Finite Group Databasebeta

← #35315559

Subgroups — #35315559

Order 21248.

372 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 95 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#24912825C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic
3#24905494C212yesmaximal
4#24905491C212yesmaximal
5C83×#1461C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7#19127725—21no—
8C2×#14581711—21no—
9#19127640C411yescharacteristic
10#19127640C2211yescharacteristic
11#19127621C411yescharacteristic
12#19127621C2211yescharacteristic
13#19127608—21no—
14#19127597—21no—
15#19127585—21no—
16#19127585—21no—
17#19127567C411yescharacteristic
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#431×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25C4×Dic166—41no—
26C4×Dic166—41no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×C4×Dic83—21no—
33C4⋊(C83⋊C8)—41no—
34C4⋊(C83⋊C8)—41no—
35#14581711—42no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95C2×C411yescharacteristic
44C83⋊#95C2×C411yescharacteristic
45C22×Q8×C83D411yescharacteristic
46C4×Q8×C83—41no—
47C4×Q8×C83—41no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—41no—
55C22⋊C8×C83—22no—
56C4×C8×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—42no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050574—42no—
64#13050574—42no—
65C2×C83⋊C8—42no—
66C4⋊Dic83—42no—
67C4⋊Dic83—41no—
68Dic83⋊C4—42no—
69Dic83⋊C4—42no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72C4×Dic83—42no—
73C2×Q8×C83—41no—
74C2×Q8×C83—41no—
75C2×Q8×C83SD1612yes—
76C2×Q8×C83Q1612yes—
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C2×C8×C83—42no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C42×C83—21no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C23×C83C2×C22⋊C411yescharacteristic
95Q8×C83—42no—
96Q8×C83—42no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#9911yescharacteristic
112C2×C4×C83#10312yes—
113C2×C4×C83#10312yes—
114C2×C4×C83C2×SD1611yescharacteristic
115C2×C4×C83C2×Q1611yescharacteristic
116C2×Dic83—22no—
117C2×Dic83—41no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—42no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127Dic166—42no—
128Dic166—42no—
129C8×C83—82no—
130C83⋊C8—82no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83C2×#9911yescharacteristic
134C22×C83C2×#10311yescharacteristic
135C22×C83#42111yescharacteristic
136C4×C83—21no—
137C4×C83—21no—
138C4×C83—21no—
139C4×C83—22no—
140C4×C83—22no—
141C4×C83—21no—
142C4×C83—41no—
143C4×C83—41no—
144C4×C83—41no—
145Dic83—42no—
146Dic83—81no—
147Dic83—81no—
148Dic83—81no—
149#11905—831nomaximal, Sylow
150C2×C83—21no—
151C2×C83—21no—
152C2×C83#152011yescharacteristic
153C2×C83#212011yescharacteristic
154C2×C83#213311yescharacteristic
155#2581—831no—
156#1801—831no—
157#1461D8311yescharacteristic
158#1354—832no—
159#1353—832no—
160C83#1190511yescharacteristic, complemented, Sylow
161C2×C4×Q8—831no—
162C2×C4×Q8C2×D8311yescharacteristic
163#431—21no—
164#423—1661no—
165C2×#101—1661no—
166#400—831no—
167#400—831no—
168#395—831no—
169#395—831no—
170#391—1661no—
171#389—1661no—
172#387—1661no—
173#387—1661no—
174#382—831no—
175#343—832no—
176#324—21no—
177#324C2×D8312yes—
178C22×Q8—831no—
179C22×Q8D33211yescharacteristic
180C2×M16—1662no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—3321no—
184C4×Q8—3321no—
185C2×C4⋊C4—831no—
186C2×C4⋊C4—1661no—
187C2×C4⋊C4—1661no—
188C2×C4⋊C4—1661no—
189C2×C4⋊C4—1661no—
190C2×C4⋊C4C4×D8311yescharacteristic
191C2×C4⋊C4C22⋊D8311yescharacteristic
192C2×C4⋊C4C22×D8311yescharacteristic
193C2×C42—1661no—
194C2×C42C4×D8311yescharacteristic
195C2×C42D33211yescharacteristic
196C2×C42C22⋊D8311yescharacteristic
197C4⋊C8—41no—
198C4⋊C8—3321no—
199C4⋊C8—3321no—
200#101—3322no—
201C22⋊C8—22no—
202C4×C8—41no—
203#95—831no—
204#95—831no—
205#95—1661no—
206#95—1661no—
207#95—1661no—
208#95—1661no—
209#95—1661no—
210#95—1661no—
211#95—1661no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—3321no—
215C2×Q8—3322no—
216C2×Q8#1305057912yes—
217C2×Q8Dic33212yes—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4—1661no—
221C22×C4—1661no—
222C22×C4(C2×D83)⋊C411yescharacteristic
223C22×C4(C2×D83)⋊C411yescharacteristic
224C22×C4(C2×D83)⋊C411yescharacteristic
225C22×C4(C2×D83)⋊C411yescharacteristic
226C22×C4C2×C4×D8311yescharacteristic
227C22×C4C2×C22⋊D8311yescharacteristic
228C22×C4C2×D33211yescharacteristic
229M16—3322no—
230M16—3322no—
231C2×C8—42no—
232C2×C8—3322no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—3321no—
237C4⋊C4—3322no—
238C4⋊C4—3322no—
239C4⋊C4—3322no—
240C4⋊C4—3321no—
241C4⋊C4—3321no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—3322no—
249C23C2×(C2×D83)⋊C411yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—3322no—
253Q8—3322no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—3321no—
268C2×C4—3321no—
269C2×C4—3321no—
270C2×C4—3321no—
271C2×C4—3321no—
272C2×C4—3321no—
273C2×C4—3322no—
274C2×C4—3321no—
275C2×C4—3321no—
276C2×C4—3321no—
277C2×C4—1662no—
278C2×C4#1458170611yescharacteristic
279C2×C4#1458170611yescharacteristic
280C2×C4#1458171712yes—
281C2×C4#1458171712yes—
282C2×C4C2×#1305057911yescharacteristic
283C2×C4C2×Dic33211yescharacteristic
284C8—82no—
285C8—6642no—
286C22—21no—
287C22—21no—
288C22C2×#1458170611yescharacteristic
289C22C2×#1458171711yescharacteristic
290C22#1912771711yescharacteristic
291C4—21no—
292C4—21no—
293C4—22no—
294C4—22no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—3322no—
301C4—6641no—
302C4—6641no—
303C4—6641no—
304C2—21no—
305C2—21no—
306C2#2490603611yescharacteristic
307C2#2490910211yescharacteristic
308C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1461C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1461D83
hypocenterC83#11905
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
160. C83#11905semidirect

Maximal subgroups

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

subgroupindex
#249128252
#249074592
#249054942
#249054912
C83×#14612
#1190583