Finite Group Databasebeta

← #34206964

Subgroups — #34206964

Order 20224.

670 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579102C211yesmaximal, characteristic, complemented
2C2×#19098799C211yesmaximal, characteristic, complemented
3C2×#19098364C211yesmaximal, characteristic, complemented
4C2×#19098363C211yesmaximal, characteristic, complemented
5C2×C22⋊(C79⋊C16)C211yesmaximal, characteristic, complemented
6#23761491C211yesmaximal, characteristic, complemented
7C2×#466×C79C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C2×#14579135—21no—
10C2×#14579102—22no—
11C2×#14579102—22no—
12C2×#14579102—22no—
13C22×C79⋊C16C2211yescharacteristic
14C22×C79⋊C16C2211yescharacteristic
15#19098799—22no—
16#19098799—22no—
17C2×#14579095C2211yescharacteristic
18C2×#14579083—21no—
19#19098364—22no—
20#19098363—22no—
21C22⋊(C79⋊C16)—22no—
22C22×D8×C79C2211yescharacteristic
23C23×C8×C79C2211yescharacteristic
24C2×#121×C79—21no—
25#466×C79—22no—
26#466×C79—22no—
27C2×#107×C79C2211yescharacteristic
28C2×#102×C79—21no—
29C22×D316—21no—
30C22×C22⋊D79—41no—
31C2×D632—21no—
32C2×D632—21no—
33C2×D632—21no—
34C2×D632—21no—
35C2×D632—21no—
36C2×D632D412yescomplemented
37#14579135—24no—
38#14579135—42no—
39C2×C4⋊Dic79—21no—
40C2×(C2×D79)⋊C4—41no—
41#14579102—28no—
42C2×C79⋊C16—21no—
43C2×C79⋊C16—21no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16—21no—
46C2×C79⋊C16—21no—
47C2×C79⋊C16D412yescomplemented
48#14579095—21no—
49#14579095D412yescomplemented
50#14579083—24no—
51C22×D4×C79—21no—
52C22×D4×C79—41no—
53C23×C4×C79D411yescharacteristic
54C2×D8×C79—21no—
55C2×D8×C79—21no—
56C2×D8×C79—21no—
57C2×D8×C79—21no—
58C2×D8×C79—21no—
59C2×D8×C79D412yescomplemented
60C22×C8×C79—21no—
61C22×C8×C79—21no—
62C22×C8×C79—21no—
63C22×C8×C79—21no—
64C22×C8×C79—21no—
65C22×C8×C79—21no—
66C22×C8×C79D411yescharacteristic
67C22×C8×C79C2311yescharacteristic
68#121×C79—24no—
69#121×C79—42no—
70C2×C4⋊C4×C79—21no—
71C2×C22⋊C4×C79—41no—
72#107×C79—21no—
73#107×C79D412yescomplemented
74#102×C79—24no—
75C23×D79—41no—
76C2×D316—22no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×C22⋊D79—44no—
83C2×C22⋊D79—82no—
84C2×C22⋊D79—82no—
85C22×Dic79—41no—
86D632—22no—
87D632—22no—
88D632—41no—
89D632—41no—
90C4⋊Dic79—22no—
91C4⋊Dic79—41no—
92(C2×D79)⋊C4—44no—
93C79⋊C16—22no—
94C79⋊C16—41no—
95C24×C79—41no—
96C24×C79D811yescharacteristic
97C2×D4×C79—22no—
98C2×D4×C79—44no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—82no—
105C2×D4×C79—82no—
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—41no—
113C22×C4×C79D811yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115D8×C79—22no—
116D8×C79—22no—
117D8×C79—41no—
118D8×C79—41no—
119C2×C8×C79—21no—
120C2×C8×C79—21no—
121C2×C8×C79—21no—
122C2×C8×C79—41no—
123C2×C8×C79—41no—
124C2×C8×C79—41no—
125C2×C8×C79—41no—
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79C2×D411yescharacteristic
128C2×C8×C79C2×D411yescharacteristic
129C2×C8×C79C2×D411yescharacteristic
130C2×C8×C79C2×D411yescharacteristic
131C2×C8×C79#4111yescharacteristic
132C4⋊C4×C79—22no—
133C4⋊C4×C79—41no—
134C22⋊C4×C79—44no—
135C23×C79—21no—
136C23×C79—21no—
137C23×C79—21no—
138C23×C79—21no—
139C23×C79—21no—
140C23×C79—21no—
141C23×C79—42no—
142C23×C79—81no—
143C23×C79—81no—
144C23×C79—81no—
145C23×C79—81no—
146C23×C79—81no—
147C23×C79—81no—
148C23×C79D1612yescomplemented
149C23×C79C2×D811yescharacteristic
150C22×D79—42no—
151C22×D79—81no—
