Finite Group Databasebeta

← #35300182

Subgroups — #35300182

Order 21248.

692 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 143 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#24915794C211yesmaximal, characteristic
2C2×#19127567C211yesmaximal, characteristic, complemented
3C2×#19127436C212yesmaximal
4#24905343C212yesmaximal, complemented
5C83×#1817C211yesmaximal, characteristic, complemented
6C2×#14581775—21no—
7C2×#14581766C2211yescharacteristic
8C22×C4⋊Dic83—21no—
9#19127603C412yescomplemented
10#19127599C412yescomplemented
11#19127592—22no—
12#19127580—21no—
13#19127567—24no—
14#19127561C2211yescharacteristic
15C2×C83⋊#95—21no—
16C2×C83⋊#95C2212yes—
17#19127436—24no—
18C22×C42×C83C2211yescharacteristic
19C2×C22⋊C8×C83C2212yes—
20#336×C83—22no—
21C22×D332C2×C411yescharacteristic
22C22×C22⋊D83—21no—
23C23×Dic83—22no—
24#14581775—42no—
25#14581775—42no—
26#14581766—22no—
27#14581766—22no—
28C2×C4⋊Dic83—24no—
29C2×C4⋊Dic83—42no—
30C2×C4⋊Dic83—42no—
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×C4⋊Dic83C2×C412yes—
33C2×Dic83⋊C4—22no—
34C2×C22⋊Dic83—21no—
35C2×(C2×D83)⋊C4—22no—
36C2×(C2×D83)⋊C4—22no—
37C2×(C2×D83)⋊C4—21no—
38C83⋊#95—24no—
39C83⋊#95—22no—
40C83⋊#95—21no—
41C83⋊#95—22no—
42C83⋊#95D414yes—
43C23×C4×C83D411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C23×C4×C83C2311yescharacteristic
46C22×C8×C83—22no—
47C2×C42×C83—22no—
48C2×C42×C83—22no—
49C2×C42×C83C2×C412yes—
50C2×C42×C83D412yes—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53C23×D83—21no—
54C2×D332—22no—
55C2×D332—42no—
56C2×C22⋊D83—41no—
57C2×C22⋊D83—41no—
58C2×C22⋊D83—41no—
59C2×C22⋊D83—41no—
60C2×C22⋊D83—41no—
61C2×C22⋊D83—41no—
62C22×Dic83—21no—
63C22×Dic83—22no—
64C22×Dic83—24no—
65C22×Dic83—44no—
66C22×Dic83—44no—
67C22×Dic83—42no—
68C22×Dic83—42no—
69C4⋊Dic83—22no—
70C4⋊Dic83—48no—
71C4⋊Dic83—42no—
72Dic83⋊C4—44no—
73C22⋊Dic83—41no—
74C22⋊Dic83—41no—
75(C2×D83)⋊C4—44no—
76(C2×D83)⋊C4—44no—
77(C2×D83)⋊C4—42no—
78C24×C83C2×D411yescharacteristic
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—42no—
86C22×C4×C83—42no—
87C22×C4×C83C22⋊C412yes—
88C22×C4×C83C22⋊C412yes—
89C22×C4×C83D812yes—
90C22×C4×C83SD1612yes—
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D412yes—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83#4112yes—
95C2×C8×C83—44no—
96C2×C8×C83—42no—
97C42×C83—24no—
98C42×C83—24no—
99C23×C83—22no—
100C23×C83—22no—
101C23×C83C2×C22⋊C411yescharacteristic
102C23×C83#12011yescharacteristic
103C23×C83#12311yescharacteristic
104C23×C83C2×D811yescharacteristic
105C23×C83C2×SD1611yescharacteristic
106C23×C83AGL(1,ℤ/8)11yescharacteristic
107C23×C83#13711yescharacteristic
108C22×D83—41no—
109C22×D83—42no—
110C22×D83—41no—
111C22×D83—41no—
112C22×D83—42no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—22no—
123C2×C4×C83—22no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83#10214yes—
129C2×C4×C83C4×D412yes—
130C2×C4×C83#12112yes—
131C22⋊D83—81no—
132C22⋊D83—81no—
133C22⋊D83—81no—
134C22⋊D83—81no—
135C2×Dic83—44no—
136C2×Dic83—44no—
137C2×Dic83—41no—
138C2×Dic83—42no—
139C2×Dic83—42no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C2×Dic83—42no—
143C2×Dic83—44no—
144C2×Dic83—84no—
145C2×Dic83—84no—
146D332—82no—
147C8×C83—82no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—22no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—22no—
156C22×C83#38611yescharacteristic
157C22×C83#41012yes—
158C22×C83C2×#10211yescharacteristic
159C22×C83#41711yescharacteristic
160C22×C83#42112yes—
161C22×C83#44711yescharacteristic
162C22×C83#44811yescharacteristic
163C22×C83#48011yescharacteristic
164C22×C83#48111yescharacteristic
165C2×D83—81no—
166C2×D83—81no—
167C2×D83—81no—
168C2×D83—82no—
169C2×D83—82no—
170C4×C83—24no—
171C4×C83—22no—
172C4×C83—22no—
173C4×C83—42no—
174Dic83—84no—
175Dic83—81no—
176Dic83—82no—
177#9156—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—21no—
181C2×C83—21no—
182C2×C83#151212yes—
183C2×C83#151712yes—
