Finite Group Databasebeta

← #34189034

Subgroups — #34189034

Order 20224.

568 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 101 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#23773408C211yesmaximal, characteristic
2#23773205C211yesmaximal, characteristic, complemented
3#23761963C211yesmaximal, characteristic, complemented
4#23761825C211yesmaximal, characteristic
5#23761795C211yesmaximal, characteristic, complemented
6#23761258C211yesmaximal, characteristic, complemented
7C79×#2536C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098893—21no—
10#19098870—22no—
11#19098858—21no—
12#19098853—21no—
13#19098848—21no—
14#19098768—21no—
15#19098714—21no—
16C4×#11929612—22no—
17#19098616C2211yescharacteristic
18#19098616C2211yescharacteristic
19#19098603—21no—
20#19098587C2211yescharacteristic
21#19098572—21no—
22#19098299—22no—
23Q8⋊(C79⋊C8)—22no—
24D4⋊(C79⋊C8)—22no—
25C2×C22⋊Q8×C79—21no—
26C2×C4×Q8×C79C2211yescharacteristic
27C2×C4×D4×C79C2211yescharacteristic
28#398×C79—21no—
29#394×C79—21no—
30#388×C79—22no—
31#382×C79C2211yescharacteristic
32#380×C79—21no—
33C2×#11929612—22no—
34C2×#11929608—21no—
35#14579161D411yescharacteristic
36#14579161D411yescharacteristic
37#14579153—21no—
38#14579140—21no—
39#14579140—21no—
40C4×Dic158—21no—
41C4×Dic158—21no—
42#14579120—21no—
43#14579094—22no—
44#14579092—22no—
45#14579092—22no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)D412yes—
50#14579085—22no—
51#14579085—22no—
52#14579084—22no—
53#14579082—22no—
54C22⋊(C79⋊C8)—21no—
55C22⋊(C79⋊C8)—21no—
56C4×C79⋊C8—21no—
57C4×C79⋊C8D412yes—
58C22×Q8×C79D411yescharacteristic
59C22×D4×C79D411yescharacteristic
60C23×C4×C79—21no—
61C22⋊Q8×C79—42no—
62C22⋊Q8×C79—42no—
63C4×Q8×C79—21no—
64C4×Q8×C79—21no—
65C4×Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79—21no—
71C2×C4⋊C4×C79—22no—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79—21no—
74C2×C4⋊C4×C79D411yescharacteristic
75C2×C4⋊C4×C79D411yescharacteristic
76C2×C22⋊C4×C79—22no—
77C2×C22⋊C4×C79—22no—
78C2×C22⋊C4×C79—21no—
79C2×C42×C79—21no—
80C2×C42×C79C2311yescharacteristic
81#95×C79—22no—
82#95×C79—21no—
83#95×C79D412yes—
84C2×Dic158—21no—
85C2×Dic158—21no—
86#11929612—44no—
87#11929608—41no—
88#11929608—41no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—41no—
92C4⋊Dic79—21no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96Dic79⋊C4—41no—
97Dic79⋊C4—41no—
98C22⋊Dic79—41no—
99C4×Dic79—41no—
100C24×C79—21no—
101C2×Q8×C79—21no—
102C2×Q8×C79—21no—
103C2×Q8×C79—21no—
104C2×Q8×C79—21no—
105C2×Q8×C79—21no—
106C2×Q8×C79SD1612yescomplemented
107C2×D4×C79—21no—
108C2×D4×C79—21no—
109C2×D4×C79—42no—
110C22×C4×C79—21no—
111C22×C4×C79—22no—
112C22×C4×C79—21no—
113C22×C4×C79—21no—
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×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C4⋊C4×C79—21no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—44no—
128C4⋊C4×C79—41no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79—41no—
131C4⋊C4×C79—41no—
132C22⋊C4×C79—44no—
133C22⋊C4×C79—44no—
134C22⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79—41no—
137C42×C79—41no—
138C42×C79C2×D411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C42×C79C2×D411yescharacteristic
141C42×C79#4111yescharacteristic
142C23×C79—22no—
143C23×C79—41no—
144C23×C79—41no—
145C23×C79—41no—
146C23×C79—41no—
147C23×C79—41no—
148C23×C79—41no—
149C23×C79#12011yescharacteristic
150Q8×C79—22no—
151Q8×C79—22no—
152Q8×C79—41no—
153Q8×C79—41no—
154D4×C79—22no—
155D4×C79—22no—
156D4×C79—41no—
157D4×C79—41no—
158C2×C4×C79—22no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—42no—
164C2×C4×C79—42no—
165C2×C4×C79—42no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—42no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—81no—
172C2×C4×C79—81no—
173C2×C4×C79—81no—
174C2×C4×C79—81no—
175C2×C4×C79#12011yescharacteristic
176C2×C4×C79#12011yescharacteristic
177C2×C4×C79#12111yescharacteristic
178C2×C4×C79#12111yescharacteristic
179C2×C4×C79#12111yescharacteristic
180C2×C4×C79#12711yescharacteristic
181C2×C4×C79C2×SD1611yescharacteristic
182C2×C4×C79#13511yescharacteristic
183C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C79#13711yescharacteristic
185C2×Dic79—41no—
186C2×Dic79—41no—
187Dic158—42no—
188Dic158—81no—
189C79⋊C8—42no—
190C79⋊C8—42no—
191C79⋊C8—81no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—42no—
195C22×C79—42no—
196C22×C79—42no—
197C22×C79—81no—
198C22×C79—81no—
199C22×C79—81no—
200C22×C79—81no—
201C22×C79#39211yescharacteristic