152C22×D79—81no—
153C22×D79—81no—
154C22×D79—81no—
155C22×D79—81no—
156C22×D79—81no—
157D4×C79—42no—
158D4×C79—42no—
159D4×C79—88no—
160D4×C79—81no—
161D4×C79—81no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—42no—
170C2×C4×C79—81no—
171C2×C4×C79—81no—
172C2×C4×C79#12011yescharacteristic
173C2×C4×C79#12011yescharacteristic
174C2×C4×C79#12111yescharacteristic
175C2×C4×C79#13511yescharacteristic
176C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
178C22⋊D79—88no—
179C2×Dic79—42no—
180C2×Dic79—81no—
181C2×Dic79—81no—
182D316—42no—
183D316—42no—
184D316—81no—
185D316—81no—
186C8×C79—41no—
187C8×C79#12011yescharacteristic
188C8×C79#12111yescharacteristic
189C8×C79#12111yescharacteristic
190C8×C79#12711yescharacteristic
191C22×C79—22no—
192C22×C79—22no—
193C22×C79—22no—
194C22×C79—41no—
195C22×C79—41no—
196C22×C79—41no—
197C22×C79—41no—
198C22×C79—82no—
199C22×C79—82no—
200C22×C79—82no—
201C22×C79—161no—
202C22×C79—161no—
203C22×C79—161no—
204C22×C79—161no—
205C22×C79#44711yescharacteristic
206C22×C79#44711yescharacteristic
207C22×C79#46611yescharacteristic
208C22×C79C2×D1611yescharacteristic
209C22×C79#50811yescharacteristic
210C22×C79#50911yescharacteristic
211C22×C79#50911yescharacteristic
212C2×D79—82no—
213C2×D79—82no—
214C2×D79—82no—
215C2×D79—161no—
216C2×D79—161no—
217C2×D79—161no—
218C2×D79—161no—
219C4×C79—41no—
220C4×C79—82no—
221C4×C79#39211yescharacteristic
222C4×C79#44911yescharacteristic
223C4×C79#44911yescharacteristic
224C4×C79#46911yescharacteristic
225Dic79—82no—
226#13304—791nomaximal, Sylow
227C2×C79—42no—
228C2×C79—162no—
229C2×C79#199911yescharacteristic
230C2×C79#218411yescharacteristic
231C2×C79#218411yescharacteristic
232C2×C79#218611yescharacteristic
233C2×C79#218611yescharacteristic
234C2×C79#221511yescharacteristic
235C2×C79#221811yescharacteristic
236D79—162no—
237C22×D16—791no—
238C2×#466—791no—
239C2×#466D7911yescharacteristic, complemented
240C2×#357—791no—
241C2×#357—791no—
242C2×C22⋊C16—791no—
243#1380—791no—
244C79#1330411yescharacteristic, complemented, Sylow
245C22×D8—791no—
246C22×D8C2×D7911yescharacteristic
247C23×C8C2×D7911yescharacteristic
248C2×#121—21no—
249C2×#121—1581no—
250C2×D16—1582no—
251C2×D16—1582no—
252C2×D16—1582no—
253C22×C16—791no—
254C22×C16—791no—
255#466—22no—
256#466—22no—
257#466—1582no—
258#466—1582no—
259C2×#107—791no—
260C2×#107C2×D7911yescharacteristic
261C2×#102—21no—
262C2×#102—1581no—
263#357—1582no—
264#357—1582no—
265C22⋊C16—1582no—
266C22×D4—21no—
267C22×D4—41no—
268C22×D4—3161no—
269C22×D4—1581no—
270C23×C4C22⋊D7911yescharacteristic
271C2×D8—21no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—792no—
277C2×D8—1581no—
278C2×D8—1581no—
279C2×D8—1581no—
280C2×D8—1581no—
281C2×D8—1581no—
282C2×D8C22⋊D7912yescomplemented
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8C22⋊D7911yescharacteristic
290C22×C8C22×D7911yescharacteristic
291#121—24no—
292#121—42no—
293#121—3162no—
294#121—1584no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—1581no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—3161no—
299D16—1588no—
300C2×C16—792no—
301C2×C16—1581no—
302C2×C16—1581no—
303C2×C16—1581no—
304C2×C16—1581no—
305C2×C16—1581no—
306#107—21no—
307#107—792no—
308#107—1581no—
309#107C22⋊D7912yescomplemented
310#102—24no—
311#102—1584no—
312C24—41no—
313C24—3161no—
314C24#1192960911yescharacteristic
315C2×D4—22no—
