Finite Group Databasebeta

← #34184437

Subgroups — #34184437

Order 20224.

594 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098844C211yesmaximal, characteristic
2C2×#19098628C211yesmaximal, characteristic
3#23764201C211yesmaximal, characteristic, complemented
4#23763397C211yesmaximal, characteristic
5C2×#19098310C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#1296C211yesmaximal, characteristic, complemented
8C22×C8×D79—21no—
9C2×#14579161C2211yescharacteristic
10C2×C4×D316C2211yescharacteristic
11#19098844—22no—
12#19098844C414yescomplemented
13#19098628—22no—
14#19098628—22no—
15C2×#14579095—21no—
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×#14579084—21no—
18C2×#14579084—21no—
19C2×#14579083—21no—
20C2×#14579081—21no—
21C2×(C2×D79)⋊C8—21no—
22C2×C4×C79⋊C8C2211yescharacteristic
23#19098325—21no—
24#19098310—22no—
25C2×C4⋊C8×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27#384×C79C2211yescharacteristic
28C22×Dic158—21no—
29C22×D316D411yescharacteristic
30C22×C4×D79—21no—
31C22×C79⋊C8—21no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34C2×C8×D79—24no—
35C2×C8×D79—42no—
36C2×C8×D79—42no—
37#14579161—21no—
38#14579161—21no—
39#14579161—21no—
40#14579161C2×C412yes—
41C4×D316—21no—
42C4×D316—21no—
43C4×D316C2×C412yes—
44C4×D316D412yes—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79—21no—
47C2×C4⋊Dic79D411yescharacteristic
48C2×(C2×D79)⋊C4—21no—
49#14579095—24no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)—21no—
52#14579084—42no—
53#14579084—42no—
54#14579083—24no—
55#14579081—42no—
56(C2×D79)⋊C8—24no—
57C4×C79⋊C8—21no—
58C4×C79⋊C8—21no—
59C22×C8×C79—21no—
60C22×C8×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C42×C79C2311yescharacteristic
65C4⋊C8×C79—21no—
66C4⋊C8×C79—21no—
67C4⋊C8×C79C2×C412yes—
68C4⋊C8×C79D412yes—
69C23×D79—21no—
70C2×Dic158—22no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×Dic158—41no—
74C2×Dic158—41no—
75C2×Dic158—41no—
76C2×D316—21no—
77C2×D316—21no—
78C2×D316—21no—
79C2×D316C22⋊C412yes—
80C2×D316SD1612yes—
81C2×D316Q1612yes—
82C22×Dic79—21no—
83C22×Dic79—41no—
84C2×C4×D79—22no—
85C2×C4×D79—22no—
86C2×C4×D79—41no—
87C2×C4×D79—41no—
88C2×C4×D79—41no—
89C2×C4×D79—41no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8D812yes—
99C8×D79—48no—
100C4⋊Dic79—22no—
101C4⋊Dic79—22no—
102C4⋊Dic79—21no—
103C4⋊Dic79—41no—
104C4⋊Dic79—41no—
105C4⋊Dic79C22⋊C412yes—
106(C2×D79)⋊C4—22no—
107(C2×D79)⋊C4—22no—
108C22×C4×C79—41no—
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79#4111yescharacteristic
112C2×C8×C79—22no—
113C2×C8×C79—22no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—81no—
123C42×C79C22×C411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79#12111yescharacteristic
128C22×D79—22no—
129C22×D79—22no—
130C22×D79—22no—
131C22×D79—41no—
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79C2×C22⋊C411yescharacteristic
141C2×C4×C79C4×D411yescharacteristic
142C2×C4×C79C4×D411yescharacteristic
143C2×C4×C79#12011yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12311yescharacteristic
146C2×C4×C79C2×D811yescharacteristic
147C2×C4×C79C2×SD1611yescharacteristic
148C2×C4×C79C2×Q1611yescharacteristic
