Finite Group Databasebeta

← #34191653

Subgroups — #34191653

Order 20224.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768574C211yesmaximal, characteristic
2#23764161C211yesmaximal, characteristic, complemented
3#23763397C211yesmaximal, characteristic
4C2×#19098312C211yesmaximal, characteristic, complemented
5C2×#19098292C211yesmaximal, characteristic
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1956C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9C2×#14579092—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×#14579081—21no—
13#19098474—22no—
14#19098464—21no—
15#19098428—21no—
16#19098420—21no—
17#19098420—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19#19098325—21no—
20#19098325—21no—
21#19098312—22no—
22#19098292—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×#103×C79—22no—
25C2×C4×C8×C79C2211yescharacteristic
26#384×C79C2211yescharacteristic
27C22×D316D411yescharacteristic
28C22×C4×D79—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31C4×D316—21no—
32C4×D316—21no—
33C4×D316—21no—
34C4×D316—21no—
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4⋊Dic79D411yescharacteristic
37C2×C4⋊Dic79D411yescharacteristic
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—22no—
40C2×(C2×D79)⋊C4—22no—
41C2×C4×Dic79—21no—
42#14579092—42no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)D412yes—
47#14579081—42no—
48C79⋊#95—22no—
49C79⋊#95—22no—
50C79⋊#95—21no—
51C22×Q8×C79D412yes—
52C22×C8×C79—21no—
53C4⋊Q8×C79—21no—
54C4⋊Q8×C79—21no—
55C4⋊Q8×C79—21no—
56C4⋊Q8×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C42×C79C2311yescharacteristic
63#103×C79—44no—
64C4×C8×C79—21no—
65C4×C8×C79—21no—
66C23×D79—21no—
67C2×D316—21no—
68C2×D316—21no—
69C2×D316—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316SD1612yes—
73C22×Dic79—21no—
74C22×Dic79—21no—
75C22×Dic79—21no—
76C2×C4×D79—41no—
77C2×C4×D79—41no—
78C2×C4×D79—41no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93C4⋊Dic79—21no—
94(C2×D79)⋊C4—41no—
95(C2×D79)⋊C4—41no—
96(C2×D79)⋊C4—82no—
97(C2×D79)⋊C4—82no—
98C4×Dic79—41no—
99C4×Dic79—41no—
100C2×Q8×C79—22no—
101C2×Q8×C79—22no—
102C2×Q8×C79—42no—
103C2×Q8×C79—42no—
104C22×C4×C79—22no—
105C22×C4×C79—41no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C2×C8×C79—22no—
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C4⋊C4×C79—42no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79—41no—
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C42×C79#4111yescharacteristic
124C42×C79#4111yescharacteristic
125C23×C79#12011yescharacteristic
126C22×D79—22no—
127C22×D79—41no—
128C22×D79—41no—
129C22×D79—41no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—41no—
133Q8×C79—44no—
134Q8×C79—82no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79—42no—
140C2×C4×C79—81no—
141C2×C4×C79—81no—
142C2×C4×C79—81no—
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12311yescharacteristic
147C2×C4×C79#12311yescharacteristic
148C2×C4×C79#12411yescharacteristic
149C2×C4×C79C2×SD1611yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79#13511yescharacteristic
152C2×C4×C79#13511yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C2×C4×C79#13711yescharacteristic
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—41no—
162C2×Dic79—41no—
163C2×Dic79—41no—
164C2×Dic79—41no—
165C2×Dic79—41no—
166D316—22no—
167D316—22no—
168D316—41no—
169D316—41no—
170C4×D79—81no—
171C4×D79—81no—
172C4×D79—81no—
173C4×D79—81no—
174C8×C79—42no—
175C79⋊C8—42no—
176C79⋊C8—81no—
177C22×C79#39411yescharacteristic
178C22×C79#44811yescharacteristic
179C22×C79#44911yescharacteristic
180C22×C79#45211yescharacteristic
181C22×C79#45211yescharacteristic
182C22×C79#45311yescharacteristic
183C22×C79#45511yescharacteristic
184C2×D79—42no—
185C2×D79—42no—
186C2×D79—42no—
187C2×D79—81no—
188C2×D79—81no—
189C2×D79—81no—
190C2×D79—81no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—82no—
194C4×C79—161no—
195C4×C79#46311yescharacteristic
196C4×C79#46411yescharacteristic
197C4×C79#46511yescharacteristic
198C4×C79#47011yescharacteristic
199C4×C79#48311yescharacteristic
200C4×C79#48511yescharacteristic
201C4×C79#48811yescharacteristic
202C4×C79#48911yescharacteristic
203Dic79—81no—
204Dic79—81no—
205Dic79—81no—
206#10412—791nomaximal, Sylow
207C2×C79#162211yescharacteristic
208C2×C79#163411yescharacteristic
209C2×C79#164311yescharacteristic
210C2×C79#164611yescharacteristic
211C2×C79#203511yescharacteristic
212C2×C79#203711yescharacteristic
213C2×C79#204211yescharacteristic
214D79—82no—
215#2563—791no—
216#1956D7911yescharacteristic, complemented
217#1948—791no—
218#1802—791no—
