Finite Group Databasebeta

← #34192080

Subgroups — #34192080

Order 20224.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098763C211yesmaximal, characteristic
2#23764337C211yesmaximal, characteristic, complemented
3#23764205C211yesmaximal, characteristic
4C2×#19098313C211yesmaximal, characteristic, complemented
5C2×#19098306C211yesmaximal, characteristic
6#23761283C211yesmaximal, characteristic, complemented
7C79×#1803C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×#14579158C2211yescharacteristic
10#19098763—22no—
11#19098763—22no—
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579081—21no—
15C2×#14579081—21no—
16C2×#14579081—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098318—21no—
19#19098318—21no—
20#19098313—22no—
21#19098306—22no—
22C2×C4×Q8×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×#103×C79—21no—
25#384×C79C2211yescharacteristic
26#340×C79—21no—
27C22×D316—21no—
28C22×D316—21no—
29C22×D316—21no—
30C2×#11929611—24no—
31C2×#11929611—42no—
32C2×#11929611—42no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8D411yescharacteristic
36C22×C79⋊C8D411yescharacteristic
37#14579158—21no—
38#14579158—21no—
39#14579158—21no—
40#14579158D412yes—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579081—24no—
46#14579081—42no—
47#14579081—42no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C22×Q8×C79D411yescharacteristic
51C22×C8×C79—21no—
52C4×Q8×C79—21no—
53C4×Q8×C79—21no—
54C4×Q8×C79—21no—
55C4×Q8×C79—21no—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79D411yescharacteristic
61C2×C42×C79—21no—
62C2×C42×C79C2311yescharacteristic
63C4⋊C8×C79—21no—
64C4⋊C8×C79—21no—
65#103×C79—42no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—22no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—81no—
82C2×D316—81no—
83#11929611—48no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8SD1612yescomplemented
96C2×Q8×C79—21no—
97C2×Q8×C79—21no—
98C2×Q8×C79—21no—
99C2×Q8×C79—21no—
100C2×Q8×C79—21no—
101C2×Q8×C79SD1612yes—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—41no—
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79#4111yescharacteristic
108C2×C8×C79—41no—
109C2×C8×C79—41no—
110C2×C8×C79—41no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
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—81no—
120C42×C79—41no—
121C42×C79—41no—
122C42×C79C2×D411yescharacteristic
123C42×C79C2×D411yescharacteristic
124C42×C79#4111yescharacteristic
125C42×C79#4111yescharacteristic
126C23×C79#12111yescharacteristic
127C22×D79—42no—
128C22×D79—81no—
129C22×D79—81no—
130C22×D79—81no—
131C22×D79—81no—
132C22×D79—81no—
133C22×D79—81no—
134Q8×C79—22no—
135Q8×C79—22no—
136Q8×C79—41no—
137Q8×C79—41no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—22no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—81no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79#12011yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12111yescharacteristic
152C2×C4×C79#12311yescharacteristic
153C2×C4×C79#12311yescharacteristic
154C2×C4×C79#12411yescharacteristic
155C2×C4×C79C2×SD1611yescharacteristic
156C2×C4×C79C2×SD1611yescharacteristic
157C2×C4×C79#13511yescharacteristic
158C2×C4×C79#13511yescharacteristic
159C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C79#13711yescharacteristic
163D316—42no—
164D316—42no—
165D316—42no—
166D316—42no—
167D316—82no—
168D316—82no—
169D316—81no—
170D316—81no—
171D316—81no—
172D316—81no—
173C8×C79—81no—
174C79⋊C8—22no—
175C79⋊C8—42no—
176C79⋊C8—41no—
177C79⋊C8—81no—
178C22×C79#39411yescharacteristic
179C22×C79#46011yescharacteristic
180C22×C79#46411yescharacteristic
181C22×C79#46511yescharacteristic
182C22×C79#46911yescharacteristic
183C22×C79#47111yescharacteristic
184C22×C79#47211yescharacteristic
185C2×D79—82no—
186C2×D79—82no—
187C2×D79—82no—
188C2×D79—161no—
189C2×D79—161no—
190C2×D79—161no—
191C2×D79—161no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—42no—
195C4×C79—81no—
196C4×C79—161no—
197C4×C79#44911yescharacteristic
198C4×C79#45011yescharacteristic
199C4×C79#46111yescharacteristic
200C4×C79#46311yescharacteristic
201C4×C79#46511yescharacteristic
202C4×C79#46811yescharacteristic
203C4×C79#48111yescharacteristic
204C4×C79#48211yescharacteristic
205#10475—791nomaximal, Sylow
206C2×C79#166611yescharacteristic
207C2×C79#167811yescharacteristic
208C2×C79#168811yescharacteristic
209C2×C79#170211yescharacteristic
210C2×C79#203411yescharacteristic
211C2×C79#205111yescharacteristic
212C2×C79#205511yescharacteristic
213D79—162no—
214C2×#460—791no—
215#1984—791no—
216#1957—791no—
