Finite Group Databasebeta

← #35322229

Subgroups — #35322229

Order 21248.

324 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127530C211yesmaximal, characteristic, complemented
2C2×#19127507C211yesmaximal, characteristic
3#24909168C211yesmaximal, characteristic
4#24908351C211yesmaximal, characteristic
5#24905558C211yesmaximal, characteristic, complemented
6#24905541C211yesmaximal, characteristic, complemented
7C83×#1772C211yesmaximal, characteristic, complemented
8C2×#14581814C2211yescharacteristic
9C22×C83⋊C16C2211yescharacteristic
10C22×C83⋊C16C2211yescharacteristic
11C2×#14581728C2211yescharacteristic
12C2×#14581720—21no—
13#19127667—21no—
14#19127530—22no—
15#19127507—22no—
16#19127504C411yescharacteristic, complemented
17#19127504C411yescharacteristic, complemented
18#19127501C411yescharacteristic, complemented
19#19127501C411yescharacteristic, complemented
20C2×#108×C83C2211yescharacteristic
21#427×C83C2211yescharacteristic
22#405×C83—21no—
23C2×C4×C8×C83C2211yescharacteristic
24#339×C83—21no—
25C2×#13050590—21no—
26#14581814—21no—
27#14581814—21no—
28#14581814—21no—
29#14581814—21no—
30C2×Dic332C2×C411yescharacteristic
31C2×#13050579—21no—
32C2×D664C2×C411yescharacteristic
33C2×#13050574—21no—
34C2×C83⋊C16—21no—
35C2×C83⋊C16—21no—
36C2×C83⋊C16—21no—
37C2×C83⋊C16—21no—
38C2×C83⋊C16—21no—
39C2×C83⋊C16C2×C411yescharacteristic
40C2×C83⋊C16C2×C411yescharacteristic
41#14581728—21no—
42#14581728—21no—
43#14581720—42no—
44(C2×D83)⋊C8—41no—
45(C2×D83)⋊C8—41no—
46C2×M16×C83—21no—
47C22×C8×C83D411yescharacteristic
48C22×C8×C83C2311yescharacteristic
49#117×C83—21no—
50C2×C42×C83D411yescharacteristic
51#108×C83—21no—
52#108×C83D412yes—
53#107×C83C2×C411yescharacteristic
54#107×C83C2×C411yescharacteristic
55#106×C83—21no—
56#97×C83—41no—
57C4×C8×C83—21no—
58C4×C8×C83—21no—
59C4×C8×C83—41no—
60#13050590—41no—
61#13050590—41no—
62#13050590—41no—
63#13050590—41no—
64C2×Dic166—21no—
65C2×D332—21no—
66C2×C22⋊D83—41no—
67C2×C4×D83—41no—
68Dic332—21no—
69Dic332—21no—
70#13050579—41no—
71#13050579—41no—
72D664—21no—
73D664—21no—
74#13050574—41no—
75#13050574—41no—
76C2×C83⋊C8—41no—
77C83⋊C16—21no—
78C83⋊C16—21no—
79C83⋊C16—41no—
80C22×C4×C83—21no—
81C22×C4×C83C2×D411yescharacteristic
82M16×C83—22no—
83M16×C83—41no—
84C2×C8×C83—21no—
85C2×C8×C83—21no—
86C2×C8×C83—41no—
87C2×C8×C83D811yescharacteristic
88C2×C8×C83D811yescharacteristic
89C2×C8×C83C22×C411yescharacteristic
90C2×C8×C83C2×D411yescharacteristic
91C2×C8×C83#4111yescharacteristic
92C2×C8×C83#4111yescharacteristic
93C2×C8×C83#4111yescharacteristic
94C2×C8×C83#4111yescharacteristic
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C22⋊C4×C83—41no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—41no—
101C23×C83#13511yescharacteristic
102C22×D83—41no—
103C2×C4×C83—22no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83C4×D411yescharacteristic
109C2×C4×C83#12111yescharacteristic
110C2×C4×C83#12311yescharacteristic
111C2×C4×C83C2×D811yescharacteristic
112C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C83#13711yescharacteristic
114C22⋊D83—81no—
115C22⋊D83—81no—
116C2×Dic83—41no—
117D332—41no—
118D332—41no—
119C4×D83—81no—
120C4×D83—81no—
121Dic166—41no—
122Dic166—41no—
123C8×C83—21no—
124C8×C83—21no—
125C8×C83—42no—
126C8×C83#11711yescharacteristic
127C8×C83C4×D411yescharacteristic
128C8×C83#12411yescharacteristic
129C8×C83#12611yescharacteristic
130C83⋊C8—81no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83C4×D811yescharacteristic
134C22×C83#46411yescharacteristic
135C22×C83#48211yescharacteristic
136C2×D83—81no—
137C2×D83—81no—
138C2×D83—81no—
139C4×C83—42no—
140C4×C83—81no—
141C4×C83—81no—
142C4×C83#38811yescharacteristic
143C4×C83#44011yescharacteristic
144C4×C83#45911yescharacteristic
145C4×C83#48011yescharacteristic
146Dic83—81no—
147#13134—831nomaximal, Sylow
148C2×C83—21no—
149C2×C83—21no—
150C2×C83#192511yescharacteristic
151C2×C83#217811yescharacteristic
