Finite Group Databasebeta

← #35313622

Subgroups — #35313622

Order 21248.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#24916106C211yesmaximal, characteristic
2#24909263C211yesmaximal, characteristic, complemented
3#24909259C211yesmaximal, characteristic, complemented
4#24905405C211yesmaximal, characteristic, complemented
5#24905388C211yesmaximal, characteristic
6#24905388C211yesmaximal, characteristic
7C83×#1323C211yesmaximal, characteristic, complemented
8#19128169C411yescharacteristic
9#19127761—21no—
10#19127761—21no—
11#19127753—21no—
12#19127753—21no—
13#19127728C411yescharacteristic
14#19127669C411yescharacteristic
15#19127667C2211yescharacteristic
16M16×Dic83C411yescharacteristic
17C2×C8×Dic83C2211yescharacteristic
18Dic83⋊C16—21no—
19Dic83⋊C16—21no—
20C22⋊(C83⋊C16)C2211yescharacteristic
21C22⋊(C83⋊C16)C2211yescharacteristic
22#19127476—21no—
23C16×Dic83—21no—
24#424×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26C22⋊C16×C83C2211yescharacteristic
27C2×#13050590C2×C411yescharacteristic
28C2×#13050574C2×C411yescharacteristic
29C22×C83⋊C8C2×C411yescharacteristic
30#14581757C811yescharacteristic
31#14581757C811yescharacteristic
32#14581754C811yescharacteristic
33#14581754C811yescharacteristic
34(C22⋊D83)⋊C4—21no—
35(C22⋊D83)⋊C4—21no—
36#14581749—21no—
37C2×C4×Dic83C2×C411yescharacteristic
38C42×D83—21no—
39C2×C83⋊C16—21no—
40C2×C83⋊C16—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)—21no—
43(C2×D83)⋊C8—21no—
44(C2×D83)⋊C8—21no—
45(C2×D83)⋊C8—21no—
46(C2×D83)⋊C8—21no—
47#14581702—21no—
48C8×Dic83—21no—
49C8×Dic83D411yescharacteristic
50C8×Dic83D411yescharacteristic
51C8×Dic83D411yescharacteristic
52C8×Dic83D411yescharacteristic
53C2×M16×C83C2×C411yescharacteristic
54C22×C8×C83C2311yescharacteristic
55#117×C83C2×C411yescharacteristic
56C2×C16×C83—21no—
57C2×C16×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60#13050590—41no—
61#13050590—41no—
62C2×Dic166C2×C811yescharacteristic
63C2×D332C2×C811yescharacteristic
64C2×C22⋊D83—21no—
65C22×Dic83M1611yescharacteristic
66C22×Dic83M1611yescharacteristic
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69C2×C4×D83—21no—
70#13050574—41no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8C22⋊C411yescharacteristic
74C2×C83⋊C8C22⋊C411yescharacteristic
75Dic83⋊C4—21no—
76Dic83⋊C4—21no—
77(C2×D83)⋊C4—21no—
78(C2×D83)⋊C4—21no—
79C4×Dic83—21no—
80C4×Dic83—21no—
81C4×Dic83C22⋊C411yescharacteristic
82C4×Dic83C22⋊C411yescharacteristic
83C4×Dic83C2×C811yescharacteristic
84C4×Dic83C2×C811yescharacteristic
85C83⋊C16—41no—
86C83⋊C16—41no—
87C22×C4×C83C22×C411yescharacteristic
88M16×C83—41no—
89C2×C8×C83—21no—
90C2×C8×C83—41no—
91C2×C8×C83C2×D411yescharacteristic
92C2×C8×C83C2×D411yescharacteristic
93C4⋊C4×C83C2×C811yescharacteristic
94C4⋊C4×C83C2×C811yescharacteristic
95C22⋊C4×C83—21no—
96C42×C83—21no—
97C16×C83—41no—
98C16×C83—41no—
99C23×C83C2×M1611yescharacteristic
100C22×D83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83C2×C22⋊C411yescharacteristic
106C2×C4×C83C22×C811yescharacteristic
107C22⋊D83—42no—
108C2×Dic83—22no—
109C2×Dic83—21no—
110C2×Dic83—41no—
111C2×Dic83C22⋊C811yescharacteristic
112C2×Dic83C22⋊C811yescharacteristic
113C2×Dic83C22⋊C811yescharacteristic
114C2×Dic83C22⋊C811yescharacteristic
115D332—41no—
116C4×D83—42no—
117C4×D83—41no—
118C4×D83—41no—
119C4×D83—42no—
120Dic166—41no—
121C8×C83—21no—
122C8×C83—21no—
123C8×C83—41no—
124C83⋊C8—21no—
125C83⋊C8—21no—
126C83⋊C8—41no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83C2×C22⋊C811yescharacteristic
131C2×D83—41no—
132C2×D83—41no—
133C2×D83—41no—
134C4×C83—21no—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83#41011yescharacteristic
