C4—12641no—
389C4#1909870511yescharacteristic
390C4#1909871711yescharacteristic
391C4#1909876411yescharacteristic
392C4#1909878611yescharacteristic
393C4#1909879211yescharacteristic
394C4#1909880911yescharacteristic
395C4#1909890011yescharacteristic
396C4#1909890611yescharacteristic
397C2—6322no—
398C2#2376280011yescharacteristic
399C2#2376285211yescharacteristic
400C2#2376290011yescharacteristic
401C2#2376293411yescharacteristic
402C2#2376460511yescharacteristic
403C2#2376469411yescharacteristic
404C2#2376471811yescharacteristic
subgroupG/H
centerC23#14579136
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#335×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#335D79
hypocenterC79#10481
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #23764361C2semidirect
5. C2×#19098311C2semidirect
6. #23761284C2semidirect
7. C2×#335×C79C2semidirect
207. C2×#335D79semidirect
210. C79#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#190988542
#237643612
#237641942
#237634012
C2×#190983112
#237612842
C2×#335×C792
#1048179