Finite Group Databasebeta

← #34177544

Subgroups — #34177544

Order 20224.

516 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764963C211yesmaximal, characteristic, complemented
2#23764933C211yesmaximal, characteristic, complemented
3#23764933C211yesmaximal, characteristic, complemented
4#23761223C211yesmaximal, characteristic, complemented
5#23761221C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic, complemented
7C79×#2682C211yesmaximal, characteristic
8C2×#14579092—21no—
9C2×#14579092—21no—
10C2×#14579082—21no—
11C2×#14579082—21no—
12#19098433C2211yescharacteristic
13#19098325—21no—
14#19098325—21no—
15#19098318C2211yescharacteristic
16#19098318C2211yescharacteristic
17#19098318C2211yescharacteristic
18C2×#123×C79—21no—
19C2×#121×C79—21no—
20#396×C79—21no—
21#394×C79—21no—
22#392×C79—21no—
23#390×C79C2211yescharacteristic
24#390×C79C2211yescharacteristic
25#389×C79C2211yescharacteristic
26#385×C79—21no—
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32C2×C4⋊Dic79—21no—
33#14579092—42no—
34#14579092—42no—
35#14579082—42no—
36#14579082—42no—
37C22×D4×C79—21no—
38C22×D4×C79—21no—
39C22×D4×C79D411yescharacteristic
40C22×D4×C79D411yescharacteristic
41C23×C4×C79—21no—
42#123×C79—42no—
43#123×C79—42no—
44#121×C79—42no—
45#121×C79—42no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C22⋊C4×C79—21no—
58C2×C22⋊C4×C79—21no—
59C2×C42×C79C2311yescharacteristic
60#95×C79—21no—
61#95×C79—21no—
62C22×Dic79—41no—
63C2×C79⋊C8—22no—
64C2×C79⋊C8—22no—
65C2×C79⋊C8—41no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C4⋊Dic79—41no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79—42no—
77C24×C79—21no—
78C24×C79—21no—
79C2×D4×C79—21no—
80C2×D4×C79—21no—
81C2×D4×C79—21no—
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79—41no—
86C2×D4×C79—42no—
87C2×D4×C79—42no—
88C2×D4×C79—42no—
89C2×D4×C79—81no—
90C2×D4×C79—81no—
91C2×D4×C79—81no—
92C2×D4×C79SD1612yes—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—42no—
98C22×C4×C79—42no—
99C22×C4×C79—81no—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C4⋊C4×C79—21no—
104C4⋊C4×C79—42no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79—41no—
107C4⋊C4×C79—41no—
108C4⋊C4×C79—41no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79SD1612yes—
111C22⋊C4×C79—42no—
112C22⋊C4×C79—42no—
113C22⋊C4×C79—41no—
114C22⋊C4×C79—41no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C22⋊C4×C79—41no—
120C22⋊C4×C79—41no—
121C22⋊C4×C79—81no—
122C42×C79—41no—
123C23×C79—22no—
124C23×C79—41no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79—41no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79—41no—
134C23×C79—41no—
135C23×C79—41no—
136C23×C79—41no—
137C23×C79#12011yescharacteristic
138D4×C79—42no—
139D4×C79—42no—
140D4×C79—81no—
141D4×C79—81no—
142D4×C79—84no—
143D4×C79—84no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—22no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—84no—
163C2×C4×C79C2×SD1611yescharacteristic
164C2×C4×C79C2×SD1611yescharacteristic
165C2×C4×C79#13511yescharacteristic
166C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C79#13711yescharacteristic
169C2×Dic79—81no—
170C2×Dic79—81no—
171C2×Dic79—81no—
172C79⋊C8—42no—
173C79⋊C8—42no—
174C79⋊C8—81no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79—42no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79—81no—
182C22×C79—81no—
183C22×C79—81no—
184C22×C79—81no—
185C22×C79—81no—
186C22×C79—81no—
187C22×C79—81no—
188C22×C79—81no—
189C22×C79#45011yescharacteristic
190C22×C79#45011yescharacteristic
191C22×C79#45211yescharacteristic
192C22×C79#45311yescharacteristic
193C22×C79#45311yescharacteristic
194C22×C79#45411yescharacteristic
195C22×C79#45811yescharacteristic
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—21no—
200C4×C79—21no—
201C4×C79—21no—
202C4×C79—42no—
203C4×C79—81no—
204C4×C79—81no—
205C4×C79—81no—
