Finite Group Databasebeta

← #34187227

Subgroups — #34187227

Order 20224.

600 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579150C211yesmaximal, characteristic
2C4⋊C8×Dic79C211yesmaximal, characteristic, complemented
3C4×Dic79⋊C8C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
5C2×#19098292C211yesmaximal, characteristic
6C2×#19098290C211yesmaximal, characteristic
7C2×Q8⋊C8×C79C211yesmaximal, characteristic, complemented
8C2×#14579150C411yescharacteristic
9C2×C4×D316C2211yescharacteristic
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×Dic79⋊C8—21no—
12C2×C4×C79⋊C8C2211yescharacteristic
13C2×C8×Dic79—21no—
14#19098470C411yescharacteristic
15#19098464C411yescharacteristic
16#19098453—21no—
17#19098453—21no—
18#19098428C411yescharacteristic
19C4×(C2×D79)⋊C4—21no—
20C4×(C2×D79)⋊C4—21no—
21C42×Dic79C2211yescharacteristic
22#19098317—21no—
23#19098317—21no—
24Q8⋊(C79⋊C8)—22no—
25#19098292—22no—
26#19098290—22no—
27C2×C4×Q8×C79C2211yescharacteristic
28C2×C4⋊C8×C79C2211yescharacteristic
29C2×C4×C8×C79C2211yescharacteristic
30Q8⋊C8×C79—22no—
31C22×D316C2×C411yescharacteristic
32C22×C4×D79—21no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35#14579150—22no—
36#14579150C814yescomplemented
37C4×D316—21no—
38C4×D316—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79C2×C411yescharacteristic
42C2×(C2×D79)⋊C4—21no—
43C2×(C2×D79)⋊C4—21no—
44C2×(C2×D79)⋊C4—21no—
45C2×(C2×D79)⋊C4—21no—
46C2×(C2×D79)⋊C4—21no—
47C2×C4×Dic79C2×C411yescharacteristic
48C2×C4×Dic79C2×C411yescharacteristic
49C2×C4×Dic79D411yescharacteristic
50C2×C4×Dic79D411yescharacteristic
51C2×C4×Dic79D411yescharacteristic
52C2×C4×Dic79D411yescharacteristic
53C4⋊(C79⋊C8)—21no—
54C4⋊(C79⋊C8)D412yes—
55Dic79⋊C8—24no—
56C4×C79⋊C8—21no—
57C4×C79⋊C8—21no—
58C8×Dic79—42no—
59C79⋊#95—21no—
60C79⋊#95—21no—
61C79⋊#95—21no—
62C79⋊#95—21no—
63C22×Q8×C79C2×C411yescharacteristic
64C22×C8×C79—21no—
65C22×C8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68C4×Q8×C79—21no—
69C4×Q8×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79C2×C411yescharacteristic
72C2×C42×C79—21no—
73C2×C42×C79C2311yescharacteristic
74C4⋊C8×C79—21no—
75C4⋊C8×C79D412yes—
76C4×C8×C79—21no—
77C4×C8×C79—21no—
78C23×D79—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×D316C2×C812yes—
85C22×Dic79—21no—
86C22×Dic79C22⋊C411yescharacteristic
87C22×Dic79C22⋊C411yescharacteristic
88C22×Dic79C22⋊C411yescharacteristic
89C22×Dic79C22⋊C411yescharacteristic
90C2×C4×D79—41no—
91C2×C4×D79—41no—
92C2×C4×D79—41no—
93C2×C4×D79—41no—
94C2×C4×D79—41no—
95C2×C4×D79—41no—
96C2×C79⋊C8—22no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C4⋊Dic79—21no—
103C4⋊Dic79—21no—
104(C2×D79)⋊C4—24no—
105(C2×D79)⋊C4—24no—
106(C2×D79)⋊C4—42no—
107(C2×D79)⋊C4—42no—
108(C2×D79)⋊C4—41no—
109(C2×D79)⋊C4—41no—
110C4×Dic79—22no—
111C4×Dic79—21no—
112C4×Dic79—21no—
113C4×Dic79—21no—
114C4×Dic79—21no—
