Finite Group Databasebeta

← #34192121

Subgroups — #34192121

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×#19098850C211yesmaximal, characteristic
2#23764194C211yesmaximal, characteristic
3#23763397C211yesmaximal, characteristic
4C2×#19098316C211yesmaximal, characteristic, complemented
5C2×#19098312C211yesmaximal, characteristic, complemented
6#23761283C211yesmaximal, characteristic, complemented
7C79×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9#19098850—22no—
10#19098850—22no—
11C2×#14579092—21no—
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C2×#14579081—21no—
16C2×#14579081—21no—
17C2×(C2×D79)⋊C8—21no—
18C2×(C2×D79)⋊C8—21no—
19#19098325—21no—
20#19098316—22no—
21#19098312—22no—
22C2×C4⋊Q8×C79C2211yescharacteristic
23C2×C4×C8×C79C2211yescharacteristic
24#384×C79C2211yescharacteristic
25#340×C79—21no—
26#340×C79—21no—
27C22×D316D411yescharacteristic
28C22×C4×D79—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31C22×C79⋊C8—21no—
32C4×D316—21no—
33C4×D316—21no—
34C4×D316—21no—
35C4×D316—21no—
36C2×C4⋊Dic79D411yescharacteristic
37C2×(C2×D79)⋊C4—21no—
38#14579092—24no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579081—24no—
46#14579081—42no—
47(C2×D79)⋊C8—42no—
48(C2×D79)⋊C8—42no—
49C22×Q8×C79—21no—
50C22×C8×C79D411yescharacteristic
51C22×C8×C79D411yescharacteristic
52C4⋊Q8×C79—21no—
53C4⋊Q8×C79—21no—
54C4⋊Q8×C79—21no—
55C4⋊Q8×C79D412yes—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C42×C79C2311yescharacteristic
62C4×C8×C79—21no—
63C4×C8×C79—21no—
64C23×D79—21no—
65C2×D316—21no—
66C2×D316—21no—
67C2×D316—21no—
68C2×D316—21no—
69C2×D316—21no—
70C2×D316Q1612yes—
71C22×Dic79—21no—
72C2×C4×D79—41no—
73C2×C4×D79—41no—
74C2×C4×D79—41no—
75C2×C4×D79—41no—
76C2×C4×D79—41no—
77C2×C4×D79—41no—
78C2×C79⋊C8—22no—
79C2×C79⋊C8—41no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89(C2×D79)⋊C4—41no—
90(C2×D79)⋊C4—41no—
91C2×Q8×C79—22no—
92C2×Q8×C79—41no—
93C2×Q8×C79—81no—
94C2×Q8×C79—81no—
95C22×C4×C79—41no—
96C22×C4×C79—41no—
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79#4111yescharacteristic
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79—21no—
104C2×C8×C79—21no—
105C2×C8×C79SD1612yes—
106C4⋊C4×C79—22no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—41no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—81no—
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—
127Q8×C79—82no—
128Q8×C79—82no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—42no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79#12011yescharacteristic
138C2×C4×C79#12111yescharacteristic
139C2×C4×C79#12111yescharacteristic
140C2×C4×C79#12311yescharacteristic
141C2×C4×C79#12311yescharacteristic
142C2×C4×C79#12411yescharacteristic
143C2×C4×C79C2×SD1611yescharacteristic
144C2×C4×C79C2×Q1611yescharacteristic
145C2×C4×C79#13511yescharacteristic
146C2×C4×C79#13511yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79#13711yescharacteristic
150C2×C4×C79#13711yescharacteristic
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154D316—22no—
155D316—22no—
156D316—41no—
157D316—41no—
158C4×D79—81no—
159C4×D79—81no—
160C4×D79—81no—
161C4×D79—81no—
162C8×C79—22no—
163C8×C79—41no—
164C79⋊C8—42no—
165C79⋊C8—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—82no—
184C4×C79—161no—
185C4×C79#44911yescharacteristic
186C4×C79#45111yescharacteristic
187C4×C79#46011yescharacteristic
188C4×C79#46411yescharacteristic
189C4×C79#46511yescharacteristic
190C4×C79#46811yescharacteristic
191C4×C79#48411yescharacteristic
192C4×C79#48511yescharacteristic
193Dic79—81no—
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#1990D7911yescharacteristic, complemented
205#1954—791no—
206#1802—791no—
207C2×#335—791no—
208C2×#333—791no—
209#1296—791no—
210C79#1048111yescharacteristic, complemented, Sylow
211C2×C4⋊Q8C2×D7911yescharacteristic
212C2×C4×D4—791no—
213#476—1582no—
214#476—1582no—
215C2×#106—1581no—
216C2×C4⋊C8—791no—
217C2×C4⋊C8—791no—
218C2×C4⋊C8—791no—
219C2×#102—1581no—
220C2×#102—1581no—
221C2×C22⋊C8—1581no—