149C2×C4×C79#13511yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79#13511yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×Dic79—22no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—42no—
158C2×Dic79—81no—
159C2×Dic79—81no—
160D316—22no—
161D316—22no—
162D316—22no—
163D316#10314yescomplemented
164C4×D79—42no—
165C4×D79—42no—
166C4×D79—42no—
167C4×D79—42no—
168Dic158—42no—
169Dic158—42no—
170Dic158—81no—
171Dic158—81no—
172C8×C79—42no—
173C8×C79—42no—
174C79⋊C8—22no—
175C79⋊C8—41no—
176C79⋊C8—81no—
177C22×C79#38611yescharacteristic
178C22×C79#45911yescharacteristic
179C22×C79#46311yescharacteristic
180C22×C79#46511yescharacteristic
181C22×C79#46711yescharacteristic
182C22×C79#47111yescharacteristic
183C22×C79#47311yescharacteristic
184C2×D79—24no—
185C2×D79—42no—
186C2×D79—42no—
187C2×D79—42no—
188C2×D79—42no—
189C2×D79—42no—
190C2×D79—42no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—161no—
194C4×C79C2×#10311yescharacteristic
195C4×C79#41611yescharacteristic
196C4×C79C4×D811yescharacteristic
197C4×C79#44211yescharacteristic
198C4×C79#44711yescharacteristic
199C4×C79#44911yescharacteristic
200C4×C79#46511yescharacteristic
201C4×C79#46711yescharacteristic
202Dic79—42no—
203Dic79—82no—
204#9219—791nomaximal, Sylow
205C2×C79#166511yescharacteristic
206C2×C79#166911yescharacteristic
207C2×C79#167911yescharacteristic
208C2×C79#168311yescharacteristic
209C2×C79#187911yescharacteristic
210C2×C79#189211yescharacteristic
211C2×C79#190811yescharacteristic
212D79—44no—
213C2×#474—791no—
214C2×C8×D4—791no—
215#1956—791no—
216#1802—791no—
217C2×#332—791no—
218#1296D7911yescharacteristic, complemented
219#1277—791no—
220C79#921911yescharacteristic, complemented, Sylow
221C23×C8—1581no—
222C2×C4⋊Q8—791no—
223C2×C4×D4—791no—
224#474—794no—
225#474—1582no—
226C8×D4—1582no—
227C8×D4—1582no—
228C2×#107—1581no—
229C2×C4⋊C8—791no—
230C2×C4⋊C8C2×D7911yescharacteristic
231C2×C4⋊C8C2×D7911yescharacteristic
232C2×#103—1581no—
233C2×#103—1581no—
234C2×#102—1581no—
235C2×#102—1581no—
236C2×C22⋊C8—1581no—
237C2×C4×C8—791no—
238#384C2×D7911yescharacteristic
239#340—1581no—
240#332—1582no—
241C22×Q8—1581no—
242C22×D4—791no—
243C23×C4—1581no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—3162no—
247C22×C8—3162no—
248C22×C8—791no—
249C22×C8—791no—
250C22×C8—1584no—
251C22×C8—1581no—
252C4⋊Q8—792no—
253C4⋊Q8—1581no—
254C4⋊Q8—1581no—
255C4⋊Q8—1581no—
256C4×D4—792no—
257C4×D4—792no—
258C4×D4—1581no—
259C4×D4—1581no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—791no—
264C2×C4⋊C4—1581no—
265C2×C4⋊C4—1581no—
266C2×C22⋊C4—1581no—
267C2×C42C22×D7911yescharacteristic
268#107—1584no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1581no—
272C4⋊C8—1581no—
273C4⋊C8C4×D7912yes—
274C4⋊C8D31612yes—
275#103—3162no—
276#103—3162no—
277#102—3162no—
278#102—1584no—
279C22⋊C8—1584no—
280C4×C8—1581no—
281C4×C8—1581no—
282C24—1581no—
283C2×Q8—3161no—
284C2×Q8—3161no—
285C2×Q8—3161no—
286C2×Q8—3161no—
287C2×Q8—3161no—
288C2×Q8—1582no—
289C2×D4—792no—
290C2×D4—792no—
291C2×D4—792no—
292C2×D4—1581no—
293C2×D4—1581no—
294C2×D4—1581no—
295C22×C4—41no—
296C22×C4—3161no—
297C22×C4—3161no—
298C22×C4—3161no—
299C22×C4—3161no—
300C22×C4—3161no—
301C22×C4—1581no—
302C22×C4—1582no—
