Finite Group Databasebeta

← #35313697

Subgroups — #35313697

Order 21248.

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

normal subgroups only

This group has 460 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#24910171C211yesmaximal, characteristic
2#24905780C212yesmaximal
3#24905409C211yesmaximal, characteristic, complemented
4#24905396C212yesmaximal, complemented
5C83×#1325C211yesmaximal, characteristic, complemented
6C2×#14581780C411yescharacteristic
7C2×#14581757C411yescharacteristic
8C2×#14581754C2211yescharacteristic
9C22⋊C8×D83—22no—
10#19127606C2211yescharacteristic
11#19127602C411yescharacteristic
12#19127591—22no—
13#19127572—22no—
14#19127547C411yescharacteristic
15#19127427C2212yes—
16#19127424—22no—
17#384×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×Dic166C2×C411yescharacteristic
20C22×D332C2×C411yescharacteristic
21C22×C4×D83D412yes—
22C2×C8×D83—42no—
23#14581780—22no—
24#14581780C814yescomplemented
25#14581757—42no—
26#14581754—42no—
27C2×Dic83⋊C4—21no—
28C2×(C2×D83)⋊C4—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4D412yes—
31C2×C4×Dic83—21no—
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83C2×C411yescharacteristic
34C22⋊(C83⋊C8)—22no—
35(C2×D83)⋊C8—42no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2311yescharacteristic
42C2×C42×C83C2×C411yescharacteristic
43C22⋊C8×C83—22no—
44C23×D83C22⋊C412yes—
45C2×Dic166—21no—
46C2×Dic166—41no—
47C2×Dic166—41no—
48C2×Dic166C2×C812yescomplemented
49C2×D332—41no—
50C2×D332—41no—
51C2×D332C2×C812yes—
52C2×D332M1612yes—
53C22×Dic83—21no—
54C22×Dic83—21no—
55C22×Dic83C22⋊C411yescharacteristic
56C22×Dic83C22⋊C411yescharacteristic
57C2×C4×D83—22no—
58C2×C4×D83—22no—
59C2×C4×D83—82no—
60C2×C83⋊C8—42no—
61C8×D83—84no—
62Dic83⋊C4—42no—
63(C2×D83)⋊C4—28no—
64(C2×D83)⋊C4—42no—
65(C2×D83)⋊C4—82no—
66C4×Dic83—22no—
67C4×Dic83—42no—
68C4×Dic83—41no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83C22×C411yescharacteristic
72C22×C4×C83C2×D412yes—
73C2×C8×C83—42no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—41no—
76C4⋊C4×C83—82no—
77C42×C83—21no—
78C42×C83C2×C812yescomplemented
79C23×C83C2×C22⋊C411yescharacteristic
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—42no—
83C22×D83—42no—
84C22×D83C22⋊C814yes—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—41no—
90C2×C4×C83—41no—
91C2×C4×C83—42no—
92C2×C4×C83—42no—
93C2×C4×C83—41no—
94C2×C4×C83C22×C811yescharacteristic
95C2×C4×C83C2×M1611yescharacteristic
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—41no—
99C2×Dic83—41no—
100C2×Dic83—41no—
101C2×Dic83—41no—
102C2×Dic83—41no—
103C2×Dic83—41no—
104C2×Dic83—41no—
105C2×Dic83#9912yes—
106D332—42no—
107D332—42no—
108C4×D83—44no—
109C4×D83—82no—
110Dic166—42no—
111Dic166—42no—
112C8×C83—82no—
113C83⋊C8—82no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83C2×C22⋊C811yescharacteristic
117C22×C83C2×#9911yescharacteristic
118C22×C83#41311yescharacteristic
119C2×D83—24no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—44no—
123C2×D83—82no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—42no—
127C4×C83—42no—
128C4×C83—81no—
129Dic83—22no—
130Dic83—41no—
131Dic83—42no—
132Dic83—81no—
133#11481—831nomaximal, Sylow
134C2×C83—21no—