222C2×C22⋊C8—1581no—
223C2×C4×C8C2×D7911yescharacteristic
224#384C2×D7911yescharacteristic
225#340—21no—
226#340—21no—
227#340—1581no—
228#335—1582no—
229#333—1582no—
230C22×Q8—21no—
231C22×D4—791no—
232C23×C4—1581no—
233C22×C8—1581no—
234C22×C8—1581no—
235C22×C8—1581no—
236C22×C8C22⋊D7911yescharacteristic
237C22×C8C22⋊D7911yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D31612yes—
242C4×D4—1581no—
243C4×D4—1581no—
244C4×D4—1581no—
245C4×D4—1581no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—791no—
252C2×C22⋊C4—1581no—
253C2×C42C22×D7911yescharacteristic
254#106—1584no—
255C4⋊C8—792no—
256C4⋊C8—1581no—
257C4⋊C8—1581no—
258C4⋊C8—1581no—
259C4⋊C8—1581no—
260C4⋊C8—1581no—
261#102—3162no—
262#102—1584no—
263C22⋊C8—3162no—
264C22⋊C8—3162no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1581no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—792no—
273C2×D4—1581no—
274C2×D4—1581no—
275C2×D4—1581no—
276C2×D4—1581no—
277C2×D4—1581no—
278C22×C4—41no—
279C22×C4—41no—
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×C22⋊D7911yescharacteristic
289C22×C4#1192962311yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—3161no—
296C2×C8—3161no—
297C2×C8—3161no—
298C2×C8—3161no—
299C2×C8—3161no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—3161no—
303C2×C8—1582no—
304C2×C8#1192961112yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—1581no—
315C4⋊C4—1581no—
316C22⋊C4—3161no—
317C22⋊C4—3161no—
318C42D4×D7911yescharacteristic
319C42C2×D31611yescharacteristic
320C42#1192962311yescharacteristic
321C42#1192962411yescharacteristic
322C23—3161no—
323C23—3161no—
324C23—3161no—
325C23—3161no—
326C23—3161no—
327C23—3161no—
328C23—1582no—
329C23#1457913211yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—3161no—
333D4—3161no—
334D4—1582no—
335D4—1582no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—3161no—
345C2×C4—3161no—
346C2×C4—3161no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4—6321no—
350C2×C4—6321no—
351C2×C4#1457913111yescharacteristic
352C2×C4#1457913311yescharacteristic
353C2×C4#1457913511yescharacteristic
354C2×C4#1457914311yescharacteristic
355C2×C4#1457914511yescharacteristic
356C2×C4#1457914711yescharacteristic
357C2×C4C2×#1192961111yescharacteristic
358C2×C4Q16×D7911yescharacteristic
359C2×C4#1457918111yescharacteristic
360C2×C4#1457918411yescharacteristic
361C2×C4#1457918611yescharacteristic
362C2×C4#1457918811yescharacteristic
363C2×C4#1457919011yescharacteristic
364C2×C4#1457919211yescharacteristic
365C8—22no—
366C8—41no—
367C8—3162no—
368C8—6321no—
369C8—6321no—
370C22—3162no—
371C22—3162no—
372C22—3162no—
373C22—6321no—
374C22—6321no—
375C22—6321no—
376C22—6321no—
377C22#1909847911yescharacteristic
378C22#1909876511yescharacteristic
379C22#1909877511yescharacteristic
380C22#1909880111yescharacteristic
381C22#1909880611yescharacteristic
382C22#1909882311yescharacteristic
383C22#1909883011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—6321no—
389C4#1909871111yescharacteristic
390C4#1909872011yescharacteristic
391C4#1909876311yescharacteristic
392C4#1909878511yescharacteristic
393C4#1909879411yescharacteristic
394C4#1909881111yescharacteristic
395C4#1909889711yescharacteristic
396C4#1909890311yescharacteristic
397C2—6322no—
398C2#2376279611yescharacteristic
399C2#2376284811yescharacteristic
400C2#2376289611yescharacteristic
401C2#2376293011yescharacteristic
402C2#2376460211yescharacteristic
403C2#2376469311yescharacteristic
404C2#2376471511yescharacteristic
subgroupG/H
centerC23#14579132
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1990C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenter#1990D79
hypocenterC79#10481
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#19098316C2semidirect
5. C2×#19098312C2semidirect
6. #23761283C2semidirect
7. C79×#1990C2semidirect
204. #1990D79semidirect
210. C79#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#190988502
#237641942
#237633972
C2×#190983162
C2×#190983122
#237612832
C79×#19902
#1048179