Finite Group Databasebeta

← #35314784

Subgroups — #35314784

Order 21248.

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

normal subgroups only

This group has 402 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#24910754C211yesmaximal, characteristic
2C4×#14581711C211yesmaximal, characteristic
3#24905494C212yesmaximal
4#24905470C212yesmaximal
5C83×#1461C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7C2×#14581711—21no—
8C4×#13050574—21no—
9#19127632—21no—
10#19127631C411yescharacteristic
11#19127631C2211yescharacteristic
12#19127621C411yescharacteristic
13#19127621C2211yescharacteristic
14#19127620—21no—
15C4×Dic83⋊C4—21no—
16C4×Dic83⋊C4—21no—
17#19127547C411yescharacteristic
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#431×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25C4×Dic166—41no—
26C4×Dic166—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊C4—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35#14581711—24no—
36#14581703—41no—
37C4×C83⋊C8—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95C2×C411yescharacteristic
44C83⋊#95C2×C411yescharacteristic
45C22×Q8×C83D411yescharacteristic
46C4×Q8×C83—41no—
47C4×Q8×C83—41no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—41no—
55C22⋊C8×C83—22no—
56C4×C8×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—42no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050574—24no—
64#13050574—42no—
65C2×C83⋊C8—42no—
66C4⋊Dic83—41no—
67Dic83⋊C4—42no—
68Dic83⋊C4—42no—
69Dic83⋊C4—42no—
70Dic83⋊C4—42no—
71Dic83⋊C4—42no—
72C4×Dic83—24no—
73C4×Dic83—41no—
74C4×Dic83—41no—
75C2×Q8×C83—22no—
76C2×Q8×C83—22no—
77C2×Q8×C83—41no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C2×C8×C83—42no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—21no—
95C23×C83C2×C22⋊C411yescharacteristic
96Q8×C83—44no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—22no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83#9911yescharacteristic
112C2×C4×C83#9911yescharacteristic
113C2×C4×C83#10412yes—
114C2×C4×C83#13512yes—
115C2×Dic83—22no—
116C2×Dic83—22no—
117C2×Dic83—22no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127Dic166—42no—
128Dic166—42no—
129C8×C83—82no—
130C83⋊C8—44no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83C2×#9911yescharacteristic
134C22×C83#41611yescharacteristic
135C22×C83C2×#10411yescharacteristic
136C4×C83—21no—
137C4×C83—22no—
138C4×C83—22no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83—41no—
142C4×C83—41no—
143C4×C83#35612yes—
144Dic83—42no—
145Dic83—42no—
146Dic83—42no—
147Dic83—81no—
148#11736—831nomaximal, Sylow
149C2×C83—21no—
150C2×C83—21no—
151C2×C83#150111yescharacteristic
152C2×C83#212011yescharacteristic
153C2×C83C2×#35611yescharacteristic
154#2399—831no—
155C4×#101—831no—
156#1461D8311yescharacteristic
157#1354—832no—
158#1346—832no—
159C83#1173611yescharacteristic, complemented, Sylow
160C2×C4×Q8—831no—
161C2×C4×Q8C2×D8311yescharacteristic
162#431—21no—
163C2×#101—1661no—
164C4×M16—1661no—
165#398—831no—
166#398—831no—
167#398—1661no—
168#395—831no—
169#395—831no—
170#395—1661no—
171C4×C4⋊C4—1661no—
172C4×C4⋊C4—1661no—
173#376—831no—
174#343—832no—
175#324—21no—
176#324C2×D8312yes—
177C22×Q8—831no—
178C22×Q8D33211yescharacteristic
179C2×M16—1662no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—3321no—
183C4×Q8—3321no—
184C2×C4⋊C4—831no—
185C2×C4⋊C4—1661no—
186C2×C4⋊C4—1661no—
187C2×C4⋊C4—1661no—
188C2×C4⋊C4—1661no—
189C2×C4⋊C4—1661no—
190C2×C4⋊C4C4×D8311yescharacteristic
191C2×C4⋊C4C22⋊D8311yescharacteristic
192C2×C4⋊C4C22×D8311yescharacteristic
193C2×C42—1661no—
194C2×C42—1661no—
195C2×C42C4×D8311yescharacteristic
196C2×C42D33211yescharacteristic
197C2×C42C22⋊D8311yescharacteristic
198C4⋊C8—41no—
199#101—1664no—
200C22⋊C8—22no—
201#97—3321no—
202C4×C8—41no—
203C4×C8—3321no—
204#95—831no—
205#95—831no—
206#95—1661no—
207#95—1661no—
208#95—1661no—
209#95—1661no—
210#95—1661no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—3321no—
215C2×Q8—3322no—
216C22×C4—1661no—
217C22×C4—1661no—
218C22×C4—1661no—
219C22×C4—1661no—
220C22×C4(C2×D83)⋊C411yescharacteristic
221C22×C4(C2×D83)⋊C411yescharacteristic
222C22×C4(C2×D83)⋊C411yescharacteristic
223C22×C4(C2×D83)⋊C411yescharacteristic
224C22×C4C2×C4×D8311yescharacteristic
225C22×C4C2×C22⋊D8311yescharacteristic
226C22×C4C2×D33211yescharacteristic
227M16—3322no—
228M16—1664no—
229C2×C8—42no—
230C2×C8—3322no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—3321no—
235C4⋊C4—3322no—
236C4⋊C4—3322no—
237C4⋊C4—3322no—
238C4⋊C4—3322no—
239C4⋊C4—3322no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—3321no—
247C42—3321no—
248C42—1664no—
249C23C2×(C2×D83)⋊C411yescharacteristic
250Q8—44no—
251Q8—3322no—
252Q8—3322no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—3321no—
268C2×C4—3321no—
269C2×C4—3321no—
270C2×C4—3321no—
271C2×C4—3321no—
272C2×C4—3321no—
273C2×C4—3321no—
274C2×C4—3321no—
275C2×C4—3321no—
276C2×C4—1662no—
277C2×C4—1662no—
278C2×C4—1662no—
279C2×C4#1458170611yescharacteristic
280C2×C4#1458170611yescharacteristic
281C2×C4#1458171812yes—
282C2×C4#1458181412yes—
283C8—82no—
284C8—3324no—
285C22—21no—
286C22—21no—
287C22C2×#1458170611yescharacteristic
288C22#1912769811yescharacteristic
289C22C2×#1458171811yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—3322no—
298C4—3322no—
299C4—3322no—
300C4—6641no—
301C4#1912749912yes—
302C2—21no—
303C2—21no—
304C2#2490595511yescharacteristic
305C2#2490910211yescharacteristic
306C2C2×#1912749911yescharacteristic
subgroupG/H
centerC2×C4#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1461C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1461D83
hypocenterC83#11736
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
159. C83#11736semidirect

Maximal subgroups

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

subgroupindex
#249107542
C4×#145817112
#249054942
#249054702
C83×#14612
#1173683