115C4×Dic79—21no—
116C4×Dic79—21no—
117C4×Dic79—41no—
118C4×Dic79C2×C812yes—
119C4×Dic79M1612yes—
120C2×Q8×C79—21no—
121C2×Q8×C79—21no—
122C2×Q8×C79—21no—
123C2×Q8×C79—21no—
124C2×Q8×C79—21no—
125C2×Q8×C79C2×C812yes—
126C22×C4×C79—21no—
127C22×C4×C79—21no—
128C22×C4×C79C22×C411yescharacteristic
129C22×C4×C79C2×D411yescharacteristic
130C22×C4×C79C2×D411yescharacteristic
131C2×C8×C79—22no—
132C2×C8×C79—22no—
133C2×C8×C79—41no—
134C2×C8×C79—41no—
135C2×C8×C79—41no—
136C2×C8×C79—41no—
137C4⋊C4×C79—21no—
138C4⋊C4×C79—21no—
139C4⋊C4×C79—41no—
140C4⋊C4×C79—41no—
141C42×C79—41no—
142C42×C79—41no—
143C42×C79C2×D411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79#4111yescharacteristic
146C42×C79#4111yescharacteristic
147C23×C79C2×C22⋊C411yescharacteristic
148C22×D79—22no—
149C22×D79—41no—
150C22×D79—41no—
151C22×D79—41no—
152C22×D79—41no—
153C22×D79—41no—
154C22×D79—41no—
155Q8×C79—22no—
156Q8×C79—22no—
157Q8×C79—41no—
158Q8×C79—41no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—22no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79C4×D411yescharacteristic
168C2×C4×C79C4×D411yescharacteristic
169C2×C4×C79#12111yescharacteristic
170C2×C4×C79#12111yescharacteristic
171C2×C4×C79#12411yescharacteristic
172C2×C4×C79#12711yescharacteristic
173C2×C4×C79C22×C811yescharacteristic
174C2×C4×C79C2×M1611yescharacteristic
175C2×C4×C79#13111yescharacteristic
176C2×C4×C79#13111yescharacteristic
177C2×C4×C79#13511yescharacteristic
178C2×C4×C79#13511yescharacteristic
179C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C79#13711yescharacteristic
181C2×Dic79—21no—
182C2×Dic79—21no—
183C2×Dic79—21no—
184C2×Dic79—21no—
185C2×Dic79—21no—
186C2×Dic79—21no—
187C2×Dic79—21no—
188C2×Dic79—21no—
189C2×Dic79—21no—
190C2×Dic79—41no—
191C2×Dic79—41no—
192C2×Dic79—41no—
193C2×Dic79C22⋊C812yes—
194C2×Dic79C22⋊C812yes—
195C2×Dic79#10412yes—
196D316—22no—
197D316—22no—
198D316—41no—
199D316—41no—
200C4×D79—81no—
201C4×D79—81no—
202C4×D79—81no—
203C4×D79—81no—
204C8×C79—42no—
205C8×C79—42no—
206C79⋊C8—42no—
207C79⋊C8—42no—
208C22×C79#39011yescharacteristic
209C22×C79C2×C22⋊C811yescharacteristic
210C22×C79#40811yescharacteristic
211C22×C79#41611yescharacteristic
212C22×C79#41711yescharacteristic
213C22×C79C2×#10411yescharacteristic
214C22×C79#42111yescharacteristic
215C2×D79—42no—
216C2×D79—42no—
217C2×D79—42no—
218C2×D79—81no—
219C2×D79—81no—
220C2×D79—81no—
221C2×D79—81no—
222C4×C79—21no—
223C4×C79—21no—
224C4×C79—42no—
225C4×C79—81no—
226C4×C79C8×D411yescharacteristic
227C4×C79C8×D411yescharacteristic
228C4×C79#43611yescharacteristic
229C4×C79#43611yescharacteristic
230C4×C79#46311yescharacteristic
231C4×C79#46311yescharacteristic
232C4×C79#46411yescharacteristic
233C4×C79#46411yescharacteristic
234Dic79—22no—
235Dic79—22no—
236Dic79—22no—
237Dic79—41no—
238Dic79—81no—
239#9675—791nomaximal, Sylow
240C2×C79#147611yescharacteristic
