Finite Group Databasebeta

← #35301781

Subgroups — #35301781

Order 21248.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#24916543C211yesmaximal, characteristic
2#24912679C211yesmaximal, characteristic
3#24907456C211yesmaximal, characteristic
4#24906148C211yesmaximal, characteristic
5#24905365C211yesmaximal, characteristic
6#24905364C211yesmaximal, characteristic
7C83×#1549C211yesmaximal, characteristic
8C2×C4×Dic166C2211yescharacteristic
9C8×Dic166—22no—
10#19127825—22no—
11#19127754C2211yescharacteristic
12#19127751—22no—
13#19127725C2211yescharacteristic
14C2×#14581710—21no—
15#19127639C414yes—
16#19127608—22no—
17#19127606—21no—
18#19127566—21no—
19C4×C4⋊Dic83C2211yescharacteristic
20#19127469—21no—
21#19127447—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25#332×C83—22no—
26C22×Dic166D411yescharacteristic
27C2×#13050574—21no—
28C4×Dic166—21no—
29C4×Dic166—21no—
30C4×Dic166—21no—
31C4×Dic166—21no—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83D411yescharacteristic
34C2×Dic83⋊C4C2×C412yes—
35C2×C4×Dic83—21no—
36C2×C4×Dic83D412yes—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)D412yes—
40Dic83⋊C8—24no—
41#14581710—42no—
42C22⋊(C83⋊C8)—21no—
43C4×C83⋊C8—21no—
44C8×Dic83—24no—
45C83⋊#95—24no—
46C83⋊#95—24no—
47C83⋊#95C2×C412yes—
48C22×Q8×C83—21no—
49C2×M16×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83C2×C412yes—
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83D412yes—
60C22⋊C8×C83—21no—
61C4×C8×C83—21no—
62C2×Dic166—21no—
63C2×Dic166—21no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166SD1612yes—
67C2×Dic166Q1612yes—
68C22×Dic83—22no—
69C22×Dic83C22⋊C412yes—
70C22×Dic83C22⋊C412yes—
71#13050574—41no—
72#13050574—41no—
73C2×C83⋊C8—22no—
74C2×C83⋊C8—41no—
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83—42no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80C4×Dic83—22no—
81C4×Dic83—22no—
82C4×Dic83—41no—
83C4×Dic83—41no—
84C2×Q8×C83—44no—
85C2×Q8×C83—41no—
86C2×Q8×C83—41no—
87C22×C4×C83—22no—
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91M16×C83—41no—
92M16×C83—41no—
93C2×C8×C83—22no—
94C2×C8×C83—41no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C42×C83C2×D411yescharacteristic
100C42×C83C2×D411yescharacteristic
101C42×C83#4111yescharacteristic
102C42×C83#4111yescharacteristic
103C23×C83C2×C22⋊C411yescharacteristic
104Q8×C83—82no—
105Q8×C83—82no—
106C2×C4×C83—22no—
107C2×C4×C83—41no—
108C2×C4×C83—44no—
109C2×C4×C83—42no—
110C2×C4×C83C4×D411yescharacteristic
111C2×C4×C83C4×D411yescharacteristic
112C2×C4×C83#12011yescharacteristic
113C2×C4×C83#12111yescharacteristic
114C2×C4×C83#12111yescharacteristic
115C2×C4×C83#12311yescharacteristic
116C2×C4×C83C2×SD1611yescharacteristic
117C2×C4×C83C2×Q1611yescharacteristic
118C2×C4×C83#13511yescharacteristic
119C2×C4×C83#13511yescharacteristic
120C2×Dic83—22no—
121C2×Dic83—22no—
122C2×Dic83—22no—
123C2×Dic83—42no—
124C2×Dic83—42no—
125C2×Dic83—42no—
126C2×Dic83—42no—
127C2×Dic83#10314yes—
128Dic166—22no—
129Dic166—22no—
130Dic166—22no—
131Dic166—22no—
132C8×C83—42no—
133C8×C83—81no—
134C83⋊C8—42no—
135C83⋊C8—81no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83#38611yescharacteristic
139C22×C83C2×#10311yescharacteristic
140C22×C83#41611yescharacteristic
141C4×C83—21no—
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—82no—
145C4×C83#45412yes—
146C4×C83#46511yescharacteristic
147C4×C83#46511yescharacteristic
148C4×C83#46711yescharacteristic
149C4×C83#46711yescharacteristic
150Dic83—24no—
151Dic83—42no—
152Dic83—82no—
153#9396—831nomaximal, Sylow
154C2×C83—21no—
155C2×C83—21no—
156C2×C83#188811yescharacteristic