217#1803D7911yescharacteristic, complemented
218C2×#334—791no—
219C2×#331—791no—
220#1296—791no—
221C79#1047511yescharacteristic, complemented, Sylow
222C22×SD16—1581no—
223C2×#127—791no—
224C2×C4×Q8C2×D7911yescharacteristic
225#460—1582no—
226#460—1582no—
227C2×C4⋊C8—791no—
228C2×C4⋊C8—791no—
229C2×C4⋊C8C2×D7911yescharacteristic
230C2×#103—21no—
231C2×#102—1581no—
232C2×#102—1581no—
233C2×#102—1581no—
234C2×C4×C8—791no—
235#384C2×D7911yescharacteristic
236#340—21no—
237#336—1581no—
238#336—1581no—
239#334—1582no—
240#331—1582no—
241C22×Q8C22⋊D7911yescharacteristic
242C22×D4—1581no—
243C22×D4—1581no—
244C22×D4—1581no—
245C2×SD16—3162no—
246C2×SD16—3162no—
247C2×SD16—1584no—
248C22×C8—21no—
249C22×C8—791no—
250C22×C8—791no—
251C22×C8—1581no—
252C22×C8—1581no—
253#127—792no—
254#127—1581no—
255#127—1581no—
256#127—1581no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D7911yescharacteristic
266C2×C42—21no—
267C2×C42C22×D7911yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—792no—
271C4⋊C8—1581no—
272C4⋊C8—1581no—
273C4⋊C8—1581no—
274#103—42no—
275#102—3162no—
276#102—3162no—
277#102—1584no—
278C4×C8—1581no—
279C4×C8—1581no—
280C24—3161no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8#1192961112yes—
287C2×D4—3161no—
288C2×D4—3161no—
289C2×D4—3161no—
290C2×D4—3161no—
291C2×D4—3161no—
292C2×D4—3161no—
293C2×D4—3161no—
294C2×D4—3161no—
295C2×D4—3161no—
296C2×D4—3161no—
297C2×D4—3161no—
298C2×D4—6321no—
299C2×D4—6321no—
300C2×D4—1582no—
301C2×D4—1582no—
302C2×D4—1582no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4D4×D7911yescharacteristic
307C22×C4C2×C22⋊D7911yescharacteristic
308C22×C4#1192962311yescharacteristic
309SD16—3168no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3161no—
314C2×C8—3161no—
315C2×C8—3161no—
316C2×C8—3161no—
317C2×C8—3161no—
318C2×C8—792no—
319C2×C8—1582no—
320C2×C8—1581no—
321C2×C8—1581no—
322C2×C8—1581no—
323C2×C8—1581no—
324C2×C8—1581no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C42—41no—
335C42—41no—
336C42D4×D7911yescharacteristic
337C42D4×D7911yescharacteristic
338C42#1192962411yescharacteristic
339C42#1192962411yescharacteristic
340C23—3162no—
341C23—6321no—
342C23—6321no—
343C23—6321no—
344C23—6321no—
345C23—6321no—
346C23—6321no—
347C23#1457913311yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—3162no—
353D4—3162no—
354D4—3162no—
355D4—3162no—
356D4—6322no—
357D4—6322no—
358D4—6321no—
359D4—6321no—
360D4—6321no—
361D4—6321no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4#1457912911yescharacteristic
374C2×C4#1457913411yescharacteristic
375C2×C4#1457913411yescharacteristic
376C2×C4#1457914311yescharacteristic
377C2×C4#1457914411yescharacteristic
378C2×C4#1457914711yescharacteristic
379C2×C4SD16×D7911yescharacteristic
380C2×C4C2×#1192961111yescharacteristic
381C2×C4#1457918111yescharacteristic
382C2×C4#1457918411yescharacteristic
383C2×C4#1457918611yescharacteristic
384C2×C4#1457918711yescharacteristic
385C2×C4#1457918811yescharacteristic
386C2×C4#1457919311yescharacteristic
387C8—81no—
388C8—3162no—
389C8—3161no—
390C8—6321no—
391C8—1582no—
392C22—6322no—
393C22—6322no—
394C22—6322no—
395C22—12641no—
396C22—12641no—
397C22—12641no—
398C22—12641no—
399C22#1909847811yescharacteristic
400C22#1909876111yescharacteristic
401C22#1909878711yescharacteristic
402C22#1909879411yescharacteristic
403C22#1909881411yescharacteristic
404C22#1909882811yescharacteristic
405C22#1909883111yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—161no—
411C4#1909870611yescharacteristic
412C4#1909871311yescharacteristic
413C4#1909876711yescharacteristic
414C4#1909878111yescharacteristic
415C4#1909879511yescharacteristic
416C4#1909881211yescharacteristic
417C4#1909888111yescharacteristic
418C4#1909888711yescharacteristic
419C2—12642no—
420C2#2376275011yescharacteristic
421C2#2376281811yescharacteristic
422C2#2376288211yescharacteristic
423C2#2376297011yescharacteristic
424C2#2376458911yescharacteristic
425C2#2376469011yescharacteristic
426C2#2376471411yescharacteristic
subgroupG/H
centerC23#14579133
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1803C2
socleC23×C79#121
prime radicalC2×#14579158C22
hypercenter#1803D79
hypocenterC79#10475
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #23764337C2semidirect
4. C2×#19098313C2semidirect
6. #23761283C2semidirect
7. C79×#1803C2semidirect
95. C2×C79⋊C8SD16semidirect
217. #1803D79semidirect
221. C79#10475semidirect

Maximal subgroups

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

subgroupindex
C2×#190987632
#237643372
#237642052
C2×#190983132
C2×#190983062
#237612832
C79×#18032
#1047579