135C2×C83—21no—
136C2×C83#145911yescharacteristic
137C2×C83#212711yescharacteristic
138C2×C83#213511yescharacteristic
139D83—44no—
140#2326—831no—
141#1459—832no—
142#1325—831no—
143#1325D8311yescharacteristic, complemented
144#1319—832no—
145C83#1148111yescharacteristic, complemented, Sylow
146C2×#124—831no—
147C2×C4×Q8—831no—
148C2×C4×D4—831no—
149C2×C22⋊C8—1662no—
150#391—831no—
151#390—831no—
152#388—1662no—
153#385—1662no—
154#384C2×D8311yescharacteristic
155#376—831no—
156#324—832no—
157#324C2×D8312yes—
158#323—1662no—
159C22×Q8—831no—
160C22×D4—831no—
161C23×C4—832no—
162C22×C8—3322no—
163#124—834no—
164#124—1662no—
165C4×Q8—3322no—
166C4×D4—3322no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—1661no—
169C2×C4⋊C4C4×D8311yescharacteristic
170C2×C4⋊C4C22×D8311yescharacteristic
171C2×C22⋊C4—832no—
172C2×C22⋊C4—1662no—
173C2×C22⋊C4—1662no—
174C2×C42—831no—
175C2×C42—831no—
176C2×C42—1661no—
177C2×C42C4×D8311yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—3322no—
180C22⋊C8—1662no—
181#95—1661no—
182#95—1661no—
183#95—1662no—
184C24—832no—
185C2×Q8—3321no—
186C2×Q8—3321no—
187C2×Q8—832no—
188C2×Q8—1661no—
189C2×D4—3321no—
190C2×D4—3321no—
191C2×D4—832no—
192C2×D4—832no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—831no—
196C22×C4—831no—
197C22×C4—6642no—
198C22×C4—1662no—
199C22×C4—1662no—
200C22×C4—1661no—
201C22×C4—1661no—
202C22×C4C2×C4×D8311yescharacteristic
203C22×C4D4×D8312yes—
204C2×C8—42no—
205C2×C8—3322no—
206C2×C8—6644no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—82no—
210C4⋊C4—3322no—
211C22⋊C4—3322no—
212C22⋊C4—6642no—
213C22⋊C4—1668no—
214C42—21no—
215C42—3321no—
216C42—3322no—
217C42—1662no—
218C42C8×D8312yescomplemented
219C23—3322no—
220C23—3322no—
221C23—834no—
222C23—1662no—
223C23—1662no—
224C23C22⋊C4×D8311yescharacteristic
225Q8—3322no—
226Q8—3322no—
227D4—3322no—
228D4—3322no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—3324no—
239C2×C4—3321no—
240C2×C4—3321no—
241C2×C4—3321no—
242C2×C4—3321no—
243C2×C4—3321no—
244C2×C4—3321no—
245C2×C4—3321no—
246C2×C4—832no—
247C2×C4—6642no—
248C2×C4—1662no—
249C2×C4—1661no—
250C2×C4C2×C8×D8311yescharacteristic
251C2×C4M16×D8311yescharacteristic
252C8—82no—
253C8—6642no—
254C22—21no—
255C22—21no—
256C22—3324no—
257C22—3322no—
258C22—3322no—
259C22—6642no—
260C22—1664no—
261C22C22⋊C8×D8311yescharacteristic
262C22#99×D8311yescharacteristic
263C22#1912768411yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—3322no—
270C4—3321no—
271C4—6641no—
272C4—1662no—
273C2—21no—
274C2—21no—
275C2—3324no—
276C2#2490578011yescharacteristic
277C2#2490912911yescharacteristic
278C2#2490915611yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1325C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1325D83
hypocenterC83#11481
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #24905409C2semidirect
4. #24905396C2semidirect
5. C83×#1325C2semidirect
24. #14581780C8semidirect
48. C2×Dic166C2×C8semidirect
78. C42×C83C2×C8semidirect
143. #1325D83semidirect
145. C83#11481semidirect
218. C42C8×D83semidirect

Maximal subgroups

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

subgroupindex
#249101712
#249057802
#249054092
#249053962
C83×#13252
#1148183