241C2×C79#147711yescharacteristic
242C2×C79#148211yescharacteristic
243C2×C79#148311yescharacteristic
244C2×C79#195411yescharacteristic
245C2×C79#198011yescharacteristic
246C2×C79#198311yescharacteristic
247D79—82no—
248C4×#124—791no—
249C4×C4⋊C8—791no—
250C4×C4⋊C8—791no—
251C2×Q8⋊C8—791no—
252C2×Q8⋊C8D7911yescharacteristic, complemented
253C2×D4⋊C8—791no—
254C2×D4⋊C8—791no—
255C79#967511yescharacteristic, complemented, Sylow
256C2×#124—791no—
257C2×C4×Q8C2×D7911yescharacteristic
258C2×C4×D4—791no—
259C2×C4⋊C8—791no—
260C2×C4⋊C8—1581no—
261C2×C4⋊C8C2×D7911yescharacteristic
262C2×C4×C8—791no—
263C2×C4×C8—1581no—
264C2×C4×C8C2×D7911yescharacteristic
265#391—791no—
266#390—791no—
267#388—1581no—
268#388—1581no—
269#382—791no—
270C4×C22⋊C4—1581no—
271C4×C22⋊C4—1581no—
272C43—791no—
273#336—1581no—
274#336—1581no—
275Q8⋊C8—22no—
276Q8⋊C8—1582no—
277D4⋊C8—1582no—
278D4⋊C8—1582no—
279C22×Q8C4×D7911yescharacteristic
280C22×D4—791no—
281C23×C4—1581no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—1581no—
285C22×C8—1581no—
286#124—794no—
287#124—1582no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—1581no—
293C4×D4—1581no—
294C4×D4—1581no—
295C4×D4—1581no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—791no—
298C2×C4⋊C4C4×D7911yescharacteristic
299C2×C22⋊C4—1581no—
300C2×C22⋊C4—1581no—
301C2×C22⋊C4—1581no—
302C2×C22⋊C4—1581no—
303C2×C22⋊C4—1581no—
304C2×C42—21no—
305C2×C42—791no—
306C2×C42—791no—
307C2×C42—791no—
308C2×C42—791no—
309C2×C42—791no—
310C2×C42—791no—
311C2×C42C22×D7911yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—792no—
314C4⋊C8—1584no—
315C4⋊C8—1581no—
316C4⋊C8C22⋊D7912yes—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—3162no—
320C4×C8—1581no—
321C4×C8—1581no—
322#95—1581no—
323#95—1581no—
324#95—1581no—
325#95—1581no—
326C24—1581no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C8×D7912yes—
333C2×D4—792no—
334C2×D4—1581no—
335C2×D4—1581no—
336C2×D4—1581no—
337C2×D4—1581no—
338C2×D4—1581no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—3161no—
342C22×C4—3161no—
343C22×C4—3161no—
344C22×C4—3161no—
345C22×C4—3161no—
346C22×C4—3161no—
347C22×C4—791no—
348C22×C4—791no—
349C22×C4—791no—
350C22×C4—791no—
351C22×C4—1581no—
352C22×C4C2×C4×D7911yescharacteristic
353C22×C4D4×D7911yescharacteristic
354C22×C4D4×D7911yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—3161no—
362C2×C8—3161no—
363C2×C8—3161no—
364C2×C8—3161no—
365C2×C8—1582no—
366C2×C8—1582no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1581no—
372C4⋊C4—1581no—
373C22⋊C4—3162no—
374C22⋊C4—3162no—
375C22⋊C4—3161no—
376C22⋊C4—3161no—
377C22⋊C4—1584no—
378C22⋊C4—1584no—
379C42—41no—
380C42—41no—
381C42—3161no—
382C42—792no—
383C42—792no—
384C42—1582no—
385C42—1581no—
386C42—1581no—
387C42—1581no—
388C42—1581no—
389C42—1581no—
390C42—1581no—
391C42D4×D7911yescharacteristic
392C42C2×C22⋊D7911yescharacteristic