184C2×C83#187411yescharacteristic
185C2×C83#189711yescharacteristic
186C2×C83#189911yescharacteristic
187D83—161no—
188#2854—831no—
189C2×#382—831no—
190#1817D8311yescharacteristic, complemented
191C2×#327—832no—
192#1294—832no—
193C83#915611yescharacteristic, complemented, Sylow
194C2×#123—1661no—
195C2×#121—831no—
196C22×C4⋊C4—1661no—
197C22×C42C2×D8311yescharacteristic
198C2×C22⋊C8C2×D8312yes—
199#390—832no—
200#389—832no—
201#388—1662no—
202#386—1661no—
203#382—1664no—
204#380—831no—
205C2×#95—832no—
206C2×#95—1661no—
207#336—22no—
208#327—1664no—
209C22×D4—831no—
210C22×D4—1661no—
211C23×C4—1662no—
212C23×C4D33211yescharacteristic
213C23×C4D33211yescharacteristic
214C23×C4C22×D8311yescharacteristic
215C22×C8—22no—
216#123—3322no—
217#123—3322no—
218#121—1662no—
219#121—1662no—
220C2×C4⋊C4—3322no—
221C2×C4⋊C4—3322no—
222C2×C4⋊C4—832no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—1664no—
225C2×C4⋊C4—1662no—
226C2×C22⋊C4—1661no—
227C2×C22⋊C4—1662no—
228C2×C22⋊C4—1662no—
229C2×C22⋊C4—1661no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D8312yes—
233C2×C42C22⋊D8312yes—
234C22⋊C8—22no—
235C22⋊C8—22no—
236#95—834no—
237#95—1664no—
238#95—1662no—
239#95—1662no—
240#95—1661no—
241C24—1661no—
242C24C2×D33211yescharacteristic
243C2×D4—3322no—
244C2×D4—3321no—
245C2×D4—3321no—
246C2×D4—3321no—
247C2×D4—3321no—
248C2×D4—3321no—
249C2×D4—3321no—
250C2×D4—1662no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—3324no—
260C22×C4—3324no—
261C22×C4—3322no—
262C22×C4—3322no—
263C22×C4—1664no—
264C22×C4—1662no—
265C22×C4—1661no—
266C22×C4(C2×D83)⋊C412yes—
267C22×C4(C2×D83)⋊C412yes—
268C22×C4D66412yes—
269C22×C4#1305057912yes—
270C22×C4C2×C4×D8311yescharacteristic
271C22×C4D4×D8312yes—
272C22×C4C2×C22⋊D8311yescharacteristic
273C22×C4#1305058912yes—
274C2×C8—44no—
275C2×C8—42no—
276C4⋊C4—3322no—
277C4⋊C4—3328no—
278C4⋊C4—3324no—
279C4⋊C4—1662no—
280C22⋊C4—3324no—
281C22⋊C4—3321no—
282C22⋊C4—3321no—
283C22⋊C4—3322no—
284C22⋊C4—3324no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—3321no—
290C23—3322no—
291C23—3321no—
292C23—3321no—
293C23—3322no—
294C23C2×(C2×D83)⋊C411yescharacteristic
295C23#1458176211yescharacteristic
296C23#1458177611yescharacteristic
297C23C2×D66411yescharacteristic
298C23C2×#1305057911yescharacteristic
299C23#1458182011yescharacteristic
300C23#1458182511yescharacteristic
301D4—6642no—
302D4—6641no—
303D4—6641no—
304D4—6641no—
305D4—6641no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3324no—
322C2×C4—3324no—
323C2×C4—3321no—
324C2×C4—3324no—
325C2×C4—3322no—
326C2×C4—3322no—
327C2×C4—3321no—
328C2×C4—3321no—
329C2×C4—3322no—
330C2×C4—6644no—
331C2×C4—6644no—
332C2×C4#1458171414yes—
333C2×C4(C22⋊D83)⋊C412yes—
334C2×C4#1458176412yes—
335C8—82no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—6641no—
345C22—6642no—
346C22—6642no—
347C22—6641no—
348C22—6641no—
349C22#1912757811yescharacteristic
350C22#1912767412yes—
351C22C2×#1458171411yescharacteristic
352C22#1912770311yescharacteristic
353C22#1912771712yes—
354C22#1912783711yescharacteristic
355C22#1912784211yescharacteristic
356C22#1912801511yescharacteristic
357C22#1912802011yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—6644no—
363C4—6641no—
364C4—6642no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—13281no—
370C2#2490600212yes—
371C2#2490602412yes—
372C2#2490777811yescharacteristic
373C2#2490790911yescharacteristic
374C2#2490791911yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1817C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#14581766C22
cosocleC23×C4×C83C23
hypercenter#1817D83
hypocenterC83#9156
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#19127567C2semidirect
4. #24905343C2semidirect
5. C83×#1817C2semidirect
9. #19127603C4semidirect
10. #19127599C4semidirect
190. #1817D83semidirect
193. C83#9156semidirect

Maximal subgroups

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

subgroupindex
#249157942
C2×#191275672
C2×#191274362
#249053432
C83×#18172
#915683