Finite Group Databasebeta

← #35313679

Subgroups — #35313679

Order 21248.

480 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 89 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#24910006C211yesmaximal, characteristic
2#24905780C212yesmaximal
3#24905406C212yesmaximal, complemented
4#24905397C211yesmaximal, characteristic, complemented
5C83×#1319C211yesmaximal, characteristic, complemented
6C2×#14581781C411yescharacteristic
7C2×Q8×Dic83C411yescharacteristic
8C2×#14581754C2211yescharacteristic
9C22⋊C8×D83—22no—
10#19127603C411yescharacteristic
11#19127593—22no—
12#19127566C2211yescharacteristic
13#19127548—22no—
14C42×Dic83C411yescharacteristic
15#19127427C2212yes—
16#19127424—22no—
17#391×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×D332C2×C411yescharacteristic
20C22×C4×D83D412yes—
21C2×C8×D83—42no—
22#14581781—22no—
23#14581781C814yescomplemented
24Q8×Dic83—41no—
25Q8×Dic83—41no—
26#14581754—41no—
27#14581754—41no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83C2×C411yescharacteristic
30C2×(C2×D83)⋊C4—22no—
31C2×(C2×D83)⋊C4—22no—
32C2×(C2×D83)⋊C4D412yes—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83—21no—
36C2×C4×Dic83C2×C411yescharacteristic
37C2×C4×Dic83C2×C411yescharacteristic
38C22⋊(C83⋊C8)—22no—
39(C2×D83)⋊C8—42no—
40C83⋊#95—22no—
41C22×Q8×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2311yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C22⋊C8×C83—22no—
45#95×C83—22no—
46C23×D83C22⋊C412yes—
47C2×D332—41no—
48C2×D332—41no—
49C2×D332C2×C812yes—
50C2×D332M1612yes—
51C22×Dic83—21no—
52C22×Dic83—21no—
53C22×Dic83C22⋊C411yescharacteristic
54C22×Dic83C22⋊C411yescharacteristic
55C2×C4×D83—22no—
56C2×C4×D83—22no—
57C2×C4×D83—44no—
58C2×C83⋊C8—42no—
59C8×D83—84no—
60C4⋊Dic83—42no—
61C4⋊Dic83—41no—
62(C2×D83)⋊C4—28no—
63(C2×D83)⋊C4—42no—
64(C2×D83)⋊C4—44no—
65C4×Dic83—21no—
66C4×Dic83—21no—
67C4×Dic83—21no—
68C4×Dic83—42no—
69C4×Dic83—42no—
70C4×Dic83—41no—
71C4×Dic83—41no—
72C4×Dic83C2×C812yescomplemented
73C2×Q8×C83—21no—
74C2×Q8×C83—41no—
75C2×Q8×C83—41no—
76C2×Q8×C83C2×C812yescomplemented
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D412yes—
81C2×C8×C83—42no—
82C4⋊C4×C83—41no—
83C42×C83—41no—
84C23×C83C2×C22⋊C411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—42no—
88C22×D83—42no—
89C22×D83C22⋊C814yes—
90Q8×C83—42no—
91Q8×C83—42no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—41no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—42no—
102C2×C4×C83C22×C811yescharacteristic
103C2×C4×C83C2×M1611yescharacteristic
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—21no—
107C2×Dic83—41no—
108C2×Dic83—41no—
109C2×Dic83—41no—
110C2×Dic83—42no—
111C2×Dic83—41no—
112C2×Dic83#9911yescharacteristic
113C2×Dic83#9911yescharacteristic
114D332—42no—
115D332—42no—
116C4×D83—44no—
117C4×D83—82no—
118C8×C83—82no—
119C83⋊C8—82no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83C2×C22⋊C811yescharacteristic
123C22×C83C2×#9911yescharacteristic
124C22×C83#41311yescharacteristic
125C2×D83—24no—
126C2×D83—42no—
127C2×D83—42no—
128C2×D83—44no—
129C2×D83—82no—
130C4×C83—21no—
131C4×C83—21no—
132C4×C83—42no—
133C4×C83—42no—
134C4×C83—42no—
135Dic83—21no—
136Dic83—42no—
137Dic83—41no—
138Dic83—81no—
139Dic83#35412yes—
140#11477—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83#145911yescharacteristic