202C22×C79#45011yescharacteristic
203C22×C79#45211yescharacteristic
204C4×C79—21no—
205C4×C79—21no—
206C4×C79—21no—
207C4×C79—21no—
208C4×C79—42no—
209C4×C79—82no—
210C4×C79—81no—
211C4×C79—81no—
212C4×C79#44811yescharacteristic
213C4×C79#44911yescharacteristic
214C4×C79#46511yescharacteristic
215C4×C79#46811yescharacteristic
216Dic79—81no—
217Dic79—81no—
218#10001—791nomaximal, Sylow
219C2×C79—21no—
220C2×C79—21no—
221C2×C79—82no—
222C2×C79#200011yescharacteristic
223C2×C79#200911yescharacteristic
224C2×C79#201011yescharacteristic
225#3101—791no—
226#3078—791no—
227#2536D7911yescharacteristic
228#1518—791no—
229#1486—791no—
230#1479—791no—
231#1286—791no—
232C79#1000111yescharacteristic, complemented, Sylow
233#533—791no—
234C2×C22⋊Q8—21no—
235C2×C4×Q8C2×D7911yescharacteristic
236C2×C4×D4C2×D7911yescharacteristic
237#483—1581no—
238#479—1582no—
239#477—1581no—
240#476—1581no—
241#475—1581no—
242#461—1581no—
243#450—1581no—
244C4×SD16—1582no—
245#431—791no—
246#431—791no—
247#428—1581no—
248#423—791no—
249#419—1581no—
250#398—21no—
251#394—21no—
252#388—22no—
253#382C2×D7911yescharacteristic
254#380—21no—
255#328—1582no—
256Q8⋊C8—1582no—
257D4⋊C8—1582no—
258C22×Q8C22⋊D7911yescharacteristic
259C22×D4C22⋊D7911yescharacteristic
260C23×C4—21no—
261C2×SD16—1582no—
262C2×M16—1581no—
263C4⋊Q8—791no—
264C4⋊Q8—791no—
265#125—1581no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—1581no—
269C22⋊Q8—1581no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—1581no—
275C4×Q8—1581no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280#117—1581no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22⋊D7911yescharacteristic
285C2×C4⋊C4C22⋊D7911yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C42—21no—
290C2×C42C22×D7911yescharacteristic
291#107—1582no—
292#106—1582no—
293#106—1582no—
294C4⋊C8—792no—
295C4⋊C8—1581no—
296C4⋊C8—1581no—
297C4⋊C8—1581no—
298#103—1582no—
299#103—1582no—
300#103—1582no—
301#102—1582no—
302C22⋊C8—1581no—
303C22⋊C8—1581no—
304C4×C8—792no—
305C4×C8—1581no—
306#95—22no—
307#95—21no—
308#95C22⋊D7912yes—
309C24—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—1581no—
316C2×Q8—1581no—
317C2×Q8#1192961212yescomplemented
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×C22⋊D7911yescharacteristic
332C22×C4C2×C22⋊D7911yescharacteristic
333C22×C4C2×C22⋊D7911yescharacteristic
334SD16—3164no—
335M16—3161no—
336M16—3161no—
337C2×C8—3161no—
338C2×C8—1582no—
339C2×C8—1582no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—3161no—
350C4⋊C4—3161no—
351C4⋊C4—1581no—
352C4⋊C4—1581no—
353C4⋊C4—1581no—
354C4⋊C4—1581no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—3161no—
360C42—41no—
361C42—41no—
362C42—3161no—
363C42D4×D7911yescharacteristic
364C42D4×D7911yescharacteristic
365C42D4×D7911yescharacteristic
366C42#1192962311yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23#1457913011yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—41no—
378Q8—41no—
379Q8—3162no—
380Q8—6321no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—3161no—
403C2×C4—3161no—
404C2×C4#1457912911yescharacteristic
405C2×C4#1457912911yescharacteristic
406C2×C4#1457913211yescharacteristic
407C2×C4#1457913311yescharacteristic
408C2×C4#1457913311yescharacteristic
409C2×C4#1457915711yescharacteristic
410C2×C4C2×#1192961211yescharacteristic
411C2×C4#1457918111yescharacteristic
412C2×C4#1457918711yescharacteristic
413C2×C4#1457919111yescharacteristic
414C8—3162no—
415C8—3162no—
416C8—6321no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#1909847111yescharacteristic
427C22#1909871411yescharacteristic
428C22#1909872511yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—6321no—
438C4—6321no—
439C4#1909870111yescharacteristic
440C4#1909870611yescharacteristic
441C4#1909879011yescharacteristic
442C4#1909880711yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2#2376440311yescharacteristic
447C2#2376446011yescharacteristic
448C2#2376446511yescharacteristic
subgroupG/H
centerC22#19098471
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2536C2
socleC22×C79#392
prime radicalC22×D4×C79D4
hypercenter#2536D79
hypocenterC79#10001
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #23773205C2semidirect
3. #23761963C2semidirect
5. #23761795C2semidirect
6. #23761258C2semidirect
106. C2×Q8×C79SD16semidirect
232. C79#10001semidirect
317. C2×Q8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237734082
#237732052
#237619632
#237618252
#237617952
#237612582
C79×#25362
#1000179