219C2×#333—791no—
220C2×D4⋊C8—791no—
221#1279—791no—
222C79#1041211yescharacteristic, complemented, Sylow
223C2×C4⋊Q8C2×D7911yescharacteristic
224C2×C4×D4—791no—
225C2×#106—1581no—
226C2×C4⋊C8—791no—
227C2×C4⋊C8—791no—
228C2×#103—22no—
229C2×#102—1581no—
230C2×C4×C8C2×D7911yescharacteristic
231#393—1582no—
232#390—1581no—
233#384C2×D7911yescharacteristic
234#382—1581no—
235#380—1581no—
236#380—1581no—
237C4×C4⋊C4—791no—
238#340—1581no—
239#340—1581no—
240#333—1582no—
241D4⋊C8—1582no—
242C22×Q8C22⋊D7912yes—
243C22×D4—791no—
244C23×C4—1581no—
245C22×C8—21no—
246C22×C8—1581no—
247C22×C8—1581no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4×D4—1581no—
253C4×D4—1581no—
254C4×D4—1581no—
255C4×D4—1581no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—791no—
262C2×C4⋊C4—791no—
263C2×C4⋊C4—791no—
264C2×C22⋊C4—1581no—
265C2×C22⋊C4—1582no—
266C2×C22⋊C4—1582no—
267C2×C42—1581no—
268C2×C42C22×D7911yescharacteristic
269#106—3162no—
270C4⋊C8—792no—
271C4⋊C8—1581no—
272C4⋊C8—1581no—
273C4⋊C8—1581no—
274#103—44no—
275#102—3162no—
276C4×C8—21no—
277C4×C8—21no—
278#95—1582no—
279#95—1582no—
280#95—1581no—
281C24—1581no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×D4—792no—
287C2×D4—1581no—
288C2×D4—1581no—
289C2×D4—1581no—
290C2×D4—1581no—
291C2×D4—1581no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—3161no—
295C22×C4—3161no—
296C22×C4—3161no—
297C22×C4—3161no—
298C22×C4—3161no—
299C22×C4—3161no—
300C22×C4—1581no—
301C22×C4—1581no—
302C22×C4—1581no—
303C22×C4D4×D7911yescharacteristic
304C22×C4D4×D7911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—3161no—
310C2×C8—3161no—
311C2×C8—3161no—
312C2×C8—3161no—
313C2×C8—3161no—
314C2×C8—1582no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—1581no—
325C4⋊C4—1581no—
326C4⋊C4—1581no—
327C4⋊C4—1581no—
328C4⋊C4—1581no—
329C4⋊C4—1581no—
330C22⋊C4—3161no—
331C22⋊C4—3161no—
332C22⋊C4—6322no—
333C22⋊C4—6322no—
334C42—3161no—
335C42—3161no—
336C42D4×D7911yescharacteristic
337C42#1192962311yescharacteristic
338C42#1192962411yescharacteristic
339C42#1192962411yescharacteristic
340C23—3161no—
341C23—3161no—
342C23—3161no—
343C23—3161no—
344C23—3161no—
345C23—3161no—
346C23—1582no—
347C23#1457912911yescharacteristic
348Q8—44no—
349Q8—82no—
350D4—3161no—
351D4—3161no—
352D4—1582no—
353D4—1582no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—3161no—
363C2×C4—3161no—
364C2×C4—3161no—
365C2×C4—3161no—
366C2×C4—3161no—
367C2×C4—3161no—
368C2×C4—3161no—
369C2×C4—3161no—
370C2×C4—3161no—
371C2×C4—6321no—
372C2×C4—6321no—
373C2×C4—6321no—
374C2×C4—6321no—
375C2×C4#1457913311yescharacteristic
376C2×C4#1457913411yescharacteristic
377C2×C4#1457913411yescharacteristic
378C2×C4#1457914311yescharacteristic
379C2×C4#1457914411yescharacteristic
380C2×C4#1457914711yescharacteristic
381C2×C4SD16×D7911yescharacteristic
382C2×C4#1457918211yescharacteristic
383C2×C4#1457918411yescharacteristic
384C2×C4#1457918411yescharacteristic
385C2×C4#1457918811yescharacteristic
386C2×C4#1457919011yescharacteristic
387C2×C4#1457919311yescharacteristic
388C2×C4#1457919311yescharacteristic
389C8—42no—
390C8—3162no—
391C8—6321no—
392C22—3162no—
393C22—3162no—
394C22—3162no—
395C22—6321no—
396C22—6321no—
397C22—6321no—
398C22—6321no—
399C22#1909847811yescharacteristic
400C22#1909870011yescharacteristic
401C22#1909870711yescharacteristic
402C22#1909872411yescharacteristic
403C22#1909872411yescharacteristic
404C22#1909873011yescharacteristic
405C22#1909874011yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—161no—
410C4—6321no—
411C4—6321no—
412C4—6321no—
413C4#1909878111yescharacteristic
414C4#1909878811yescharacteristic
415C4#1909879511yescharacteristic
416C4#1909882011yescharacteristic
417C4#1909889211yescharacteristic
418C4#1909890311yescharacteristic
419C4#1909891911yescharacteristic
420C4#1909892411yescharacteristic
421C2—6322no—
422C2#2376246911yescharacteristic
423C2#2376254311yescharacteristic
424C2#2376260411yescharacteristic
425C2#2376262511yescharacteristic
426C2#2376459411yescharacteristic
427C2#2376461011yescharacteristic
428C2#2376463711yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1956C2
socleC23×C79#120
prime radicalC22×D316D4
hypercenter#1956D79
hypocenterC79#10412
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. #23764161C2semidirect
4. C2×#19098312C2semidirect
6. #23761240C2semidirect
7. C79×#1956C2semidirect
216. #1956D79semidirect
222. C79#10412semidirect

Maximal subgroups

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

subgroupindex
#237685742
#237641612
#237633972
C2×#190983122
C2×#190982922
#237612402
C79×#19562
#1041279