Finite Group Databasebeta

← C83×#7778 #35235602

Subgroups — C83×#7778 #35235602

Order 21248.

574 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C83×#2682C211yesmaximal, characteristic
2C83×#2058C211yesmaximal, characteristic, complemented
3C83×#2023C211yesmaximal, characteristic, complemented
4C83×#2013C211yesmaximal, characteristic, complemented
5C83×#1271C211yesmaximal, characteristic, complemented
6C83×#1270C211yesmaximal, characteristic, complemented
7C83×#1270C211yesmaximal, characteristic, complemented
8C22×SD16×C83—21no—
9C2×C4⋊Q8×C83C2211yescharacteristic
10C2×#123×C83—21no—
11C2×#121×C83—21no—
12C2×#106×C83—21no—
13C2×#103×C83—21no—
14C2×#102×C83—21no—
15#396×C83—21no—
16#394×C83—21no—
17#392×C83—21no—
18#390×C83C2211yescharacteristic
19#390×C83C2211yescharacteristic
20#389×C83C2211yescharacteristic
21#385×C83—21no—
22#340×C83—21no—
23#340×C83—21no—
24#336×C83C2211yescharacteristic
25#336×C83C2211yescharacteristic
26#336×C83C2211yescharacteristic
27C22×Q8×C83—21no—
28C22×D4×C83—21no—
29C22×D4×C83—21no—
30C22×D4×C83D411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83—21no—
33C2×SD16×C83—42no—
34C2×SD16×C83—42no—
35C2×SD16×C83—42no—
36C22×C8×C83—21no—
37C22×C8×C83—21no—
38C22×C8×C83—21no—
39C4⋊Q8×C83—42no—
40#123×C83—42no—
41#123×C83—42no—
42#121×C83—42no—
43#121×C83—42no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83—21no—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—21no—
56C2×C22⋊C4×C83—21no—
57C2×C22⋊C4×C83—21no—
58C2×C22⋊C4×C83—21no—
59C2×C42×C83C2311yescharacteristic
60#106×C83—42no—
61#103×C83—42no—
62#102×C83—42no—
63#95×C83—21no—
64#95×C83—21no—
65C24×C83—21no—
66C24×C83—21no—
67C2×Q8×C83—22no—
68C2×Q8×C83—41no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×Q8×C83—41no—
72C2×Q8×C83—41no—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83—41no—
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83—41no—
80C2×D4×C83—42no—
81C2×D4×C83—42no—
82C2×D4×C83—42no—
83C2×D4×C83—81no—
84C2×D4×C83—81no—
85C2×D4×C83—81no—
86C2×D4×C83SD1612yescomplemented
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—42no—
92C22×C4×C83—42no—
93C22×C4×C83—41no—
94C22×C4×C83—81no—
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98SD16×C83—48no—
99C2×C8×C83—22no—
100C2×C8×C83—22no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83—41no—
107C2×C8×C83—41no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—22no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C4⋊C4×C83—41no—
114C4⋊C4×C83—42no—
115C4⋊C4×C83—41no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—41no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83SD1612yescomplemented
120C22⋊C4×C83—42no—
121C22⋊C4×C83—42no—
122C22⋊C4×C83—41no—
123C22⋊C4×C83—41no—
124C22⋊C4×C83—41no—
125C22⋊C4×C83—41no—
126C22⋊C4×C83—41no—
127C22⋊C4×C83—41no—
128C22⋊C4×C83—41no—
129C22⋊C4×C83—41no—
130C22⋊C4×C83—81no—
131C42×C83—41no—
132C23×C83—22no—
133C23×C83—41no—
134C23×C83—41no—
135C23×C83—41no—
136C23×C83—41no—
137C23×C83—41no—
138C23×C83—41no—
139C23×C83—41no—
140C23×C83—41no—
141C23×C83—41no—
142C23×C83—41no—
143C23×C83—41no—
144C23×C83—41no—
145C23×C83—41no—
146C23×C83#12011yescharacteristic
147Q8×C83—42no—
148Q8×C83—42no—
149Q8×C83—81no—
150Q8×C83—81no—
151D4×C83—42no—
152D4×C83—42no—
153D4×C83—81no—
154D4×C83—81no—
155D4×C83—84no—
156D4×C83—84no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—21no—
160C2×C4×C83—21no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—22no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—41no—
171C2×C4×C83—41no—
172C2×C4×C83—41no—
173C2×C4×C83—41no—
174C2×C4×C83—42no—
175C2×C4×C83—41no—
176C2×C4×C83—84no—
177C2×C4×C83—81no—
178C2×C4×C83—81no—
179C2×C4×C83C2×SD1611yescharacteristic
180C2×C4×C83C2×SD1611yescharacteristic
181C2×C4×C83#13511yescharacteristic
182C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
185C8×C83—42no—
186C8×C83—42no—
187C8×C83—81no—
188C22×C83—42no—
189C22×C83—42no—
190C22×C83—42no—
191C22×C83—81no—
192C22×C83—81no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83—81no—
197C22×C83—81no—
198C22×C83—81no—
199C22×C83—81no—
200C22×C83—81no—
201C22×C83—81no—
202C22×C83#44911yescharacteristic
203C22×C83#45011yescharacteristic
204C22×C83#45011yescharacteristic
205C22×C83#45311yescharacteristic
206C22×C83#45511yescharacteristic
207C22×C83#45611yescharacteristic