138C4×C83#41311yescharacteristic
139Dic83—21no—
140Dic83—21no—
141Dic83—21no—
142Dic83—41no—
143Dic83#35012yes—
144#11464—831nomaximal, Sylow
145C2×C83—41no—
146C2×C83#145811yescharacteristic
147C2×C83C2×#35011yescharacteristic
148C2×C83#211711yescharacteristic
149D83—81no—
150#2922—831no—
151#2163—831no—
152#2162—831no—
153#1323—831no—
154#1323D8311yescharacteristic, complemented
155#1315—831no—
156#1315—831no—
157C83#1146411yescharacteristic, complemented, Sylow
158C4×#41—831no—
159#433—1661no—
160#433—1661no—
161#431—1661no—
162#431—1661no—
163#424—831no—
164#424C2×D8311yescharacteristic
165#408—831no—
166#408—831no—
167C4×M16—831no—
168C2×C4×C8—831no—
169C4⋊C16—1661no—
170C4⋊C16—1661no—
171C22⋊C16—831no—
172C22⋊C16—831no—
173C22⋊C16C2×D8311yescharacteristic
174C22⋊C16C2×D8311yescharacteristic
175#346—1661no—
176C4×C16—1661no—
177C2×#41—831no—
178C2×M16—831no—
179C2×M16C4×D8311yescharacteristic
180C22×C8—831no—
181C22×C8C22×D8311yescharacteristic
182C4×Q8—831no—
183C4×Q8—831no—
184C4×D4—831no—
185C4×D4—831no—
186C4×D4—1661no—
187C4×D4—1661no—
188#117—1661no—
189#117C4×D8311yescharacteristic
190C2×C42—831no—
191C2×C42—1661no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1661no—
195C2×C16—1661no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—1661no—
199C4⋊C8—1661no—
200C22⋊C8—1661no—
201C22⋊C8—1661no—
202C22⋊C8—1661no—
203C22⋊C8—1661no—
204#97—1661no—
205C4×C8—831no—
206C4×C8—831no—
207C4×C8—831no—
208C4×C8—831no—
209C4×C8—1661no—
210#41—3321no—
211#41—3321no—
212C2×Q8—831no—
213C2×D4—831no—
214C2×D4—1661no—
215C22×C4—831no—
216C22×C4—831no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4C2×C4×D8311yescharacteristic
221M16—41no—
222M16—3321no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—3321no—
226C2×C8—831no—
227C2×C8—831no—
228C2×C8—1661no—
229C2×C8D4×D8311yescharacteristic
230C2×C8D4×D8311yescharacteristic
231C4⋊C4—1661no—
232C4⋊C4—1661no—
233C4⋊C4C8×D8311yescharacteristic
234C4⋊C4C8×D8311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1661no—
237C22⋊C4—1661no—
238C42—21no—
239C42—831no—
240C42—831no—
241C42—831no—
242C42—831no—
243C42—1661no—
244C42—1661no—
245C16—41no—
246C16—41no—
247C16—3321no—
248C16—3321no—
249C23—1661no—
250C23M16×D8311yescharacteristic
251Q8—3321no—
252D4—3321no—
253D4—3322no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—3322no—
259C2×C4—3322no—
260C2×C4—3321no—
261C2×C4—3321no—
262C2×C4—3321no—
263C2×C4—831no—
264C2×C4—831no—
265C2×C4—831no—
266C2×C4—831no—
267C2×C4—1661no—
268C2×C4—1662no—
269C2×C4C22⋊C4×D8311yescharacteristic
270C2×C4C2×C8×D8311yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C8—3321no—
275C8—1661no—
276C8—1661no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—3321no—
281C22—3321no—
282C22—3321no—
283C22C22⋊C8×D8311yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—3321no—
288C4—832no—
289C4—1661no—
290C4—1661no—
291C4—1661no—
292C4#1912767311yescharacteristic
293C4#1912768411yescharacteristic
294C2—41no—
295C2—6641no—
296C2#2490577611yescharacteristic
297C2#350×D8311yescharacteristic
298C2#2490908911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C22×C8
FrattiniC22×C8C22×D83
FittingC83×#1323C2
socleC22×C83C2×C22⋊C8
prime radicalC2×#13050590C2×C4
cosocleC22×C8×C83C23
hypercenter#1323D83
hypocenterC83#11464
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #24909263C2semidirect
3. #24909259C2semidirect
4. #24905405C2semidirect
7. C83×#1323C2semidirect
154. #1323D83semidirect
157. C83#11464semidirect

Maximal subgroups

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

subgroupindex
#249161062
#249092632
#249092592
#249054052
#249053882
C83×#13232
#1146483