316C2×D4—44no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—3164no—
325C2×D4—3161no—
326C2×D4—3161no—
327C2×D4—3161no—
328C2×D4—3161no—
329C2×D4—3161no—
330C2×D4—6322no—
331C2×D4—6322no—
332C2×D4—1582no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—3161no—
341C22×C4#1192960911yescharacteristic
342C22×C4C2×C22⋊D7911yescharacteristic
343D8—22no—
344D8—22no—
345D8—41no—
346D8—41no—
347D8—3161no—
348D8—3161no—
349D8—1582no—
350D8—1582no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8D4×D7911yescharacteristic
359C2×C8D4×D7911yescharacteristic
360C2×C8D4×D7911yescharacteristic
361C2×C8C2×C22⋊D7911yescharacteristic
362C2×C8C2×C22⋊D7911yescharacteristic
363C2×C8#1192962311yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—3161no—
367C4⋊C4—1582no—
368C22⋊C4—44no—
369C22⋊C4—3164no—
370C16—3161no—
371C16—1582no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—3162no—
386C23—6321no—
387C23—6321no—
388C23—6321no—
389C23—6321no—
390C23—6321no—
391C23—6321no—
392C23#1457910212yescomplemented
393C23C2×#1192960911yescharacteristic
394D4—42no—
395D4—42no—
396D4—88no—
397D4—81no—
398D4—81no—
399D4—3162no—
400D4—3162no—
401D4—6328no—
402D4—6321no—
403D4—6321no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—3162no—
415C2×C4—6321no—
416C2×C4—6321no—
417C2×C4#1457912911yescharacteristic
418C2×C4#1457913011yescharacteristic
419C2×C4#1457913311yescharacteristic
420C2×C4#1457918111yescharacteristic
421C2×C4#1457918611yescharacteristic
422C2×C4#1457918711yescharacteristic
423C8—41no—
424C8#1457912911yescharacteristic
425C8#1457913211yescharacteristic
426C8#1457913311yescharacteristic
427C8#1457915711yescharacteristic
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—6322no—
443C22—6322no—
444C22—6322no—
445C22—12641no—
446C22—12641no—
447C22—12641no—
448C22—12641no—
449C22#1909869611yescharacteristic
450C22#1909869711yescharacteristic
451C22#1909879711yescharacteristic
452C22C2×#1457910211yescharacteristic
453C22#1909899311yescharacteristic
454C22#1909899811yescharacteristic
455C22#1909899911yescharacteristic
456C4—41no—
457C4—82no—
458C4—6322no—
459C4#1909847111yescharacteristic
460C4#1909870611yescharacteristic
461C4#1909870811yescharacteristic
462C4#1909881411yescharacteristic
463C2—42no—
464C2—162no—
465C2—12642no—
466C2#2376439611yescharacteristic
467C2#2376531111yescharacteristic
468C2#2376531211yescharacteristic
469C2#2376532111yescharacteristic
470C2#2376532311yescharacteristic
471C2#2376548411yescharacteristic
472C2#2376550111yescharacteristic
subgroupG/H
centerC23C2×#11929609
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC2×#466×C79C2
socleC23×C79C2×D8
hypercenterC2×#466D79
hypocenterC79#13304
LCSG ▷ C22×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#14579102C2semidirect
2. C2×#19098799C2semidirect
3. C2×#19098364C2semidirect
4. C2×#19098363C2semidirect
5. C2×C22⋊(C79⋊C16)C2semidirect
6. #23761491C2semidirect
7. C2×#466×C79C2semidirect
36. C2×D632D4semidirect
47. C2×C79⋊C16D4semidirect
49. #14579095D4semidirect
59. C2×D8×C79D4semidirect
73. #107×C79D4semidirect
148. C23×C79D16semidirect
239. C2×#466D79semidirect
244. C79#13304semidirect
282. C2×D8C22⋊D79semidirect
309. #107C22⋊D79semidirect
392. C23#14579102semidirect

Maximal subgroups

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

subgroupindex
C22×#145791022
C2×#190987992
C2×#190983642
C2×#190983632
C2×C22⋊(C79⋊C16)2
#237614912
C2×#466×C792
#1330479