393C42#1192962311yescharacteristic
394C42#1192962411yescharacteristic
395C23—3161no—
396C23—3161no—
397C23—3161no—
398C23—3161no—
399C23—3161no—
400C23—3161no—
401C23—1582no—
402C23C22⋊C4×D7911yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407D4—3161no—
408D4—3161no—
409D4—1582no—
410D4—1582no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—3161no—
420C2×C4—3161no—
421C2×C4—3161no—
422C2×C4—792no—
423C2×C4—792no—
424C2×C4—792no—
425C2×C4—6321no—
426C2×C4—6321no—
427C2×C4—6321no—
428C2×C4—6321no—
429C2×C4—1581no—
430C2×C4—1581no—
431C2×C4—1581no—
432C2×C4—1581no—
433C2×C4—1581no—
434C2×C4—1581no—
435C2×C4—1581no—
436C2×C4—1581no—
437C2×C4—1581no—
438C2×C4(C22⋊D79)⋊C411yescharacteristic
439C2×C4C4×C22⋊D7911yescharacteristic
440C2×C4#1457913311yescharacteristic
441C2×C4#1457913411yescharacteristic
442C2×C4#1457914711yescharacteristic
443C2×C4#1457915711yescharacteristic
444C2×C4C2×C8×D7911yescharacteristic
445C2×C4M16×D7911yescharacteristic
446C2×C4#1457916711yescharacteristic
447C2×C4#1457916911yescharacteristic
448C2×C4#1457918211yescharacteristic
449C2×C4#1457918411yescharacteristic
450C2×C4#1457918811yescharacteristic
451C2×C4#1457919311yescharacteristic
452C8—42no—
453C8—42no—
454C8—3162no—
455C8—3162no—
456C22—3162no—
457C22—3162no—
458C22—3162no—
459C22—6321no—
460C22—6321no—
461C22—6321no—
462C22—6321no—
463C22#1909846311yescharacteristic
464C22C22⋊C8×D7911yescharacteristic
465C22#1909852811yescharacteristic
466C22#1909855811yescharacteristic
467C22#1909856311yescharacteristic
468C22#104×D7911yescharacteristic
469C22#1909857811yescharacteristic
470C4—21no—
471C4—21no—
472C4—42no—
473C4—81no—
474C4—3161no—
475C4—6321no—
476C4—1582no—
477C4—1582no—
478C4—1582no—
479C4(C22⋊D79)⋊C811yescharacteristic
480C4C8×C22⋊D7911yescharacteristic
481C4#1909863811yescharacteristic
482C4#1909863911yescharacteristic
483C4#1909878011yescharacteristic
484C4#1909878111yescharacteristic
485C4#1909878711yescharacteristic
486C4#1909878811yescharacteristic
487C2—6322no—
488C2#2376178111yescharacteristic
489C2#2376178611yescharacteristic
490C2#2376180711yescharacteristic
491C2#2376181211yescharacteristic
492C2#2376419211yescharacteristic
493C2#2376431711yescharacteristic
494C2#2376432911yescharacteristic
subgroupG/H
centerC22×C4D4×D79
commutatorC2×C4×C79C22×C8
FrattiniC2×C42C22×D79
FittingC2×Q8⋊C8×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenterC2×Q8⋊C8D79
hypocenterC79#9675
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×Dic79C2semidirect
3. C4×Dic79⋊C8C2semidirect
4. C2×Q8⋊(C79⋊C8)C2semidirect
7. C2×Q8⋊C8×C79C2semidirect
36. #14579150C8semidirect
252. C2×Q8⋊C8D79semidirect
255. C79#9675semidirect

Maximal subgroups

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

subgroupindex
C4×#145791502
C4⋊C8×Dic792
C4×Dic79⋊C82
C2×Q8⋊(C79⋊C8)2
C2×#190982922
C2×#190982902
C2×Q8⋊C8×C792
#967579