152C2×C83#217911yescharacteristic
153D83—161no—
154C2×#368—831no—
155C2×#359—831no—
156#2138—831no—
157#1974—831no—
158#1772D8311yescharacteristic, complemented
159#1382—831no—
160#1374—831no—
161C83#1313411yescharacteristic, complemented, Sylow
162C2×#135—831no—
163C22×C16—831no—
164C22×C16—831no—
165C2×#108—831no—
166C2×#108C2×D8311yescharacteristic
167#427C2×D8311yescharacteristic
168C2×#104—1661no—
169#408—1661no—
170#405—21no—
171C2×C4×C8C2×D8311yescharacteristic
172#368—1662no—
173#359—1662no—
174#358—831no—
175#358—831no—
176#357—831no—
177#357—831no—
178#339—21no—
179C2×#41—1661no—
180#135—1661no—
181#135—1661no—
182#135—1661no—
183#135—1661no—
184C2×Q16—831no—
185C2×SD16—1661no—
186C2×D8—831no—
187C2×M16—21no—
188C2×M16—1661no—
189C22×C8C22⋊D8311yescharacteristic
190C22×C8C22×D8311yescharacteristic
191#117—21no—
192C2×C42C22⋊D8311yescharacteristic
193C2×C16—831no—
194C2×C16—831no—
195C2×C16—1661no—
196C2×C16—1661no—
197C2×C16—1661no—
198C2×C16—1661no—
199C2×C16—1661no—
200#108—21no—
201#108—1661no—
202#108—1661no—
203#108D33212yes—
204#107C4×D8311yescharacteristic
205#107C4×D8311yescharacteristic
206#106—21no—
207#104—3322no—
208C22⋊C8—3321no—
209C22⋊C8—3321no—
210#97—41no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—41no—
214#41—3321no—
215#41—3321no—
216#41—3321no—
217#41—3321no—
218C2×Q8—1661no—
219C2×D4—3321no—
220C2×D4—1661no—
221C22×C4—21no—
222C22×C4—3321no—
223C22×C4C2×C22⋊D8311yescharacteristic
224Q16—1661no—
225Q16—1661no—
226SD16—3321no—
227SD16—3321no—
228D8—1661no—
229D8—1661no—
230M16—22no—
231M16—41no—
232M16—3321no—
233M16—3321no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—3321no—
238C2×C8#1305057511yescharacteristic
239C2×C8#1305057511yescharacteristic
240C2×C8C2×C4×D8311yescharacteristic
241C2×C8C2×D33211yescharacteristic
242C2×C8#1305059011yescharacteristic
243C2×C8#1305059011yescharacteristic
244C2×C8#1305059011yescharacteristic
245C2×C8#1305059011yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—21no—
248C22⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—41no—
252C16—3321no—
253C16—1661no—
254C16—1661no—
255C23—3321no—
256C23#1458181211yescharacteristic
257Q8—3321no—
258Q8—3321no—
259D4—3321no—
260D4—3321no—
261D4—6641no—
262D4—6641no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—3321no—
269C2×C4—6641no—
270C2×C4—6641no—
271C2×C4C4×C22⋊D8311yescharacteristic
272C2×C4#1458176611yescharacteristic
273C2×C4#1458177511yescharacteristic
274C2×C4C2×#1305057511yescharacteristic
275C2×C4#1458181711yescharacteristic
276C2×C4#1458182311yescharacteristic
277C8—21no—
278C8—21no—
279C8—42no—
280C8—6641no—
281C8#1458174911yescharacteristic
282C8C4×D33211yescharacteristic
283C8#1458178011yescharacteristic
284C8#1458178611yescharacteristic
285C22—21no—
286C22—21no—
287C22—6641no—
288C22—6641no—
289C22—6641no—
290C22C4×#1305057511yescharacteristic
291C22#1912792311yescharacteristic
292C22#1912802511yescharacteristic
293C4—42no—
294C4—81no—
295C4—81no—
296C4—6641no—
297C4#1912759211yescharacteristic
298C4#1912779911yescharacteristic
299C4#1912789611yescharacteristic
300C4#1912801411yescharacteristic
301C2—21no—
302C2—21no—
303C2—13281no—
304C2#2490808311yescharacteristic
305C2#2490934211yescharacteristic
306C2#2490934711yescharacteristic
subgroupG/H
centerC2×C4C2×#13050575
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1772C2
socleC22×C83C4×D8
prime radicalC2×#14581814C22
cosocleC22×C8×C83C23
hypercenter#1772D83
hypocenterC83#13134
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
1. C2×#19127530C2semidirect
5. #24905558C2semidirect
6. #24905541C2semidirect
7. C83×#1772C2semidirect
16. #19127504C4semidirect
17. #19127504C4semidirect
18. #19127501C4semidirect
19. #19127501C4semidirect
158. #1772D83semidirect
161. C83#13134semidirect

Maximal subgroups

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

subgroupindex
C2×#191275302
C2×#191275072
#249091682
#249083512
#249055582
#249055412
C83×#17722
#1313483