206Dic79—161no—
207#7825—791nomaximal, Sylow
208C2×C79—82no—
209C2×C79—161no—
210C2×C79#161311yescharacteristic
211C2×C79#161611yescharacteristic
212C2×C79#161811yescharacteristic
213C2×C79#162911yescharacteristic
214C2×C79#163111yescharacteristic
215C2×C79#163511yescharacteristic
216C2×C79#165611yescharacteristic
217#2682D7911yescharacteristic
218#2100—791no—
219#2094—791no—
220#2094—791no—
221#1271—791no—
222#1270—791no—
223#1270—791no—
224C79#782511yescharacteristic, complemented, Sylow
225C2×#123—21no—
226C2×#121—21no—
227C2×#106—1581no—
228C2×#106—1581no—
229C2×#102—1581no—
230C2×#102—1581no—
231#396—21no—
232#394—21no—
233#392—21no—
234#390C2×D7911yescharacteristic
235#390C2×D7911yescharacteristic
236#389C2×D7911yescharacteristic
237#385—21no—
238#384—791no—
239#340—1581no—
240#340—1581no—
241#336—791no—
242#336—791no—
243#336—791no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C22⋊D7911yescharacteristic
247C22×D4C22⋊D7911yescharacteristic
248C23×C4—21no—
249C22×C8—1581no—
250C22×C8—1581no—
251C22×C8—1581no—
252#123—42no—
253#123—42no—
254#121—42no—
255#121—42no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—1581no—
259C2×C4⋊C4—1581no—
260C2×C4⋊C4—1581no—
261C2×C4⋊C4C22⋊D7911yescharacteristic
262C2×C4⋊C4C22⋊D7911yescharacteristic
263C2×C4⋊C4C22⋊D7911yescharacteristic
264C2×C4⋊C4C22⋊D7911yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42C22×D7911yescharacteristic
273#106—3162no—
274#106—3162no—
275#102—3162no—
276#102—3162no—
277#95—21no—
278#95—21no—
279C24—21no—
280C24—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4#1192961212yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—81no—
302C22×C4—3161no—
303C22×C4C2×C22⋊D7911yescharacteristic
304C22×C4C2×C22⋊D7911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306C2×C8—3161no—
307C2×C8—3161no—
308C2×C8—3161no—
309C2×C8—3161no—
310C2×C8—3161no—
311C2×C8—3161no—
312C2×C8—3161no—
313C2×C8—1582no—
314C2×C8—1582no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—3161no—
323C4⋊C4—3161no—
324C4⋊C4—3161no—
325C4⋊C4—3161no—
326C4⋊C4—3162no—
327C4⋊C4#1192961212yes—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C42—41no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#1457913011yescharacteristic
355D4—42no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—84no—
360D4—84no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—84no—
380C2×C4—6321no—
381C2×C4—6321no—
382C2×C4—6321no—
383C2×C4C2×#1192961211yescharacteristic
384C2×C4C2×#1192961211yescharacteristic
385C2×C4#1457918111yescharacteristic
386C2×C4#1457918711yescharacteristic
387C2×C4#1457918711yescharacteristic
388C2×C4#1457919111yescharacteristic
389C8—3162no—
390C8—3162no—
391C8—6321no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1909871411yescharacteristic
407C22#1909871411yescharacteristic
408C22#1909872511yescharacteristic
409C22#1909873111yescharacteristic
410C22#1909873111yescharacteristic
411C22#1909873611yescharacteristic
412C22#1909875311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—12641no—
424C2—82no—
425C2—161no—
426C2#2376242011yescharacteristic
427C2#2376243711yescharacteristic
428C2#2376244811yescharacteristic
429C2#2376251611yescharacteristic
430C2#2376252711yescharacteristic
431C2#2376255111yescharacteristic
432C2#2376269311yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#2682C2
socleC23×C79#120
hypercenter#2682D79
hypocenterC79#7825
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2682

Complements

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

NH ≅ G/Ntype
1. #23764963C2semidirect
2. #23764933C2semidirect
3. #23764933C2semidirect
4. #23761223C2semidirect
5. #23761221C2semidirect
6. #23761221C2semidirect
224. C79#7825semidirect

Maximal subgroups

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

subgroupindex
#237649632
#237649332
#237612232
#237612212
C79×#26822
#782579