Finite Group Databasebeta

← #35307757

Subgroups — #35307757

Order 21248.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#24913577C211yesmaximal, characteristic
2#24907770C212yesmaximal
3C2×#19127437C212yesmaximal
4#24905306C211yesmaximal, characteristic
5C83×#1442C211yesmaximal, characteristic
6C2×#14581771C2212yes—
7C2×#14581715—22no—
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127633—22no—
10#19127622—22no—
11#19127561—21no—
12#19127561—21no—
13#19127463—22no—
14#19127437—24no—
15C22×C4⋊C4×C83C2211yescharacteristic
16C2×#95×C83—21no—
17C2×#95×C83—21no—
18C2×#95×C83C2211yescharacteristic
19C2×#95×C83C2211yescharacteristic
20C22×Dic166D412yes—
21C22×C83⋊C8—22no—
22#14581771—22no—
23#14581771—22no—
24#14581771—22no—
25#14581771—22no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83D412yes—
31C2×Dic83⋊C4—22no—
32C2×C22⋊Dic83—22no—
33#14581715—44no—
34C22⋊(C83⋊C8)—22no—
35C22⋊(C83⋊C8)—22no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C23×C4×C83—21no—
39C23×C4×C83—21no—
40C23×C4×C83D411yescharacteristic
41C23×C4×C83D411yescharacteristic
42C23×C4×C83C2311yescharacteristic
43C2×C4⋊C4×C83—22no—
44C2×C4⋊C4×C83—42no—
45#95×C83—22no—
46#95×C83—21no—
47#95×C83—42no—
48#95×C83—42no—
49#95×C83D412yes—
50C2×Dic166—22no—
51C2×Dic166—22no—
52C2×Dic166—42no—
53C2×Dic166—42no—
54C22×Dic83—22no—
55C22×Dic83—22no—
56C2×C83⋊C8—42no—
57C2×C83⋊C8—42no—
58C2×C83⋊C8—42no—
59C4⋊Dic83—22no—
60C4⋊Dic83—22no—
61C4⋊Dic83—81no—
62C4⋊Dic83—81no—
63C4⋊Dic83—81no—
64C4⋊Dic83—81no—
65Dic83⋊C4—42no—
66Dic83⋊C4—42no—
67C22⋊Dic83—42no—
68C22⋊Dic83—42no—
69C24×C83C2×D411yescharacteristic
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83—41no—
77C22×C4×C83—41no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83—41no—
83C22×C4×C83—41no—
84C22×C4×C83—41no—
85C22×C4×C83—41no—
86C22×C4×C83—41no—
87C22×C4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83—41no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83SD1612yes—
95C22×C4×C83Q1612yes—
96C22×C4×C83C2×D412yes—
97C22×C4×C83C2×D411yescharacteristic
98C22×C4×C83#4111yescharacteristic
99C22×C4×C83#4112yes—
100C4⋊C4×C83—24no—
101C4⋊C4×C83—42no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83#12011yescharacteristic
107C23×C83#12111yescharacteristic
108C23×C83#12411yescharacteristic
109C23×C83C2×SD1611yescharacteristic
110C23×C83C2×Q1611yescharacteristic
111C23×C83#13711yescharacteristic
112C23×C83#13711yescharacteristic
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—42no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83—42no—
132C2×C4×C83—81no—
133C2×C4×C83—81no—
134C2×C4×C83—81no—
135C2×C4×C83—81no—
136C2×C4×C83—81no—
137C2×C4×C83—81no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83—81no—
141C2×C4×C83—81no—
142C2×C4×C83#12112yes—
143C2×C4×C83#12312yes—
144C2×Dic83—42no—
145C2×Dic83—42no—
146C2×Dic83—42no—
147C2×Dic83—42no—
148C2×Dic83—42no—
149C2×Dic83—42no—
150Dic166—44no—
151Dic166—82no—
152C83⋊C8—82no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83#39411yescharacteristic
166C22×C83#44811yescharacteristic
167C22×C83#45111yescharacteristic
168C22×C83#45411yescharacteristic
169C22×C83#45511yescharacteristic
170C22×C83B(4,𝔽2)11yescharacteristic
171C22×C83#45811yescharacteristic
172C22×C83#46111yescharacteristic
173C22×C83#46211yescharacteristic
174C22×C83#48711yescharacteristic
175C22×C83#49111yescharacteristic
176C4×C83—22no—
177C4×C83—22no—
178C4×C83—42no—
179C4×C83—42no—
180C4×C83—82no—
181C4×C83—82no—
182Dic83—82no—
183Dic83—82no—
184#10407—831nomaximal, Sylow
185C2×C83—21no—
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—21no—
189C2×C83#159911yescharacteristic
190C2×C83#160211yescharacteristic
