Finite Group Databasebeta

← #34183165

Subgroups — #34183165

Order 20224.

484 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772066C211yesmaximal, characteristic, complemented
2#23761851C211yesmaximal, characteristic, complemented
3#23761799C211yesmaximal, characteristic, complemented
4#23761782C211yesmaximal, characteristic, complemented
5#23761297C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#3035C211yesmaximal, characteristic
8#19098624—22no—
9#19098620—21no—
10#19098616—21no—
11#19098616C2211yescharacteristic
12#19098587C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14#19098530—21no—
15#19098519—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17Q8⋊(C79⋊C8)—21no—
18D4⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—22no—
21D4⋊(C79⋊C8)—21no—
22C2×#135×C79—21no—
23#532×C79C2211yescharacteristic
24C4×#41×C79C2211yescharacteristic
25#464×C79—21no—
26#463×C79—21no—
27#462×C79C411yescharacteristic
28#461×C79C411yescharacteristic
29#460×C79C411yescharacteristic
30#459×C79C411yescharacteristic
31C2×C4⋊C8×C79C2211yescharacteristic
32#416×C79—21no—
33C2×#11929608—21no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)D412yes—
40C4⋊(C79⋊C8)D411yescharacteristic
41C4⋊(C79⋊C8)D411yescharacteristic
42C22⋊(C79⋊C8)—22no—
43C22⋊(C79⋊C8)—22no—
44C22⋊(C79⋊C8)—21no—
45#14579072—22no—
46C4×C79⋊C8—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C2×#41×C79—21no—
51C2×#41×C79D411yescharacteristic
52#135×C79—42no—
53#135×C79—42no—
54C2×Q16×C79—21no—
55C2×SD16×C79—21no—
56C2×SD16×C79—21no—
57C2×D8×C79—21no—
58C22×C8×C79—21no—
59C4⋊Q8×C79C2×C411yescharacteristic
60#127×C79C2×C411yescharacteristic
61#124×C79—21no—
62#121×C79—21no—
63#121×C79—21no—
64C4×Q8×C79C2×C411yescharacteristic
65C4×Q8×C79D411yescharacteristic
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79C2×C411yescharacteristic
71C4×D4×C79D411yescharacteristic
72#117×C79—21no—
73#117×C79D411yescharacteristic
74C2×C42×C79—21no—
75C2×C42×C79C2311yescharacteristic
76C4⋊C8×C79—21no—
77C4⋊C8×C79C2×C411yescharacteristic
78C4⋊C8×C79C2×C411yescharacteristic
79#103×C79—21no—
80#103×C79—21no—
81#102×C79—21no—
82#102×C79—21no—
83#11929608—41no—
84#11929608—41no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94#41×C79—21no—
95#41×C79—21no—
96#41×C79—21no—
97#41×C79—21no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101#41×C79—41no—
102C2×Q8×C79—21no—
103C2×Q8×C79C22⋊C411yescharacteristic
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79—21no—
108C2×D4×C79—41no—
109C2×D4×C79C22⋊C411yescharacteristic
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—41no—
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79#4111yescharacteristic
116C22×C4×C79#4111yescharacteristic
117Q16×C79—42no—
118SD16×C79—42no—
119SD16×C79—42no—
120D8×C79—42no—
121C2×C8×C79—21no—
122C2×C8×C79—21no—
123C2×C8×C79—41no—
124C2×C8×C79—41no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79C22⋊C411yescharacteristic
130C4⋊C4×C79C22⋊C411yescharacteristic
131C22⋊C4×C79—21no—
132C22⋊C4×C79—21no—
133C22⋊C4×C79—21no—
134C22⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79—21no—
137C42×C79—21no—
138C42×C79—21no—
139C42×C79C22×C411yescharacteristic
140C42×C79C2×D411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C23×C79—21no—
144C23×C79—41no—
145C23×C79#12311yescharacteristic
146Q8×C79—21no—
147Q8×C79—21no—
148Q8×C79—41no—
149Q8×C79—41no—
150D4×C79—21no—
151D4×C79—21no—
152D4×C79—42no—
153D4×C79—41no—
154D4×C79—41no—
155D4×C79—41no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—81no—
160D4×C79—81no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—81no—
173C2×C4×C79—81no—
174C2×C4×C79—81no—
175C2×C4×C79C2×C22⋊C411yescharacteristic
176C2×C4×C79C4×D411yescharacteristic
177C2×C4×C79C4×D411yescharacteristic
178C2×C4×C79#12011yescharacteristic
179C2×C4×C79#12111yescharacteristic
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#13511yescharacteristic
182C2×C4×C79#13511yescharacteristic
183C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
185C8×C79—41no—
186C8×C79—41no—
187C79⋊C8—42no—
188C79⋊C8—41no—
189C79⋊C8—41no—
190C79⋊C8—42no—
191C79⋊C8—81no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—41no—
195C22×C79—41no—
196C22×C79—41no—
197C22×C79—81no—
198C22×C79—81no—
199C22×C79—81no—
200C22×C79#38611yescharacteristic
201C22×C79#48211yescharacteristic
202C22×C79#48211yescharacteristic
203C4×C79—21no—
204C4×C79—21no—
205C4×C79—21no—
206C4×C79—41no—
207C4×C79—41no—
208C4×C79—41no—
209C4×C79—81no—
210C4×C79#41611yescharacteristic
211C4×C79#41811yescharacteristic
212C4×C79#44311yescharacteristic
213C4×C79#44411yescharacteristic
214C4×C79#44911yescharacteristic