157C2×C83#189211yescharacteristic
158C2×C83#191511yescharacteristic
159#2982—831no—
160#2568—831no—
161#1801—831no—
162#1549—831no—
163#1549D8311yescharacteristic
164#1303—831no—
165#1303—831no—
166C83#939611yescharacteristic, complemented, Sylow
167C2×C4⋊Q8C2×D8311yescharacteristic
168C2×C4×Q8—831no—
169C8×Q8—1662no—
170C8×Q8—1662no—
171#431—831no—
172#431—1662no—
173#431C2×D8311yescharacteristic
174#423—831no—
175#423C2×D8311yescharacteristic
176C2×#101—1661no—
177#400—834no—
178#391—1662no—
179#391—1661no—
180#382—1661no—
181C4×C4⋊C4—831no—
182#343—1661no—
183#332—22no—
184#332—1662no—
185C22×Q8—21no—
186C22×Q8—831no—
187C2×M16—21no—
188C2×M16—1661no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4×Q8—1661no—
194C4×Q8—1661no—
195C4×Q8—1661no—
196C4×Q8—1661no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—832no—
199C2×C4⋊C4—831no—
200C2×C4⋊C4—1661no—
201C2×C4⋊C4C4×D8312yes—
202C2×C42—832no—
203C2×C42—1661no—
204C2×C42C22×D8311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—832no—
208C4⋊C8—1664no—
209C4⋊C8—1661no—
210C4⋊C8—1661no—
211C4⋊C8C22⋊D8312yes—
212#101—3322no—
213C22⋊C8—21no—
214C22⋊C8—1661no—
215C4×C8—21no—
216C4×C8—1664no—
217C4×C8—1661no—
218#95—832no—
219#95—1664no—
220#95—1664no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—832no—
225C2×Q8—832no—
226C2×Q8—1661no—
227C2×Q8—1661no—
228C2×Q8—1661no—
229C2×Q8—1661no—
230C22×C4—22no—
231C22×C4—832no—
232C22×C4—832no—
233C22×C4—1662no—
234C22×C4C2×C4×D8311yescharacteristic
235C22×C4D4×D8311yescharacteristic
236C22×C4D4×D8311yescharacteristic
237M16—41no—
238M16—41no—
239M16—3321no—
240M16—3321no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—3321no—
244C2×C8—1662no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—3322no—
250C4⋊C4—1661no—
251C4⋊C4—1661no—
252C4⋊C4—1662no—
253C4⋊C4—1662no—
254C42—3321no—
255C42—3321no—
256C42—1662no—
257C42—1662no—
258C42D4×D8311yescharacteristic
259C42C2×C22⋊D8311yescharacteristic
260C42#1305058911yescharacteristic
261C42#1305059011yescharacteristic
262C23C22⋊C4×D8311yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—1662no—
266Q8—1662no—
267Q8—1662no—
268Q8—1662no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—3322no—
274C2×C4—3322no—
275C2×C4—3322no—
276C2×C4—3322no—
277C2×C4—834no—
278C2×C4—1662no—
279C2×C4—1662no—
280C2×C4—1662no—
281C2×C4(C22⋊D83)⋊C411yescharacteristic
282C2×C4C4×C22⋊D8311yescharacteristic
283C2×C4#1458176011yescharacteristic
284C2×C4#1458176311yescharacteristic
285C2×C4#1458176511yescharacteristic
286C2×C4#1458177411yescharacteristic
287C2×C4SD16×D8311yescharacteristic
288C2×C4Q16×D8311yescharacteristic
289C2×C4#1458181311yescharacteristic
290C2×C4#1458181511yescharacteristic
291C8—42no—
292C8—81no—
293C8—3322no—
294C8—6641no—
295C22—21no—
296C22—21no—
297C22#1912757711yescharacteristic
298C22#103×D8311yescharacteristic
299C22#1912769611yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—3322no—
305C4—6642no—
306C4—1664no—
307C4#1912787212yes—
308C4#1912792711yescharacteristic
309C4#1912793311yescharacteristic
310C4#1912793911yescharacteristic
311C4#1912794111yescharacteristic
312C2—21no—
313C2—21no—
314C2#2490786311yescharacteristic
315C2#2490788311yescharacteristic
316C2#2490801211yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1549C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1549D83
hypocenterC83#9396
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
166. C83#9396semidirect

Maximal subgroups

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

subgroupindex
#249165432
#249126792
#249074562
#249061482
#249053652
#249053642
C83×#15492
#939683