Finite Group Databasebeta

← #35314718

Subgroups — #35314718

Order 21248.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#24913284C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905489C214yesmaximal, complemented
4C83×#1742C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic
6#19127754—21no—
7#19127640C2212yes—
8#19127637C412yes—
9#19127621—21no—
10#19127613—21no—
11#19127599—21no—
12#19127567C411yescharacteristic
13#19127561—21no—
14#19127427C2212yes—
15#19127425—21no—
16C22×M16×C83—21no—
17C2×#117×C83C2211yescharacteristic
18#343×C83—21no—
19#343×C83C2212yes—
20C22×Dic166—21no—
21#14581792—41no—
22#14581792—41no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83C2×C412yes—
26C2×Dic83⋊C4—22no—
27C2×C22⋊Dic83—22no—
28C4⋊(C83⋊C8)—41no—
29C22⋊(C83⋊C8)—22no—
30C4×C83⋊C8—41no—
31C83⋊#95—21no—
32C83⋊#95—22no—
33C83⋊#95—22no—
34C83⋊#95C2×C412yes—
35C23×C4×C83D411yescharacteristic
36C2×M16×C83—24no—
37C2×M16×C83—22no—
38C2×M16×C83—41no—
39C2×M16×C83—42no—
40C22×C8×C83—41no—
41#117×C83—41no—
42#117×C83—41no—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C4⋊C4×C83C2311yescharacteristic
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83D411yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×Dic166—42no—
50C2×Dic166—82no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—41no—
55C4⋊Dic83—41no—
56C4⋊Dic83—42no—
57C4⋊Dic83—42no—
58Dic83⋊C4—82no—
59C22⋊Dic83—82no—
60C24×C83C22⋊C411yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83—22no—
64C22×C4×C83C22⋊C411yescharacteristic
65C22×C4×C83C22⋊C411yescharacteristic
66C22×C4×C83C22⋊C411yescharacteristic
67C22×C4×C83SD1612yes—
68C22×C4×C83Q1612yes—
69C22×C4×C83C22×C411yescharacteristic
70C22×C4×C83C2×D411yescharacteristic
71C22×C4×C83C2×D411yescharacteristic
72M16×C83—24no—
73M16×C83—44no—
74M16×C83—42no—
75C2×C8×C83—42no—
76C2×C8×C83—44no—
77C4⋊C4×C83—41no—
78C4⋊C4×C83—41no—
79C22⋊C4×C83—41no—
80C22⋊C4×C83—41no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83—21no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—41no—
88C23×C83—41no—
89C23×C83#10312yes—
90C23×C83C2×C22⋊C411yescharacteristic
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83#9911yescharacteristic
106C2×C4×C83#9911yescharacteristic
107C2×C4×C83#10312yes—
108C2×C4×C83#10412yes—
109C2×C4×C83C2×SD1611yescharacteristic
110C2×C4×C83C2×Q1611yescharacteristic
111C2×Dic83—42no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117Dic166—84no—
118C8×C83—44no—
119C83⋊C8—82no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—22no—
123C22×C83—42no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—81no—
127C22×C83C2×#9911yescharacteristic
128C22×C83C2×#10311yescharacteristic
129C22×C83C2×#10411yescharacteristic
130C4×C83—21no—
131C4×C83—21no—
132C4×C83—41no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83#32812yes—
137C4×C83#32812yes—
138Dic83—82no—
139Dic83—82no—
140#11723—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83—42no—
144C2×C83C2×#32811yescharacteristic
145C2×C83#212311yescharacteristic
146C2×C83#213411yescharacteristic
147#2628—831no—
148#1742D8311yescharacteristic
149#1458—831no—
150#1353—834no—
151C83#1172311yescharacteristic, complemented, Sylow
152C22×M16—21no—
153C2×C4⋊Q8—831no—
154C2×#117C2×D8311yescharacteristic
155#431—1661no—
156#400—832no—
157#399—832no—
158#395—1661no—
159#393—1661no—
160#389—1661no—
161#382—831no—
162#380—1661no—
163#343—21no—
164#343C2×D8312yes—
165#324—832no—
166#323—1661no—
167C22×Q8—1661no—
168C23×C4C22⋊D8311yescharacteristic
169C2×M16—24no—
170C2×M16—22no—
171C2×M16—41no—
172C2×M16—42no—
173C22×C8—41no—
174C4⋊Q8—3321no—
175C4⋊Q8—3321no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—832no—
179C2×C4⋊C4—1661no—
180C2×C4⋊C4—1661no—
181C2×C4⋊C4—1662no—
182C2×C4⋊C4C22⋊D8311yescharacteristic
183C2×C4⋊C4C22×D8311yescharacteristic
184C2×C22⋊C4—1662no—
185C2×C22⋊C4C4×D8311yescharacteristic
186C2×C22⋊C4D33211yescharacteristic
187C2×C42C4×D8311yescharacteristic
188C2×C42D33211yescharacteristic
189C4⋊C8—3321no—
190C22⋊C8—1662no—
191C4×C8—3321no—
192#95—832no—
193#95—1661no—
194#95—1662no—
195#95—1662no—
196C24(C2×D83)⋊C411yescharacteristic
197C2×Q8—3322no—
198C2×Q8—6642no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—1662no—
203C22×C4—1662no—
204C22×C4(C2×D83)⋊C411yescharacteristic
205C22×C4(C2×D83)⋊C411yescharacteristic
206C22×C4(C2×D83)⋊C411yescharacteristic
207C22×C4#1305057812yes—
208C22×C4#1305058012yes—
209C22×C4C2×C4×D8311yescharacteristic
210C22×C4C2×C22⋊D8311yescharacteristic
211C22×C4C2×D33211yescharacteristic
212M16—24no—
213M16—44no—
214M16—42no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—3322no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—3322no—
221C4⋊C4—3321no—
222C4⋊C4—3321no—
223C4⋊C4—3322no—
224C4⋊C4—6642no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—6642no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1458171512yes—
237C23C2×(C2×D83)⋊C411yescharacteristic
238Q8—6644no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—3322no—
254C2×C4—3322no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—3322no—
259C2×C4#1458170611yescharacteristic
260C2×C4#1458170611yescharacteristic
261C2×C4#1458171512yes—
262C2×C4#1458172012yes—
263C2×C4C2×#1305057811yescharacteristic
264C2×C4C2×#1305058011yescharacteristic
265C8—44no—
266C8—6642no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1458170611yescharacteristic
275C22C2×#1458171511yescharacteristic
276C22C2×#1458172011yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—6642no—
284C4—6642no—
285C4#1912743712yes—
286C4#1912743712yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1912743711yescharacteristic
291C2#2490911811yescharacteristic
292C2#2490915411yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1742C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1742D83
hypocenterC83#11723
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #24905489C2semidirect
151. C83#11723semidirect

Maximal subgroups

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

subgroupindex
#249132842
#249057772
#249054892
C83×#17422
#1172383