Finite Group Databasebeta

← #35315035

Subgroups — #35315035

Order 21248.

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

normal subgroups only

This group has 444 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#24911780C211yesmaximal, characteristic
2C4×#14581709C211yesmaximal, characteristic
3#24905491C214yesmaximal, complemented
4C83×#1465C211yesmaximal, characteristic
5C22×C4⋊Dic83C411yescharacteristic
6C2×#14581709—21no—
7C4×#13050574—21no—
8#19127640C2212yes—
9#19127637C412yes—
10#19127599—21no—
11#19127581—21no—
12#19127573—21no—
13#19127561—21no—
14#19127561C411yescharacteristic
15#19127470C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17#407×C83—21no—
18#324×C83C2212yes—
19#323×C83—21no—
20C23×Dic83—21no—
21C2×#13050574—22no—
22C2×C4⋊Dic83—21no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—41no—
25C2×C4⋊Dic83—41no—
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—22no—
28C2×C22⋊Dic83—21no—
29C2×C22⋊Dic83—22no—
30#14581709—24no—
31#14581703—41no—
32C4×C83⋊C8—41no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C83⋊#95C2×C412yes—
36C22×D4×C83D411yescharacteristic
37C23×C4×C83C2×C411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C2×M16×C83—41no—
40C4×D4×C83—41no—
41C4×D4×C83—41no—
42C2×C4⋊C4×C83C2311yescharacteristic
43C2×C22⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C22⋊C8×C83—22no—
47C22⋊C8×C83—41no—
48C22×Dic83—22no—
49C22×Dic83—22no—
50C22×Dic83—42no—
51C22×Dic83—82no—
52#13050574—24no—
53#13050574—42no—
54C2×C83⋊C8—42no—
55C4⋊Dic83—42no—
56C4⋊Dic83—41no—
57C4⋊Dic83—41no—
58C4⋊Dic83—42no—
59Dic83⋊C4—82no—
60C22⋊Dic83—42no—
61C22⋊Dic83—82no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C22⋊C411yescharacteristic
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—41no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22×C411yescharacteristic
76C22×C4×C83C2×D411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78M16×C83—82no—
79C2×C8×C83—42no—
80C4⋊C4×C83—41no—
81C22⋊C4×C83—41no—
82C22⋊C4×C83—41no—
83C42×C83—21no—
84C42×C83—21no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—21no—
88C23×C83—21no—
89C23×C83—21no—
90C23×C83—41no—
91C23×C83—41no—
92C23×C83—41no—
93C23×C83—41no—
94C23×C83#10412yes—
95C23×C83C2×C22⋊C411yescharacteristic
96D4×C83—44no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—81no—
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#9911yescharacteristic
112C2×C4×C83#13712yes—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—84no—
120C8×C83—82no—
121C83⋊C8—44no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—22no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—41no—
131C22×C83—41no—
132C22×C83—41no—
133C22×C83—81no—
134C22×C83C2×#9911yescharacteristic
135C22×C83#41911yescharacteristic
136C22×C83C2×#10411yescharacteristic
137C4×C83—21no—
138C4×C83—22no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83—41no—
142C4×C83#35212yes—
143Dic83—82no—
144Dic83—82no—
145#11790—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83—42no—
149C2×C83—81no—
150C2×C83#150811yescharacteristic
151C2×C83C2×#35211yescharacteristic
152C2×C83#213411yescharacteristic
153#2487—831no—
154C4×#100—831no—
155#1465D8311yescharacteristic
156#1353—834no—
157C83#1179011yescharacteristic, complemented, Sylow
158C2×C4×D4C2×D8311yescharacteristic
159C22×C4⋊C4—831no—
160C2×#100—1661no—
161#407—21no—
162C4×M16—1661no—
163#400—832no—
164#399—832no—
165#389—1661no—
166#386—1661no—
167#385—1661no—
168#380—831no—
169#380—1661no—
170#343—832no—
171#324C2×D8312yes—
172#323—21no—
173C22×D4D33211yescharacteristic
174C23×C4—1661no—
175C23×C4C4×D8311yescharacteristic
176C23×C4C22⋊D8311yescharacteristic
177C2×M16—41no—
178C2×M16—1662no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—3321no—
182C2×C4⋊C4—3321no—
183C2×C4⋊C4—832no—
184C2×C4⋊C4—1661no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4—1662no—
187C2×C4⋊C4C22×D8311yescharacteristic
188C2×C22⋊C4—1661no—
189C2×C22⋊C4—1662no—
190C2×C22⋊C4C4×D8311yescharacteristic
191C2×C22⋊C4C22⋊D8311yescharacteristic
192C2×C42D33211yescharacteristic
193#100—1664no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196#97—3321no—
197C4×C8—3321no—
198#95—832no—
199#95—1662no—
200#95—1662no—
201C24(C2×D83)⋊C411yescharacteristic
202C24(C2×D83)⋊C411yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—3322no—
213C22×C4—6642no—
214C22×C4—1662no—
215C22×C4—1662no—
216C22×C4(C2×D83)⋊C411yescharacteristic
217C22×C4(C2×D83)⋊C411yescharacteristic
218C22×C4C2×C4×D8311yescharacteristic
219C22×C4C2×C22⋊D8311yescharacteristic
220C22×C4C2×D33211yescharacteristic
221M16—82no—
222M16—3322no—
223M16—1664no—
224C2×C8—42no—
225C2×C8—3322no—
226C4⋊C4—41no—
227C4⋊C4—3322no—
228C4⋊C4—3322no—
229C4⋊C4—3321no—
230C4⋊C4—3321no—
231C4⋊C4—6642no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—3322no—
235C22⋊C4—6642no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1458171812yes—
248C23C2×(C2×D83)⋊C411yescharacteristic
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4—3322no—
267C2×C4—3322no—
268C2×C4—3322no—
269C2×C4—6644no—
270C2×C4#1458170611yescharacteristic
271C2×C4#1458170611yescharacteristic
272C2×C4#1458182512yes—
273C8—82no—
274C8—3324no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22C2×#1458170611yescharacteristic
288C22#1912771111yescharacteristic
289C22C2×#1458171811yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—6642no—
296C4—6642no—
297C4#1912749012yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2490598411yescharacteristic
303C2C2×#1912749011yescharacteristic
304C2#2490915411yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1465C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C4⋊C4×C83C23
hypercenter#1465D83
hypocenterC83#11790
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #24905491C2semidirect
157. C83#11790semidirect

Maximal subgroups

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

subgroupindex
#249117802
C4×#145817092
#249054912
C83×#14652
#1179083