215C4×C79#44911yescharacteristic
216#8996—791nomaximal, Sylow
217C2×C79—21no—
218C2×C79—21no—
219C2×C79—81no—
220C2×C79—161no—
221C2×C79#187511yescharacteristic
222C2×C79#187611yescharacteristic
223C2×C79#188111yescharacteristic
224#3035D7911yescharacteristic
225#2925—791no—
226#1492—791no—
227#1480—791no—
228#1476—791no—
229#1301—791no—
230#1277—791no—
231C79#899611yescharacteristic, complemented, Sylow
232C2×#135—21no—
233#532C2×D7911yescharacteristic
234C4×#41C2×D7911yescharacteristic
235#464—21no—
236#463—21no—
237#462Dic7911yescharacteristic
238#461Dic7911yescharacteristic
239#460Dic7911yescharacteristic
240#459Dic7911yescharacteristic
241#433—1582no—
242#432—1581no—
243#431—791no—
244#431—1581no—
245#423—791no—
246C2×C4⋊C8—791no—
247C2×C4⋊C8C2×D7911yescharacteristic
248#416—21no—
249#408—1581no—
250#405—1581no—
251C2×C4×C8—791no—
252Q8⋊C8—1581no—
253D4⋊C8—1581no—
254D4⋊C8—1581no—
255D4⋊C8—1582no—
256D4⋊C8—1581no—
257C2×#41—21no—
258C2×#41C22⋊D7911yescharacteristic
259#135—42no—
260#135—42no—
261C2×Q16—21no—
262C2×SD16—21no—
263C2×SD16—21no—
264C2×D8—21no—
265C2×M16—1581no—
266C22×C8—21no—
267C22×C8—1581no—
268C22×C8—1581no—
269C4⋊Q8C2×Dic7911yescharacteristic
270#127C2×Dic7911yescharacteristic
271#124—21no—
272#121—21no—
273#121—21no—
274C4×Q8C2×Dic7911yescharacteristic
275C4×Q8C22⋊D7911yescharacteristic
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4C2×Dic7911yescharacteristic
281C4×D4C22⋊D7911yescharacteristic
282#117—21no—
283#117C22⋊D7911yescharacteristic
284C2×C42—21no—
285C2×C42C22×D7911yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—791no—
288C4⋊C8—791no—
289C4⋊C8—792no—
290C4⋊C8—1581no—
291C4⋊C8—1581no—
292C4⋊C8—1581no—
293C4⋊C8C2×Dic7911yescharacteristic
294C4⋊C8C2×Dic7911yescharacteristic
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—1581no—
300C22⋊C8—1582no—
301C22⋊C8—1582no—
302#97—1582no—
303C4×C8—1582no—
304C4×C8—1581no—
305C4×C8—1581no—
306C4×C8—1581no—
307#41—21no—
308#41—21no—
309#41—21no—
310#41—21no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315C2×Q8—21no—
316C2×Q8C22⋊Dic7911yescharacteristic
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4C22⋊Dic7911yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4C2×C22⋊D7911yescharacteristic
328C22×C4#1192962311yescharacteristic
329C22×C4#1192962311yescharacteristic
330Q16—42no—
331SD16—42no—
332SD16—42no—
333D8—42no—
334M16—3161no—
335M16—3161no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—3161no—
341C2×C8—3161no—
342C2×C8—3161no—
343C2×C8—3161no—
344C2×C8—3161no—
345C2×C8—1582no—
346C2×C8—1581no—
347C2×C8—1582no—
348C2×C8—1581no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4C22⋊Dic7911yescharacteristic
354C4⋊C4C22⋊Dic7911yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42C22×Dic7911yescharacteristic
364C42D4×D7911yescharacteristic
365C42D4×D7911yescharacteristic
366C42C2×C22⋊D7911yescharacteristic
367C23—21no—
368C23—41no—
369C23#1457914211yescharacteristic
370Q8—21no—
371Q8—21no—
372Q8—41no—
373Q8—41no—
374D4—21no—
375D4—21no—
376D4—42no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382D4—41no—
383D4—81no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4C2×C22⋊Dic7911yescharacteristic
400C2×C4D4×Dic7911yescharacteristic
401C2×C4D4×Dic7911yescharacteristic
402C2×C4#1457912911yescharacteristic
403C2×C4#1457913311yescharacteristic
404C2×C4#1457913311yescharacteristic
405C2×C4#1457918111yescharacteristic
406C2×C4#1457918111yescharacteristic
407C2×C4#1457918711yescharacteristic
408C2×C4#1457918711yescharacteristic
409C8—41no—
410C8—41no—
411C8—3162no—
412C8—3161no—
413C8—3161no—
414C8—3162no—
415C8—6321no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#1909844311yescharacteristic
425C22#1909888411yescharacteristic
426C22#1909888411yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4#1909855911yescharacteristic
435C4#1909856811yescharacteristic
436C4#1909868011yescharacteristic
437C4#1909868511yescharacteristic
438C4#1909870611yescharacteristic
439C4#1909870611yescharacteristic
440C2—21no—
441C2—21no—
442C2—81no—
443C2—161no—
444C2#2376372311yescharacteristic
445C2#2376372911yescharacteristic
446C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3035C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3035D79
hypocenterC79#8996
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #23772066C2semidirect
2. #23761851C2semidirect
3. #23761799C2semidirect
4. #23761782C2semidirect
5. #23761297C2semidirect
6. #23761234C2semidirect
231. C79#8996semidirect

Maximal subgroups

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

subgroupindex
#237720662
#237618512
#237617992
#237617822
#237612972
#237612342
C79×#30352
#899679