191C2×C83#160811yescharacteristic
192C2×C83#161711yescharacteristic
193C2×C83#204011yescharacteristic
194C2×C83#204311yescharacteristic
195C2×C83#204411yescharacteristic
196#2659—831no—
197#1872—832no—
198C2×#328—832no—
199#1442D8311yescharacteristic
200#1280—831no—
201C83#1040711yescharacteristic, complemented, Sylow
202C2×C22⋊Q8—832no—
203C22×C4⋊C4C2×D8311yescharacteristic
204C2×#103—1662no—
205C2×C22⋊C8—832no—
206#398—1662no—
207#395—1662no—
208#380—1661no—
209#380—1661no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95C2×D8311yescharacteristic
213C2×#95C2×D8311yescharacteristic
214#340—1662no—
215#328—1664no—
216C22×Q8—832no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22⋊D8311yescharacteristic
220C23×C4C22⋊D8311yescharacteristic
221C23×C4C22×D8311yescharacteristic
222C22×C8—1662no—
223C22⋊Q8—1662no—
224C22⋊Q8—1662no—
225C22⋊Q8—1662no—
226C22⋊Q8—1662no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—832no—
230C2×C4⋊C4—1662no—
231C2×C4⋊C4—1661no—
232C2×C4⋊C4—1661no—
233C2×C4⋊C4—1661no—
234C2×C4⋊C4—1661no—
235C2×C22⋊C4—1662no—
236#103—3324no—
237C22⋊C8—1662no—
238C22⋊C8—1662no—
239#95—22no—
240#95—21no—
241#95—42no—
242#95—42no—
243#95—1662no—
244#95—1662no—
245#95D33212yes—
246C24C2×C22⋊D8311yescharacteristic
247C2×Q8—3322no—
248C2×Q8—3322no—
249C2×Q8—1662no—
250C2×Q8—1662no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—1662no—
276C22×C4—1662no—
277C22×C4#1305057812yes—
278C22×C4#1305058012yes—
279C22×C4D4×D8312yes—
280C22×C4C2×D33211yescharacteristic
281C22×C4#1305059011yescharacteristic
282C22×C4#1305059112yes—
283C2×C8—3322no—
284C2×C8—3322no—
285C2×C8—3322no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—3322no—
289C4⋊C4—3322no—
290C4⋊C4—6641no—
291C4⋊C4—6641no—
292C4⋊C4—6641no—
293C4⋊C4—6641no—
294C4⋊C4—1662no—
295C4⋊C4—1662no—
296C22⋊C4—3322no—
297C22⋊C4—3322no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1458176011yescharacteristic
303C23#1458176611yescharacteristic
304C23#1458178111yescharacteristic
305C23C2×#1305057811yescharacteristic
306C23C2×#1305058011yescharacteristic
307C23#1458182211yescharacteristic
308C23#1458182211yescharacteristic
309Q8—3324no—
310Q8—6642no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—3322no—
341C2×C4—3322no—
342C2×C4—3322no—
343C2×C4—3322no—
344C2×C4—3322no—
345C2×C4—3322no—
346C2×C4#1458176712yes—
347C2×C4#1458177712yes—
348C8—6642no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1912761811yescharacteristic
362C22#1912783911yescharacteristic
363C22#1912785611yescharacteristic
364C22#1912787311yescharacteristic
365C22#1912787811yescharacteristic
366C22#1912788711yescharacteristic
367C22#1912789011yescharacteristic
368C22#1912790611yescharacteristic
369C22#1912791111yescharacteristic
370C22#1912805311yescharacteristic
371C22#1912807111yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—82no—
377C4—82no—
378C4—6642no—
379C4—6642no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#2490640211yescharacteristic
385C2#2490641911yescharacteristic
386C2#2490645311yescharacteristic
387C2#2490650411yescharacteristic
388C2#2490869211yescharacteristic
389C2#2490870911yescharacteristic
390C2#2490871411yescharacteristic
subgroupG/H
centerC23#14581760
commutator, cosocleC23×C4×C83C23
FrattiniC23×C4C22×D83
FittingC83×#1442C2
socleC23×C83#120
prime radicalC24×C83C2×D4
hypercenter#1442D83
hypocenterC83#10407
LCSG ▷ C23×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1442

Complements

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

NH ≅ G/Ntype
201. C83#10407semidirect

Maximal subgroups

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

subgroupindex
#249135772
#249077702
C2×#191274372
#249053062
C83×#14422
#1040783