303C22×C4—1582no—
304C22×C4D4×D7911yescharacteristic
305C22×C4D4×D7911yescharacteristic
306C22×C4#1192962411yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3168no—
314C2×C8—3161no—
315C2×C8—3161no—
316C2×C8—3161no—
317C2×C8—792no—
318C2×C8—1581no—
319C2×C8—1581no—
320C2×C8—1581no—
321C2×C8—1581no—
322C2×C8—1581no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—81no—
328C4⋊C4—3161no—
329C4⋊C4—3161no—
330C4⋊C4—792no—
331C4⋊C4—1582no—
332C4⋊C4—1582no—
333C4⋊C4—1581no—
334C22⋊C4—1582no—
335C22⋊C4—1582no—
336C42C2×C4×D7911yescharacteristic
337C42D4×D7911yescharacteristic
338C42C2×D31611yescharacteristic
339C42#1192962511yescharacteristic
340C23—3161no—
341C23—3161no—
342C23—3161no—
343C23—3161no—
344C23—1582no—
345C23—1582no—
346C23—1582no—
347C23#1457913411yescharacteristic
348Q8—3162no—
349Q8—3162no—
350Q8—6321no—
351Q8—6321no—
352D4—794no—
353D4—1582no—
354D4—1582no—
355D4—1582no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—3161no—
362C2×C4—3161no—
363C2×C4—3162no—
364C2×C4—3162no—
365C2×C4—3162no—
366C2×C4—3162no—
367C2×C4—3162no—
368C2×C4—6321no—
369C2×C4—6321no—
370C2×C4—1582no—
371C2×C4C22⋊C4×D7911yescharacteristic
372C2×C4#1457912311yescharacteristic
373C2×C4C4×D31611yescharacteristic
374C2×C4#1457913111yescharacteristic
375C2×C4#1457913611yescharacteristic
376C2×C4#1457914611yescharacteristic
377C2×C4D8×D7911yescharacteristic
378C2×C4SD16×D7911yescharacteristic
379C2×C4Q16×D7911yescharacteristic
380C2×C4#1457918011yescharacteristic
381C2×C4#1457918211yescharacteristic
382C2×C4#1457918411yescharacteristic
383C2×C4#1457918811yescharacteristic
384C2×C4#1457918811yescharacteristic
385C8—42no—
386C8—42no—
387C8—3161no—
388C8—6321no—
389C8—1582no—
390C22—3162no—
391C22—3162no—
392C22—3162no—
393C22—3162no—
394C22—3162no—
395C22—3162no—
396C22—1584no—
397C22#1909844111yescharacteristic
398C22#1909875711yescharacteristic
399C22#1909878111yescharacteristic
400C22#1909879111yescharacteristic
401C22#1909880311yescharacteristic
402C22#1909882511yescharacteristic
403C22#1909883711yescharacteristic
404C4—21no—
405C4—21no—
406C4—161no—
407C4—3162no—
408C4—6322no—
409C4#103×D7911yescharacteristic
410C4#1909855611yescharacteristic
411C4#1909864511yescharacteristic
412C4#1909867411yescharacteristic
413C4#1909869811yescharacteristic
414C4#1909871111yescharacteristic
415C4#1909879311yescharacteristic
416C4#1909880511yescharacteristic
417C2—3164no—
418C2#2376274011yescharacteristic
419C2#2376276411yescharacteristic
420C2#2376282611yescharacteristic
421C2#2376285411yescharacteristic
422C2#2376374911yescharacteristic
423C2#2376382211yescharacteristic
424C2#2376391611yescharacteristic
subgroupG/H
centerC23#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1296C2
socleC23×C79#121
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1296D79
hypocenterC79#9219
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #23764201C2semidirect
5. C2×#19098310C2semidirect
6. #23761234C2semidirect
7. C79×#1296C2semidirect
12. #19098844C4semidirect
163. D316#103semidirect
218. #1296D79semidirect
220. C79#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#190988442
C2×#190986282
#237642012
#237633972
C2×#190983102
#237612342
C79×#12962
#921979