208C22×C83B(4,𝔽2)11yescharacteristic
209C4×C83—21no—
210C4×C83—21no—
211C4×C83—21no—
212C4×C83—21no—
213C4×C83—21no—
214C4×C83—21no—
215C4×C83—42no—
216C4×C83—82no—
217C4×C83—81no—
218C4×C83—81no—
219C4×C83—81no—
220#7778C8311yesmaximal, characteristic, direct product factor, Sylow
221C2×C83—82no—
222C2×C83—161no—
223C2×C83#160011yescharacteristic
224C2×C83#160111yescharacteristic
225C2×C83#160411yescharacteristic
226C2×C83#162911yescharacteristic
227C2×C83#164211yescharacteristic
228C2×C83#165411yescharacteristic
229C2×C83#166111yescharacteristic
230#2682C2×C8311yescharacteristic
231#2058C2×C8311yescharacteristic, complemented
232#2023C2×C8311yescharacteristic, complemented
233#2013C2×C8311yescharacteristic, complemented
234#1271C2×C8311yescharacteristic, complemented
235#1270C2×C8311yescharacteristic, complemented
236#1270C2×C8311yescharacteristic, complemented
237C83#777811yescharacteristic, direct product factor, Sylow
238C22×SD16—21no—
239C2×C4⋊Q8C22×C8311yescharacteristic
240C2×#123—21no—
241C2×#121—21no—
242C2×#106—21no—
243C2×#103—21no—
244C2×#102—21no—
245#396—21no—
246#394—21no—
247#392—21no—
248#390C22×C8311yescharacteristic
249#390C22×C8311yescharacteristic
250#389C22×C8311yescharacteristic
251#385—21no—
252#340—21no—
253#340—21no—
254#336C22×C8311yescharacteristic
255#336C22×C8311yescharacteristic
256#336C22×C8311yescharacteristic
257C22×Q8—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4D4×C8311yescharacteristic
261C22×D4D4×C8311yescharacteristic
262C23×C4—21no—
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16—42no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C4⋊Q8—42no—
270#123—42no—
271#123—42no—
272#121—42no—
273#121—42no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4D4×C8311yescharacteristic
279C2×C4⋊C4D4×C8311yescharacteristic
280C2×C4⋊C4D4×C8311yescharacteristic
281C2×C4⋊C4D4×C8311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42C23×C8311yescharacteristic
290#106—42no—
291#103—42no—
292#102—42no—
293#95—21no—
294#95—21no—
295C24—21no—
296C24—21no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4SD16×C8312yescomplemented
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4C2×D4×C8311yescharacteristic
326C22×C4C2×D4×C8311yescharacteristic
327C22×C4C2×D4×C8311yescharacteristic
328SD16—48no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4SD16×C8312yescomplemented
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—81no—
361C42—41no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23#120×C8311yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—42no—
382D4—42no—
383D4—81no—
384D4—81no—
385D4—84no—
386D4—84no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C2×SD16×C8311yescharacteristic
410C2×C4C2×SD16×C8311yescharacteristic
411C2×C4#135×C8311yescharacteristic
412C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
413C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
414C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
415C8—42no—
416C8—42no—
417C8—81no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#449×C8311yescharacteristic
433C22#450×C8311yescharacteristic
434C22#450×C8311yescharacteristic
435C22#453×C8311yescharacteristic
436C22#455×C8311yescharacteristic
437C22#456×C8311yescharacteristic
438C22B(4,𝔽2)×C8311yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—81no—
450C2—82no—
451C2—161no—
452C2C83×#160011yescharacteristic
453C2C83×#160111yescharacteristic
454C2C83×#160411yescharacteristic
455C2C83×#162911yescharacteristic
456C2C83×#164211yescharacteristic
457C2C83×#165411yescharacteristic
458C2C83×#166111yescharacteristic
subgroupG/H
center, socleC23×C83#120
commutator, Frattini, cosocleC2×C42C23×C83
prime radicalC83×#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2058C2semidirect
3. C83×#2023C2semidirect
4. C83×#2013C2semidirect
5. C83×#1271C2semidirect
6. C83×#1270C2semidirect
7. C83×#1270C2semidirect
86. C2×D4×C83SD16semidirect
119. C4⋊C4×C83SD16semidirect
220. #7778C83direct (N × H)
231. #2058C2×C83semidirect
232. #2023C2×C83semidirect
233. #2013C2×C83semidirect
234. #1271C2×C83semidirect
235. #1270C2×C83semidirect
236. #1270C2×C83semidirect
237. C83#7778direct (N × H)
316. C2×D4SD16×C83semidirect
349. C4⋊C4SD16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#26822
C83×#20582
C83×#20232
C83×#20132
C83×#12712
C83×#12702
#777883