144C2×C83C2×#35411yescharacteristic
145C2×C83#213211yescharacteristic
146D83—44no—
147C4×#124—831no—
148#1459—832no—
149#1324—832no—
150#1319—831no—
151#1319D8311yescharacteristic, complemented
152C83#1147711yescharacteristic, complemented, Sylow
153C2×#124—831no—
154C2×C4×Q8—831no—
155C2×C4×D4—831no—
156C2×C22⋊C8—1662no—
157#391C2×D8311yescharacteristic
158#390—831no—
159#388—1662no—
160#382—831no—
161C4×C22⋊C4—1662no—
162C43—831no—
163#324—832no—
164#324C2×D8312yes—
165#323—1662no—
166C22×Q8C4×D8311yescharacteristic
167C22×D4—831no—
168C23×C4—832no—
169C22×C8—3322no—
170#124—834no—
171#124—1662no—
172C4×Q8—3321no—
173C4×Q8—3321no—
174C4×D4—3321no—
175C4×D4—3321no—
176C2×C4⋊C4—831no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4C22×D8311yescharacteristic
179C2×C22⋊C4—832no—
180C2×C22⋊C4—1662no—
181C2×C22⋊C4—1662no—
182C2×C42—831no—
183C2×C42—831no—
184C2×C42—1661no—
185C2×C42—1661no—
186C2×C42—1661no—
187C2×C42C4×D8311yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—3322no—
190C22⋊C8—1662no—
191#95—22no—
192#95—1662no—
193C24—832no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8C8×D8312yescomplemented
198C2×D4—3321no—
199C2×D4—3321no—
200C2×D4—832no—
201C2×D4—832no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—3324no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—1662no—
208C22×C4—1662no—
209C22×C4—1661no—
210C22×C4—1661no—
211C22×C4C2×C4×D8311yescharacteristic
212C22×C4D4×D8312yes—
213C2×C8—42no—
214C2×C8—3322no—
215C2×C8—6644no—
216C4⋊C4—41no—
217C4⋊C4—3321no—
218C4⋊C4—3322no—
219C22⋊C4—3324no—
220C22⋊C4—3322no—
221C22⋊C4—1668no—
222C42—41no—
223C42—3322no—
224C42—3321no—
225C42—3321no—
226C42—3322no—
227C42—832no—
228C42—1661no—
229C42—1661no—
230C42—1661no—
231C23—3322no—
232C23—3322no—
233C23—834no—
234C23—1662no—
235C23—1662no—
236C23C22⋊C4×D8311yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—3322no—
240D4—3322no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—3324no—
252C2×C4—3321no—
253C2×C4—3321no—
254C2×C4—3321no—
255C2×C4—3322no—
256C2×C4—3321no—
257C2×C4—831no—
258C2×C4—831no—
259C2×C4—6642no—
260C2×C4—1662no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4C2×C8×D8311yescharacteristic
264C2×C4M16×D8311yescharacteristic
265C8—82no—
266C8—6642no—
267C22—21no—
268C22—21no—
269C22—3324no—
270C22—3322no—
271C22—3322no—
272C22—6642no—
273C22—1664no—
274C22C22⋊C8×D8311yescharacteristic
275C22#99×D8311yescharacteristic
276C22#1912768211yescharacteristic
277C4—21no—
278C4—21no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—3322no—
283C4—3321no—
284C4—832no—
285C4—6641no—
286C4—1661no—
287C2—21no—
288C2—21no—
289C2—3324no—
290C2#2490577811yescharacteristic
291C2#354×D8311yescharacteristic
292C2#2490914711yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1319C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1319D83
hypocenterC83#11477
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #24905406C2semidirect
4. #24905397C2semidirect
5. C83×#1319C2semidirect
23. #14581781C8semidirect
72. C4×Dic83C2×C8semidirect
76. C2×Q8×C83C2×C8semidirect
151. #1319D83semidirect
152. C83#11477semidirect
197. C2×Q8C8×D83semidirect

Maximal subgroups

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

subgroupindex
#249100062
#249057802
#249054062
#249053972
C83×#13192
#1147783